Finite Group Databasebeta

← #29646915

Subgroups — #29646915

Order 15616.

1216 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459305C211yesmaximal, characteristic
2#21454785C211yesmaximal, characteristic, complemented
3#21454487C211yesmaximal, characteristic, complemented
4#21444286C212yesmaximal, complemented
5#21444260C212yesmaximal, complemented
6#21444243C211yesmaximal, characteristic, complemented
7#21444234C212yesmaximal, complemented
8#21444218C211yesmaximal, characteristic, complemented
9#21444208C212yesmaximal, complemented
10C2×#16839491C211yesmaximal, characteristic, complemented
11C61×#1492C211yesmaximal, characteristic, complemented
12C2×#41×D61—21no—
13#16840432C411yescharacteristic
14#16840427C411yescharacteristic
15#16840420—21no—
16#16840415C411yescharacteristic
17#16840412C412yes—
18#16840405—21no—
19#16840401—22no—
20#16840397C411yescharacteristic
21#16840396C412yes—
22#16840309C2212yes—
23#16840271—22no—
24#16840269—21no—
25C4×#11758222C2211yescharacteristic
26C2×C4×D244C2211yescharacteristic
27C4×D4×D61C2212yes—
28#16839892—21no—
29#16839841—21no—
30#16839833—22no—
31#16839822—22no—
32#16839818C2212yescomplemented
33#16839815C2211yescharacteristic, complemented
34#16839814C2212yescomplemented
35#16839813C2212yescomplemented
36C2×C4⋊#92418C2211yescharacteristic, complemented
37#16839730—21no—
38#16839720—21no—
39C4×#11758189C2212yescomplemented
40#16839491C2211yescharacteristic
41#16839491C2211yescharacteristic
42#16839491C2211yescharacteristic, complemented
43#16839491C2211yescharacteristic, complemented
44D4⋊#92418C2212yescomplemented
45D4⋊#92418C2212yescomplemented
46#16839489C2212yes—
47#16839489C2212yescomplemented
48C61×#532C2211yescharacteristic
49C61×#423C2211yescharacteristic, complemented
50C2×C4×C8×C61C2211yescharacteristic, complemented
51C61×D4⋊C8C2212yescomplemented
52C61×D4⋊C8C2212yescomplemented
53C2×#11758226—21no—
54C2×#11758222—22no—
55C2×#11758222D411yescharacteristic
56C2×#11758224—22no—
57C2×#11758224—21no—
58#41×D61—44no—
59#41×D61—44no—
60C2×Q8×D61—21no—
61C22×D244C2×C411yescharacteristic
62C2×D4×D61—22no—
63C2×D4×D61—21no—
64C2×D4×D61C2×C412yes—
65C22×C4×D61—21no—
66C22×C4×D61—22no—
67C2×#11758189—22no—
68C22×#92418—21no—
69#14519776C2×C412yes—
70#14519773C2×C412yes—
71#14519763—22no—
72#14519762—22no—
73#14519759—22no—
74#14519758—22no—
75#14519755—22no—
76#14519754—22no—
77#14519753—21no—
78#14519753—21no—
79#14519752C2×C412yes—
80#14519750—22no—
81#14519750—22no—
82#14519749—22no—
83#14519748—21no—
84#14519748—21no—
85#14519748C2×C412yes—
86#14519747—22no—
87#14519745—22no—
88#14519745—22no—
89C4×Dic122C2×C411yescharacteristic
90C4×Dic122D411yescharacteristic
91C4×D244—21no—
92C4×D244—21no—
93C4×D244C2×C411yescharacteristic
94C4×D244C2×C411yescharacteristic
95C4×D244C2×C411yescharacteristic
96C4×D244D411yescharacteristic
97C4×D244C2311yescharacteristic
98C4×D244C2311yescharacteristic
99D4×Dic61C2×C412yes—
100#14519739C2×C412yes—
101C4×C22⋊D61—22no—
102C4×C22⋊D61—22no—
