Finite Group Databasebeta

← #29646652

Subgroups — #29646652

Order 15616.

1268 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#532×D61C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3C2×#16839813C211yesmaximal, characteristic, complemented
4#21444286C214yesmaximal, complemented
5#21444271C211yesmaximal, characteristic, complemented
6#21444270C211yesmaximal, characteristic, complemented
7#21444269C211yesmaximal, characteristic, complemented
8#21444208C214yesmaximal, complemented
9C61×#1492C211yesmaximal, characteristic, complemented
10C2×#41×D61—21no—
11#16840313C411yescharacteristic
12#16840312C2211yescharacteristic
13#16840311—21no—
14#16840310C411yescharacteristic
15#16840309—21no—
16#16840309C2212yes—
17#16840308—21no—
18#16840307—21no—
19#16840307C412yes—
20C4⋊Q8×D61C411yescharacteristic
21#127×D61C411yescharacteristic
22#124×D61—21no—
23#121×D61—21no—
24#121×D61C412yes—
25C4×D4×D61—21no—
26C4×D4×D61C2212yes—
27C2×C42×D61C2211yescharacteristic
28#16839818—21no—
29#16839818C2212yescomplemented
30#16839814—21no—
31#16839814C2212yes—
32#16839813—21no—
33#16839813—21no—
34#16839813C2212yes—
35#16839813C2212yescomplemented
36#16839786C2211yescharacteristic, complemented
37C2×#14519706—21no—
38#16839725—21no—
39C2×#14519689—21no—
40M16×Dic61—21no—
41C2×C4×#92418C2211yescharacteristic, complemented
42#16839491C2214yescomplemented
43D4⋊#92418C2214yescomplemented
44#16839489C2214yescomplemented
45C61×#532C2211yescharacteristic
46C61×#423C2211yescharacteristic, complemented
47C2×C4×C8×C61C2211yescharacteristic, complemented
48C61×D4⋊C8C2214yescomplemented
49C2×#11758226—21no—
50C2×#11758222—21no—
51C2×#11758222—22no—
52C2×#11758224—22no—
53C2×#11758224—21no—
54#41×D61—44no—
55#41×D61—44no—
56C2×Q8×D61—21no—
57C2×D4×D61—22no—
58C2×D4×D61—21no—
59C2×D4×D61—21no—
60C2×D4×D61C2×C412yes—
61C22×C4×D61—22no—
62C22×C4×D61C2×C411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C2×#11758194—21no—
66C22×#92418—21no—
67#14519777C2×C411yescharacteristic
68#14519776—21no—
69#14519776C2×C412yes—
70#14519775—21no—
71#14519774C2×C411yescharacteristic
72#14519773—21no—
73#14519773—21no—
74#14519773C2×C412yes—
75#14519766—21no—
76#14519765—21no—
77#14519764—21no—
78#14519763—22no—
79#14519763—22no—
80#14519752—21no—
81#14519752C2×C412yes—
82#14519751—21no—
83#14519751C2×C412yes—
84#14519750—22no—
85#14519750—22no—
86#14519749—22no—
87#14519749—22no—
88#14519748—21no—
89#14519748C2×C412yes—
90C4×D244—21no—
91C4×D244C2312yes—
92D4×Dic61—21no—
93D4×Dic61C2×C412yes—
94#14519739—21no—
95#14519739C2×C412yes—
96C4×C22⋊D61—22no—
97C4×C22⋊D61—22no—
98#14519737—22no—
99#14519737—22no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61C2×C412yes—
102C22⋊C4×D61—22no—
103C22⋊C4×D61—22no—
104C2×C4×Dic61C2×C411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C42×D61C2×C411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61C2311yescharacteristic
114C42×D61C2311yescharacteristic
115C4⋊#92418—21no—
116C4⋊#92418C2312yes—
117#14519706—22no—
