Finite Group Databasebeta

← #29716624

Subgroups — #29716624

Order 15616.

1064 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840490C211yesmaximal, characteristic, complemented
2C2×#16840062C211yesmaximal, characteristic
3C2×#16839862C211yesmaximal, characteristic
4#21447157C211yesmaximal, characteristic, complemented
5#21447147C211yesmaximal, characteristic, complemented
6#21446965C211yesmaximal, characteristic
7#21446955C211yesmaximal, characteristic
8#21446787C211yesmaximal, characteristic, complemented
9#21446760C211yesmaximal, characteristic, complemented
10#21446177C211yesmaximal, characteristic
11#21446177C211yesmaximal, characteristic
12#21446086C211yesmaximal, characteristic, complemented
13#21446072C211yesmaximal, characteristic, complemented
14#21445821C211yesmaximal, characteristic, complemented
15C61×#2538C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17#16840490—24no—
18#16840490—24no—
19C2×#14519777C2211yescharacteristic
20C2×#14519775—21no—
21C2×#14519769C2211yescharacteristic
22C2×#14519769C2211yescharacteristic
23C2×#14519767—21no—
24C2×#14519767—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×Dic122C2211yescharacteristic
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519742—21no—
29#16840062—24no—
30#16839862—24no—
31C2×#14519710C2211yescharacteristic
32C2×#14519710C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×C4⋊#92418C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519698C2211yescharacteristic
40C2×#14519698C2211yescharacteristic
41C2×#14519688C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839517C2211yescharacteristic
52C2×C22⋊Q8×C61—21no—
53C2×C4×Q8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#396—21no—
56C61×#396—21no—
57C61×#390C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#384C2211yescharacteristic
62C61×#380—21no—
63C22×Dic122—21no—
64C22×Dic122C2311yescharacteristic
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200Q814yescomplemented
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519777—22no—
73#14519777—22no—
74#14519775—28no—
75#14519769—22no—
76#14519769—22no—
77#14519769—22no—
78#14519769D414yes—
79#14519767—28no—
80#14519767—28no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85C4×Dic122—22no—
86C4×Dic122D414yes—
87#14519742—28no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102#14519710—22no—
103#14519710—22no—
104#14519708—22no—
105C4⋊#92418—22no—
106#14519700—22no—
107#14519700—22no—
108#14519700—22no—
109#14519698—22no—
110#14519698—22no—
111#14519698—22no—
112#14519688—22no—
113C22×Q8×C61D411yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C23×C4×C61—21no—
116C22⋊Q8×C61—44no—
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C22⋊C4×C61—21no—
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C2×C42×C61C2311yescharacteristic
138C2×C42×C61C2311yescharacteristic
139#95×C61—21no—
140#95×C61—21no—
141C24×C61—21no—
142C2×Q8×C61—22no—
143C2×Q8×C61—22no—
144C2×Q8×C61—41no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×Q812yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—24no—
172C2×Dic122—42no—
173C2×Dic122—42no—
174C2×Dic122C2×D412yes—
175C2×Dic122C2×Q812yes—
176C2×Dic122#4112yes—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61—42no—
189C4⋊C4×C61—81no—
190C22⋊C4×C61—42no—
191C22⋊C4×C61—42no—
192C22⋊C4×C61—42no—
193C22⋊C4×C61—81no—
194C22⋊C4×C61—81no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61C2×D412yes—
200C42×C61C2×D412yes—
201#11758200—28no—
202C4⋊Dic61—24no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61C2×D412yes—
208C4⋊Dic61C2×D412yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214#11758196—24no—
215#11758196—24no—
216C4×Dic61—24no—
217C4×Dic61—24no—
218C4×Dic61—24no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×Q812yes—
231C23×C61—22no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61C22×D411yescharacteristic
239Q8×C61—44no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—41no—
243D4×C61—41no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—84no—
269C2×C4×C61#12012yes—
270C2×C4×C61#12012yes—
271C2×C4×C61AGL(1,ℤ/8)12yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×Q811yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C612−1+411yescharacteristic
277C2×C4×C612−1+411yescharacteristic
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290Dic122—22no—
291Dic122—22no—
292Dic122—22no—
293Dic122—48no—
294Dic122C22⋊Q814yescomplemented
295#92418—22no—
296#92418—41no—
297#92418—41no—
298#92418—41no—
299#24346—611nomaximal, Sylow
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61C2×#12011yescharacteristic
308C22×C61#54811yescharacteristic
309C22×C61D4×Q811yescharacteristic
310C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C61#57511yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57811yescharacteristic
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61C2×C22⋊Q811yescharacteristic
324C4×C61#53611yescharacteristic
325C4×C61#54811yescharacteristic
326C4×C61D4×Q811yescharacteristic
327Dic61—24no—
328Dic61—42no—
329Dic61—42no—
330Dic61—42no—
331C2×#557—611no—
332C2×#478—611no—
333C2×#440—611no—
334#2538D6111yescharacteristic
335#2059—611no—
336#2057—611no—
337#2025—611no—
338#2023—611no—
339#1992—611no—
340#1985—611no—
341#1880—611no—