103C4×C22⋊D61C2×C412yes—
104C4×C22⋊D61D412yes—
105#14519737—22no—
106#14519736D411yescharacteristic
107#14519734D412yes—
108C2×C4⋊Dic61C2×C411yescharacteristic
109C4⋊C4×D61C2×C412yes—
110C2×#11758198—21no—
111C22⋊C4×D61—22no—
112C42×D61C2312yes—
113C4⋊#92418—21no—
114C4⋊#92418—21no—
115C4⋊#92418C2311yescharacteristic
116C4⋊#92418C2311yescharacteristic
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519689—22no—
121#14519689—22no—
122#14519687—22no—
123#14519687—22no—
124C4×#92418C2312yes—
125C8×Dic61—22no—
126C2×#41×C61—21no—
127C2×M16×C61—21no—
128C22×C8×C61—21no—
129C4⋊Q8×C61C2×C411yescharacteristic
130#127×C61C2×C411yescharacteristic
131#124×C61—21no—
132#121×C61—21no—
133#121×C61C2×C412yes—
134C4×D4×C61—21no—
135C4×D4×C61C2312yes—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139C4×C8×C61—21no—
140C4×C8×C61C2311yescharacteristic
141C4×C8×C61C2311yescharacteristic
142C23×D61—21no—
143C23×D61—22no—
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⋊D61C22⋊C412yes—
160C22×Dic61—21no—
161C22×Dic61—22no—
162#11758226—44no—
163Q8×D61—44no—
164#11758224—44no—
165#11758224—48no—
166D4×D61—24no—
167D4×D61—42no—
168D4×D61—44no—
169D4×D61—48no—
170#11758222—22no—
171#11758222—22no—
172#11758222—44no—
173#11758222—44no—
174C2×D244—22no—
175C2×D244—22no—
176C2×D244—21no—
177C2×D244—41no—
178C2×D244C22⋊C411yescharacteristic
179C2×D244C22×C411yescharacteristic
180C2×D244C22×C411yescharacteristic
181C2×C4×D61—22no—
182C2×C4×D61—21no—
183C2×C4×D61—21no—
184C2×C4×D61—21no—
185C2×C4×D61—21no—
186C2×C4×D61—22no—
187C2×C4×D61—44no—
188C2×C4×D61—44no—
189C2×C4×D61—42no—
190C2×C4×D61—42no—
191C2×C4×D61C22×C412yes—
192C2×C4×D61C2×D412yes—
193C2×C4×D61C2×D412yes—
194C2×Dic122—22no—
195C2×Dic122C22⋊C411yescharacteristic
196M16×C61—42no—
197C2×C8×C61—21no—
198C2×C8×C61—21no—
199C2×C8×C61—22no—
200C2×C8×C61—42no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61C22×C412yes—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C42×C61C22×C411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2411yescharacteristic
209C22⋊Dic61—22no—
210C22⋊Dic61—22no—
211C22⋊Dic61C22⋊C412yes—
212#11758198—21no—
213#11758198—21no—
214#11758198—21no—
215#11758198—21no—
216#11758198—22no—
217#11758198—22no—
218#11758198C22⋊C412yes—
219#11758198C22⋊C412yes—
220C4⋊Dic61—21no—
221C4⋊Dic61C22⋊C411yescharacteristic
222C4⋊Dic61C22⋊C411yescharacteristic
223C4⋊Dic61C22×C411yescharacteristic
224C4⋊Dic61C22×C411yescharacteristic
225#11758196—22no—
226#11758196—22no—
227#11758196C22⋊C412yes—
228#11758196C22⋊C412yes—
229C4×Dic61C22×C412yes—
230C4×Dic61C2×D412yes—
231C4×Dic61C2×D412yes—
232C2×#92418—21no—
233C2×#92418—22no—
234C2×#92418—21no—
235C2×#92418—42no—
236#11758189—44no—
237C23×C61—22no—
238C23×C61C2×C22⋊C411yescharacteristic
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—44no—
242C22×D61—42no—
243C22×D61—44no—
244C22×D61—42no—
245C22×D61—82no—
246C22×D61—81no—
247C22×D61C2×C22⋊C412yes—
248Q8×C61—42no—
249D4×C61—22no—
250D4×C61—22no—
251D4×C61—44no—
252D4×C61—42no—