118#14519706—22no—
119#14519689—22no—
120#14519689—22no—
121#14519689—22no—
122#14519689—22no—
123#14519687—22no—
124C4×#92418—21no—
125C4×#92418C2312yes—
126C8×Dic61—22no—
127C2×#41×C61—21no—
128C2×M16×C61—21no—
129C22×C8×C61—21no—
130C4⋊Q8×C61C2×C411yescharacteristic
131#127×C61C2×C411yescharacteristic
132#124×C61—21no—
133#121×C61—21no—
134#121×C61C2×C412yes—
135C4×D4×C61—21no—
136C4×D4×C61C2312yes—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2312yes—
140C4×C8×C61—21no—
141C4×C8×C61C2312yes—
142C23×D61—22no—
143C23×D61C22⋊C411yescharacteristic
144#41×C61—42no—
145#41×C61—42no—
146C2×Q8×C61—21no—
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C22×C412yes—
151C22×C4×C61—22no—
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C22×Dic61—22no—
161C22×Dic61C22⋊C411yescharacteristic
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164#11758226—44no—
165Q8×D61—44no—
166#11758224—44no—
167#11758224—48no—
168D4×D61—28no—
169D4×D61—44no—
170D4×D61—44no—
171D4×D61—48no—
172#11758222—44no—
173#11758222—44no—
174#11758222—42no—
175#11758222—42no—
176C2×D244—22no—
177C2×D244—21no—
178C2×D244—21no—
179C2×D244C22×C412yes—
180C2×C4×D61—22no—
181C2×C4×D61—22no—
182C2×C4×D61—24no—
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—42no—
187C2×C4×D61—44no—
188C2×C4×D61—44no—
189C2×C4×D61C22⋊C411yescharacteristic
190C2×C4×D61C22⋊C411yescharacteristic
191C2×C4×D61C22⋊C411yescharacteristic
192C2×C4×D61C22⋊C411yescharacteristic
193C2×C4×D61C22⋊C411yescharacteristic
194C2×C4×D61C22⋊C411yescharacteristic
195C2×C4×D61C22×C411yescharacteristic
196C2×C4×D61C22×C411yescharacteristic
197C2×C4×D61C2×D411yescharacteristic
198C2×C4×D61C2×D411yescharacteristic
199C2×C4×D61C2×D411yescharacteristic
200C2×C4×D61C2×D411yescharacteristic
201C2×Dic122—21no—
202C2×Dic122—22no—
203M16×C61—42no—
204C2×C8×C61—22no—
205C2×C8×C61—22no—
206C2×C8×C61—42no—
207C4⋊C4×C61—21no—
208C4⋊C4×C61C22×C412yes—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C42×C61C22×C411yescharacteristic
212C42×C61C2×D411yescharacteristic
213C42×C61C2×D411yescharacteristic
214C42×C61C2411yescharacteristic
215C22⋊Dic61—22no—
216C22⋊Dic61—22no—
217#11758198—22no—
218#11758198—22no—
219#11758198—22no—
220#11758198—22no—
221C4⋊Dic61—21no—
222C4⋊Dic61C22×C412yes—
223#11758196—22no—
224#11758196—22no—
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22⋊C411yescharacteristic
231C4×Dic61C22×C411yescharacteristic
232C4×Dic61C22×C411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237#11758194—42no—
238C2×#92418—22no—
239C2×#92418—22no—
240C2×#92418—42no—
241C23×C61—22no—
242C23×C61C2×C22⋊C411yescharacteristic
243C22×D61—22no—
244C22×D61—22no—
245C22×D61—22no—
246C22×D61—41no—
247C22×D61—44no—
248C22×D61—44no—
249C22×D61—44no—
250C22×D61C2×C22⋊C411yescharacteristic
251C22×D61C2×C22⋊C411yescharacteristic
252Q8×C61—42no—
253D4×C61—24no—
254D4×C61—44no—
255D4×C61—42no—
256D4×C61—42no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61AGL(1,ℤ/8)12yes—