342#1880—611no—
343#1865—611no—
344#1863—611no—
345#1803—611no—
346C2×C61—82no—
347C2×C61#240911yescharacteristic
348C2×C61#300611yescharacteristic
349C2×C61#301311yescharacteristic
350C2×C61#336811yescharacteristic
351C2×C61#337711yescharacteristic
352C2×C61#338611yescharacteristic
353C2×C61#338811yescharacteristic
354C22×SD16—611no—
355#557—1224no—
356#557—1224no—
357C2×C4⋊Q8—611no—
358C2×C4⋊Q8—1221no—
359C2×#125—611no—
360C2×#125—611no—
361C2×#125—1221no—
362C2×#125—1221no—
363C2×C22⋊Q8—21no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8—611no—
367C2×C4×Q8—1221no—
368C2×C4×Q8C2×D6111yescharacteristic
369C2×C4×D4C2×D6111yescharacteristic
370#478—1224no—
371#440—1224no—
372C2×#107—611no—
373C2×#107—611no—
374C2×#106—611no—
375C2×C4⋊C8—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#97—611no—
383#396—21no—
384#396—21no—
385#390C2×D6111yescharacteristic
386#390C2×D6111yescharacteristic
387#389C2×D6111yescharacteristic
388#389C2×D6111yescharacteristic
389#384C2×D6111yescharacteristic
390#380—21no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#336—611no—
398#336—611no—
399#336—611no—
400#336—611no—
401C61#2434611yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8—1221no—
404C22×Q8C22⋊D6111yescharacteristic
405C22×D4C22×D6111yescharacteristic
406C23×C4—21no—
407C2×SD16—614no—
408C2×SD16—1222no—
409C2×SD16—1222no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—611no—
413C22×C8—611no—
414C4⋊Q8—1222no—
415C4⋊Q8—1222no—
416C4⋊Q8—1228no—
417#125—614no—
418#125—1222no—
419#125—1222no—
420#125—1228no—
421#125—1228no—
422#125—1222no—
423C22⋊Q8—44no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—614no—
427C4×Q8—1222no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×Q8—1222no—
431C4×Q8—1228no—
432C4×Q8—1222no—
433C4×D4—22no—
434C4×D4—22no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4C22⋊D6111yescharacteristic
451C2×C4⋊C4C22⋊D6111yescharacteristic
452C2×C4⋊C4C22⋊D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C22⋊C4—21no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—21no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42—1221no—
464C2×C42C22×D6111yescharacteristic
465C2×C42C22×D6111yescharacteristic
466C2×C42C22×D6111yescharacteristic
467#107—1222no—
468#107—1222no—
469#106—1222no—
470C4⋊C8—1222no—
471#103—1222no—
472#103—1222no—
473#103—1222no—
474#102—1222no—
475#102—1222no—
476#102—1222no—
477#97—1222no—
478#95—21no—
479#95—21no—
480C24—21no—
481C2×Q8—22no—
482C2×Q8—22no—
483C2×Q8—41no—
484C2×Q8—612no—
485C2×Q8—612no—
486C2×Q8—612no—
487C2×Q8—1224no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—2442no—
492C2×Q8—2442no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4Q8×D6112yes—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516SD16—1228no—
517C2×C8—612no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—81no—
542C4⋊C4—612no—
543C4⋊C4—612no—
544C4⋊C4—612no—
545C4⋊C4—1224no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1224no—
554C4⋊C4—1224no—
555C4⋊C4—1224no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—81no—
560C22⋊C4—81no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—1224no—
566C42—1224no—
567C42—1224no—
568C42D4×D6112yes—
569C42D4×D6112yes—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C22×C22⋊D6111yescharacteristic
578Q8—44no—
579Q8—614no—
580Q8—1222no—
581Q8—1222no—
582Q8—1222no—
583Q8—2448no—
584D4—22no—
585D4—22no—
586D4—41no—
587D4—41no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—84no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4#1451974512yes—
626C2×C4#1451974512yes—
627C2×C4#1451980312yes—
628C2×C4C2×D4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×Q8×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4#1451982411yescharacteristic
633C2×C4#1451982511yescharacteristic
634C8—1222no—
635C8—2441no—
636C8—2441no—
637C8—2441no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C2×#1451974511yescharacteristic
646C22#1684041911yescharacteristic
647C22Q8×C22⋊D6111yescharacteristic
648C22C2×#1451980311yescharacteristic
649C22#1684056011yescharacteristic
650C22#1684057311yescharacteristic
651C22#1684057711yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—1224no—
662C4—2442no—
663C4—2442no—
664C4—2442no—
665C4C22⋊Q8×D6111yescharacteristic
666C4#1684033511yescharacteristic
667C4#1684042211yescharacteristic
668C4#1684042511yescharacteristic
669C2—82no—
670C2#2144918011yescharacteristic
671C2#2145506911yescharacteristic
672C2#2145513111yescharacteristic
673C2#2145854411yescharacteristic
674C2#2145865511yescharacteristic
675C2#2145874011yescharacteristic
676C2#2145876411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2538D61
hypocenterC61#24346
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#16840490C2semidirect
4. #21447157C2semidirect
5. #21447147C2semidirect
8. #21446787C2semidirect
9. #21446760C2semidirect
12. #21446086C2semidirect
13. #21446072C2semidirect
14. #21445821C2semidirect
67. C2×#11758200Q8semidirect
294. Dic122C22⋊Q8semidirect
401. C61#24346semidirect

Maximal subgroups

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

subgroupindex
C2×#168404902
C2×#168400622
C2×#168398622
#214471572
#214471472
#214469652
#214469552
#214467872
#214467602
#214461772
#214460862
#214460722
#214458212
C61×#25382
#2434661