253D4×C61—81no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—22no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61AGL(1,ℤ/8)12yes—
264C2×C4×C61AGL(1,ℤ/8)12yes—
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C22⋊D61—24no—
269C22⋊D61—44no—
270C22⋊D61—44no—
271C22⋊D61—44no—
272C22⋊D61—82no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—44no—
276C2×Dic61—42no—
277C2×Dic61C2×C22⋊C412yes—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280D244—21no—
281D244—21no—
282D244—21no—
283D244—22no—
284D244—42no—
285D244—44no—
286D244#13112yes—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—42no—
292C4×D61—44no—
293C4×D61—44no—
294C4×D61—41no—
295C4×D61—41no—
296C4×D61—41no—
297C4×D61—41no—
298C4×D61—42no—
299C4×D61—42no—
300Dic122—22no—
301Dic122—44no—
302C8×C61—22no—
303C8×C61—41no—
304C8×C61—42no—
305#92418—42no—
306#92418—41no—
307#92418—41no—
308#17511—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61#41812yes—
315C22×C61C22×C22⋊C411yescharacteristic
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—41no—
325C2×D61—88no—
326C2×D61—84no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—42no—
331C4×C61#41812yes—
332C4×C61C2×#13111yescharacteristic
333C4×C61#56811yescharacteristic
334C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
335C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
336Dic61—22no—
337Dic61—22no—
338Dic61—22no—
339Dic61—41no—
340Dic61—41no—
341D4×#41—611no—
342#2965—611no—
343#2925—611no—
344#1496—612no—
345#1492D6111yescharacteristic, complemented
346#1490—612no—
347#1486—611no—
348#1484—612no—
349#1480—611no—
350#1478—612no—
351C2×D4⋊C8—611no—
352C2×C61—22no—
353C2×C61—82no—
354C2×C61#288011yescharacteristic
355C2×C61C2×#41811yescharacteristic
356C2×C61#290111yescharacteristic
357D61—42no—
358D61—42no—
359D61—81no—
360C22×#41—1221no—
361#550—611no—
362D4×Q8—611no—
363#548—1221no—
364#547—611no—
365#547—612no—
366#546—1222no—
367#546—1221no—
368D42—611no—
369D42—612no—
370#532—612no—
371#532C2×D6111yescharacteristic
372#525—1222no—
373#525—1221no—
374C4×#41—611no—
375C2×C4×D4—612no—
376C2×C4×D4—611no—
377#446—1221no—
378#436—1221no—
379#435—1222no—
380#433—1222no—
381#432—612no—
382#431—611no—
383#431—612no—
384#431—612no—
385#423C2×D6111yescharacteristic, complemented
386C2×C4⋊C8—611no—
387#408—1221no—
388#405—1221no—
389C4×M16—612no—
390C2×C4×C8C2×D6111yescharacteristic, complemented
391D4⋊C8—612no—
392D4⋊C8—612no—
393D4⋊C8—612no—
394D4⋊C8—612no—
395D4⋊C8—611no—
396D4⋊C8—611no—
397D4⋊C8—611no—
398D4⋊C8—611no—
399D4⋊C8C2×D6112yescomplemented
400D4⋊C8C2×D6112yescomplemented
401C61#1751111yescharacteristic, complemented, Sylow
402C2×#41—21no—
403C2×#41—611no—
404C2×#41—1222no—
405C2×#41—1222no—
406C2×#41—1221no—
407C2×#41—1221no—
408C2×#41—2444no—
409C2×#41—2444no—
410C22×Q8—1221no—
411C22×D4—612no—
412C22×D4—611no—
413C22×D4—1222no—
414C22×D4—1221no—
415C23×C4—1222no—
416C23×C4—1221no—
417C2×M16—21no—
418C2×M16—1222no—
419C22×C8—21no—
420C22×C8—1221no—
421C4⋊Q8—612no—