266C2×C4×C61AGL(1,ℤ/8)12yes—
267C2×C4×C61C23×C411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C22⋊D61—44no—
271C22⋊D61—44no—
272C22⋊D61—44no—
273C22⋊D61—44no—
274C22⋊D61—44no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—44no—
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C2×C22⋊C411yescharacteristic
285C2×Dic61C2×C22⋊C411yescharacteristic
286D244—24no—
287D244—44no—
288D244—42no—
289D244—42no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—21no—
293C4×D61—21no—
294C4×D61—21no—
295C4×D61—21no—
296C4×D61—22no—
297C4×D61—22no—
298C4×D61—22no—
299C4×D61—22no—
300C4×D61—22no—
301C4×D61—41no—
302C4×D61—41no—
303C4×D61—44no—
304C4×D61—44no—
305C4×D61—42no—
306C4×D61—42no—
307C4×D61#13114yes—
308Dic122—44no—
309Dic122—42no—
310C8×C61—42no—
311C8×C61—42no—
312#92418—42no—
313#92418—42no—
314#17478—611nomaximal, Sylow
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—42no—
318C22×C61—42no—
319C22×C61—42no—
320C22×C61#41812yes—
321C22×C61C22×C22⋊C411yescharacteristic
322C2×D61—21no—
323C2×D61—21no—
324C2×D61—21no—
325C2×D61—21no—
326C2×D61—21no—
327C2×D61—44no—
328C2×D61—42no—
329C2×D61—42no—
330C2×D61—42no—
331C2×D61—88no—
332C2×D61#40812yes—
333C4×C61—21no—
334C4×C61—21no—
335C4×C61—42no—
336C4×C61#41812yes—
337C4×C61#42112yes—
338C4×C61C2×#13112yes—
339C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
340C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
341Dic61—21no—
342Dic61—21no—
343Dic61—21no—
344Dic61—21no—
345Dic61—21no—
346Dic61—42no—
347Dic61#40812yes—
348C2×#532—611no—
349#2920—611no—
350C2×#431—611no—
351#1496—614no—
352#1492—611no—
353#1492—611no—
354#1492—611no—
355#1492D6111yescharacteristic, complemented
356#1478—614no—
357C2×C61—22no—
358C2×C61—82no—
359C2×C61C2×#40811yescharacteristic
360C2×C61C2×#41811yescharacteristic
361C2×C61C2×#42111yescharacteristic
362D61—22no—
363D61—41no—
364D61—82no—
365C22×#41—1221no—
366#532—611no—
367#532—611no—
368#532—612no—
369#532—612no—
370#532—611no—
371#532—1221no—
372#532—1221no—
373#532—1221no—
374#532—1221no—
375#532C2×D6111yescharacteristic
376C2×C4⋊Q8—611no—
377C2×#127—611no—
378C2×#124—1221no—
379C2×#121—612no—
380C2×#121—1221no—
381C2×C4×D4—612no—
382C2×C4×D4—1221no—
383C22×C42—611no—
384#432—612no—
385#432—1221no—
386#431—612no—
387#431—612no—
388#431—612no—
389#431—1221no—
390#431—1221no—
391#431—1221no—
392#423—611no—
393#423C2×D6111yescharacteristic, complemented
394C2×C4⋊C8—1221no—
395#407—1221no—
396C2×C22⋊C8—1221no—
397C4×M16—1221no—
398C2×C4×C8—611no—
399C2×C4×C8C2×D6111yescharacteristic, complemented
400D4⋊C8—614no—
401D4⋊C8—614no—
402D4⋊C8—614no—
403D4⋊C8C2×D6114yescomplemented
404C61#1747811yescharacteristic, complemented, Sylow
405C2×#41—21no—
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1221no—
409C2×#41—1221no—
410C2×#41—1221no—
411C2×#41—2444no—
412C2×#41—2444no—
413C22×Q8—1221no—
414C22×D4—612no—
415C22×D4—1221no—
416C22×D4—1221no—
417C22×D4—1222no—
418C23×C4—611no—
419C23×C4—611no—
420C23×C4—611no—
421C23×C4—1222no—
422C2×M16—21no—
423C2×M16—1221no—
424C22×C8—21no—
425C22×C8—1221no—
426C4⋊Q8—612no—
427C4⋊Q8—611no—
428C4⋊Q8—1221no—
429C4⋊Q8—1221no—
430C4⋊Q8C4×D6111yescharacteristic
431#127—612no—
432#127—611no—
433#127—1221no—
434#127—1221no—
435#127C4×D6111yescharacteristic
436#124—21no—
437#124—1222no—
438#124—1222no—
439#124—1221no—
440#124—1221no—
441#124—1221no—
442#121—21no—
443#121—612no—
444#121—612no—
445#121—612no—
446#121—1222no—
447#121—1222no—
448#121—1221no—
449#121—1222no—
450#121—1221no—
451#121—1222no—
452#121—1221no—
453#121C4×D6112yes—
454C4×D4—21no—
455C4×D4—612no—
456C4×D4—612no—
457C4×D4—612no—
458C4×D4—1222no—
459C4×D4—1222no—
460C4×D4—1221no—
461C4×D4—1222no—
462C4×D4—1222no—
463C4×D4—1221no—
464C4×D4—1221no—
465C4×D4C22×D6112yes—
466C2×C4⋊C4—612no—
467C2×C4⋊C4—1221no—
468C2×C22⋊C4—1222no—
469C2×C22⋊C4—1222no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42—611no—
477C2×C42—611no—
478C2×C42—611no—
479C2×C42—611no—
480C2×C42—611no—
481C2×C42C22×D6111yescharacteristic
482C4⋊C8—21no—
483C4⋊C8—612no—
484C4⋊C8—1222no—
485C4⋊C8—1222no—
486C4⋊C8—1221no—
487C4⋊C8C22×D6112yes—
488C22⋊C8—1222no—
489C22⋊C8—1222no—
490C22⋊C8—1222no—
491C22⋊C8—1222no—
492#97—1222no—
493C4×C8—21no—
494C4×C8—612no—
495C4×C8—1222no—
496C4×C8—1221no—
497C4×C8C22×D6112yes—
498C24—611no—
499C24—1222no—
500#41—42no—
501#41—42no—
502#41—2444no—
503#41—2444no—
504#41—2448no—
505#41—2444no—
506#41—2444no—
507#41—2442no—
508#41—2442no—
509C2×Q8—21no—
510C2×Q8—1222no—
511C2×Q8—1221no—
512C2×Q8—2444no—
513C2×D4—22no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4—612no—
517C2×D4—1222no—
518C2×D4—1228no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—1222no—
522C2×D4—1222no—
523C2×D4—1222no—
524C2×D4—1221no—
525C2×D4—1221no—
526C2×D4—2444no—
527C2×D4—2444no—
528C2×D4—2448no—
529C2×D4C2×C4×D6112yes—
530C22×C4—22no—
531C22×C4—611no—
532C22×C4—611no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—611no—
538C22×C4—611no—
539C22×C4—611no—
540C22×C4—611no—
541C22×C4—611no—
542C22×C4—611no—
543C22×C4—611no—
544C22×C4—611no—
545C22×C4—611no—
546C22×C4—1224no—
547C22×C4—1222no—
548C22×C4—1222no—
549C22×C4—1222no—
550C22×C4—1222no—
551C22×C4—1222no—
552C22×C4—1222no—
553C22×C4—2442no—
554C22×C4—2444no—
555C22×C4—2444no—
556C22×C4C2×C4×D6111yescharacteristic
557C22×C4D4×D6111yescharacteristic
558C22×C4D4×D6111yescharacteristic
559M16—42no—
560M16—2442no—
561C2×C8—22no—
562C2×C8—22no—
563C2×C8—42no—
564C2×C8—1222no—
565C2×C8—1222no—
566C2×C8—2442no—
567C4⋊C4—21no—
568C4⋊C4—612no—
569C4⋊C4—1222no—
570C4⋊C4—1222no—
571C4⋊C4—1221no—
572C4⋊C4C2×C4×D6112yes—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—1222no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C22⋊C4—1222no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C42—611no—
582C42—611no—
583C42—611no—
584C42—611no—
585C42—611no—
586C42—611no—
587C42—611no—
588C42—611no—
589C42—611no—
590C42—611no—
591C42—611no—
592C42—611no—
593C42C2×C4×D6111yescharacteristic
594C42D4×D6111yescharacteristic