422C4⋊Q8C4×D6111yescharacteristic
423#127—612no—
424#127C4×D6111yescharacteristic
425#124—21no—
426#124—1222no—
427#123—1222no—
428#123—1222no—
429#123—1222no—
430C22⋊Q8—1222no—
431C22⋊Q8—1222no—
432C22⋊Q8—1221no—
433C22⋊Q8—1221no—
434#121—21no—
435#121—612no—
436#121—612no—
437#121—1222no—
438#121—1222no—
439#121—1221no—
440#121—1221no—
441#121—1222no—
442#121C4×D6112yes—
443#120—1222no—
444#120—1222no—
445#120—1222no—
446C4×Q8—611no—
447C4×Q8—611no—
448C4×D4—21no—
449C4×D4—612no—
450C4×D4—612no—
451C4×D4—612no—
452C4×D4—612no—
453C4×D4—611no—
454C4×D4—611no—
455C4×D4—611no—
456C4×D4—611no—
457C4×D4—611no—
458C4×D4—611no—
459C4×D4—1222no—
460C4×D4—1222no—
461C4×D4—1222no—
462C4×D4—1221no—
463C4×D4—1221no—
464C4×D4C22×D6112yes—
465#117—611no—
466#117—612no—
467C2×C4⋊C4—611no—
468C2×C4⋊C4—612no—
469C2×C22⋊C4—1222no—
470C2×C22⋊C4—1221no—
471C2×C42—612no—
472C2×C42C22×D6111yescharacteristic
473C4⋊C8—21no—
474C4⋊C8—611no—
475C4⋊C8—611no—
476C4⋊C8—1222no—
477C4⋊C8—1221no—
478C4⋊C8—1221no—
479C4⋊C8C22×D6112yes—
480C22⋊C8—1222no—
481C22⋊C8—1222no—
482C22⋊C8—1222no—
483C22⋊C8—1221no—
484#97—1222no—
485#97—1222no—
486C4×C8—21no—
487C4×C8—612no—
488C4×C8—1222no—
489C4×C8C22×D6111yescharacteristic
490C4×C8C22×D6111yescharacteristic
491C24—1222no—
492C24—1221no—
493#41—42no—
494#41—42no—
495#41—1222no—
496#41—1222no—
497#41—2444no—
498#41—2444no—
499#41—2448no—
500#41—2444no—
501#41—2444no—
502C2×Q8—21no—
503C2×Q8—611no—
504C2×Q8—1222no—
505C2×Q8—2444no—
506C2×D4—22no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—612no—
510C2×D4—611no—
511C2×D4—611no—
512C2×D4—611no—
513C2×D4—1222no—
514C2×D4—1222no—
515C2×D4—1222no—
516C2×D4—1222no—
517C2×D4—1222no—
518C2×D4—1224no—
519C2×D4—1222no—
520C2×D4—1221no—
521C2×D4—2441no—
522C2×D4—2442no—
523C2×D4—2448no—
524C2×D4—2444no—
525C2×D4C2×C4×D6112yes—
526C22×C4—22no—
527C22×C4—612no—
528C22×C4—612no—
529C22×C4—612no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—1222no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—1221no—
537C22×C4—1221no—
538C22×C4—2444no—
539C22×C4—2444no—
540C22×C4—2442no—
541C22×C4—2442no—
542C22×C4C2×C4×D6111yescharacteristic
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545M16—42no—
546M16—2444no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—22no—
550C2×C8—42no—
551C2×C8—1222no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—2442no—
555C4⋊C4—21no—
556C4⋊C4—612no—
557C4⋊C4—612no—
558C4⋊C4—611no—
559C4⋊C4—611no—
560C4⋊C4—611no—
561C4⋊C4—611no—
562C4⋊C4—1222no—
563C4⋊C4—1222no—
564C4⋊C4—1221no—
565C4⋊C4C2×C4×D6112yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—612no—
569C22⋊C4—612no—
570C22⋊C4—612no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C22⋊C4—1221no—
576C22⋊C4—1221no—
577C22⋊C4—1221no—
578C22⋊C4—1221no—
579C42—612no—
580C42—612no—
581C42—612no—
582C42C2×C4×D6111yescharacteristic
583C42D4×D6111yescharacteristic
584C42D4×D6111yescharacteristic
585C42C23×D6111yescharacteristic
586C23—22no—
587C23—612no—
588C23—1221no—
589C23—1221no—
590C23—4881no—
591C23—4882no—
592C23—2442no—
593C23—2444no—
594C23—2444no—
595C23—2442no—
596C23C22⋊C4×D6111yescharacteristic
597Q8—42no—
598Q8—1222no—
599Q8—2444no—
600D4—22no—
601D4—22no—
602D4—44no—
603D4—42no—
604D4—81no—
605D4—612no—
606D4—1224no—
607D4—1221no—
608D4—1221no—
609D4—1221no—
610D4—1222no—
611D4—4882no—
612D4—2442no—
613D4—2444no—
614D4—2444no—
615D4—2444no—
616D4—2444no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—612no—
624C2×C4—612no—
625C2×C4—612no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—2442no—
633C2×C4—2444no—
634C2×C4—2444no—
635C2×C4—2444no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4C22⋊C4×D6111yescharacteristic
644C2×C4C22⋊C4×D6111yescharacteristic
645C2×C4C22⋊C4×D6111yescharacteristic
646C2×C4#1451980112yes—
647C2×C4#1451980412yes—
648C2×C4C22×C4×D6111yescharacteristic
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4C2×D4×D6111yescharacteristic
651C8—22no—
652C8—41no—
653C8—42no—
654C8—2442no—
655C8—2441no—
656C8—2441no—
657C22—21no—
658C22—21no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—1222no—
663C22—1222no—
664C22—1222no—
665C22—1222no—
666C22—4884no—
667C22—4888no—
668C22—2441no—
669C22—2441no—
670C22—2441no—
671C22—2441no—
672C22—2441no—
673C22#1683976512yes—
674C22C2×C22⋊C4×D6111yescharacteristic
675C4—21no—
676C4—21no—
677C4—21no—
678C4—42no—
679C4—1222no—
680C4—1222no—
681C4—1222no—
682C4—2441no—
683C4—2441no—
684C4#1683976512yes—
685C4#131×D6111yescharacteristic
686C4#1684052711yescharacteristic
687C4C2×#1451980111yescharacteristic
688C4C2×#1451980411yescharacteristic
689C2—22no—
690C2—82no—
691C2—4881no—
692C2—2442no—
693C2—2442no—
694C2#2145426711yescharacteristic
695C2C2×#1683976511yescharacteristic
696C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1492C2
socleC22×C61C22×C22⋊C4
prime radical#21459305C2
cosocleC42×C61C24
hypercenter#1492D61
hypocenterC61#17511
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. #21454785C2semidirect
3. #21454487C2semidirect
4. #21444286C2semidirect
5. #21444260C2semidirect
6. #21444243C2semidirect
7. #21444234C2semidirect
8. #21444218C2semidirect
9. #21444208C2semidirect
10. C2×#16839491C2semidirect
11. C61×#1492C2semidirect
32. #16839818C22semidirect
33. #16839815C22semidirect
34. #16839814C22semidirect
35. #16839813C22semidirect
36. C2×C4⋊#92418C22semidirect
39. C4×#11758189C22semidirect
42. #16839491C22semidirect
43. #16839491C22semidirect
44. D4⋊#92418C22semidirect
45. D4⋊#92418C22semidirect
47. #16839489C22semidirect
49. C61×#423C22semidirect
50. C2×C4×C8×C61C22semidirect
51. C61×D4⋊C8C22semidirect
52. C61×D4⋊C8C22semidirect
345. #1492D61semidirect
385. #423C2×D61semidirect
390. C2×C4×C8C2×D61semidirect
399. D4⋊C8C2×D61semidirect
400. D4⋊C8C2×D61semidirect
401. C61#17511semidirect

Maximal subgroups

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

subgroupindex
#214593052
#214547852
#214544872
#214442862
#214442602
#214442432
#214442342
#214442182
#214442082
C2×#168394912
C61×#14922
#1751161