595C42D4×D6111yescharacteristic
596C42C23×D6111yescharacteristic
597C23—22no—
598C23—611no—
599C23—611no—
600C23—1222no—
601C23—1222no—
602C23—1222no—
603C23—2441no—
604C23—2444no—
605C23—2444no—
606C23—2444no—
607C23C22⋊C4×D6111yescharacteristic
608Q8—42no—
609Q8—2444no—
610Q8—2442no—
611D4—24no—
612D4—44no—
613D4—42no—
614D4—42no—
615D4—1224no—
616D4—2444no—
617D4—2444no—
618D4—2444no—
619D4—2444no—
620D4—2444no—
621D4—2444no—
622D4—2442no—
623D4—2442no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—611no—
630C2×C4—614no—
631C2×C4—611no—
632C2×C4—611no—
633C2×C4—611no—
634C2×C4—611no—
635C2×C4—611no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—1222no—
642C2×C4—1222no—
643C2×C4—1222no—
644C2×C4—1222no—
645C2×C4—1221no—
646C2×C4—1221no—
647C2×C4—1221no—
648C2×C4—1221no—
649C2×C4—1222no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4—2444no—
653C2×C4—2444no—
654C2×C4—2441no—
655C2×C4—2444no—
656C2×C4—2442no—
657C2×C4—2442no—
658C2×C4C22⋊C4×D6111yescharacteristic
659C2×C4C22⋊C4×D6111yescharacteristic
660C2×C4C22⋊C4×D6111yescharacteristic
661C2×C4#1451980112yes—
662C2×C4#1451980412yes—
663C2×C4C22×C4×D6111yescharacteristic
664C2×C4C2×D4×D6111yescharacteristic
665C2×C4C2×D4×D6111yescharacteristic
666C8—42no—
667C8—42no—
668C8—2442no—
669C8—2442no—
670C22—21no—
671C22—21no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—612no—
676C22—1221no—
677C22—1221no—
678C22—1221no—
679C22—1221no—
680C22—1221no—
681C22—4888no—
682C22—2444no—
683C22—2442no—
684C22—2442no—
685C22—2442no—
686C22#1683976512yes—
687C22C2×C22⋊C4×D6111yescharacteristic
688C4—21no—
689C4—21no—
690C4—42no—
691C4—612no—
692C4—1221no—
693C4—1221no—
694C4—1221no—
695C4—1221no—
696C4—1221no—
697C4—2442no—
698C4#1683976512yes—
699C4#1683977712yes—
700C4#131×D6112yes—
701C4C2×#1451980111yescharacteristic
702C4C2×#1451980411yescharacteristic
703C2—22no—
704C2—82no—
705C2—1222no—
706C2—4882no—
707C2—2441no—
708C2#408×D6111yescharacteristic
709C2C2×#1683976511yescharacteristic
710C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1492C2
socleC22×C61C22×C22⋊C4
prime radical#532×D61C2
cosocleC42×C61C24
hypercenter#1492D61
hypocenterC61#17478
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454451C2semidirect
3. C2×#16839813C2semidirect
4. #21444286C2semidirect
5. #21444271C2semidirect
6. #21444270C2semidirect
7. #21444269C2semidirect
8. #21444208C2semidirect
9. C61×#1492C2semidirect
29. #16839818C22semidirect
35. #16839813C22semidirect
36. #16839786C22semidirect
41. C2×C4×#92418C22semidirect
42. #16839491C22semidirect
43. D4⋊#92418C22semidirect
44. #16839489C22semidirect
46. C61×#423C22semidirect
47. C2×C4×C8×C61C22semidirect
48. C61×D4⋊C8C22semidirect
355. #1492D61semidirect
393. #423C2×D61semidirect
399. C2×C4×C8C2×D61semidirect
403. D4⋊C8C2×D61semidirect
404. C61#17478semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#532×D612
#214544512
C2×#168398132
#214442862
#214442712
#214442702
#214442692
#214442082
C61×#14922
#1747861