Finite Group Databasebeta

← #29661518

Subgroups — #29661518

Order 15616.

880 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3C2×#16840132C211yesmaximal, characteristic, complemented
4#21445153C214yesmaximal, complemented
5#21445142C214yesmaximal, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3559C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172C2212yes—
12#16840132—22no—
13#16840132C2214yes—
14#16840121C2212yes—
15#16840121C2214yes—
16#16840112—21no—
17#16840108—21no—
18#16840095—21no—
19#16840078—21no—
20#16840047—21no—
21#16840042—21no—
22#16839819C2211yescharacteristic
23#16839763—22no—
24C2×#14519700—21no—
25C2×#14519700—21no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2214yes—
29#16839504C2212yes—
30#16839502C2212yes—
31#16839502C2214yes—
32C61×#563C2212yes—
33C61×#562C2212yes—
34C61×#554—21no—
35C61×#553—21no—
36C61×#553—21no—
37C61×#547—22no—
38C61×#547—21no—
39C61×#544—21no—
40C61×#540—21no—
41C61×#539—22no—
42C61×#539—21no—
43C61×#537—21no—
44C61×#532C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758194—21no—
49C22×#92418—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777C2312yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56#14519710—22no—
57#14519708—22no—
58C4⋊#92418—21no—
59#14519701—24no—
60#14519701—22no—
61#14519700—24no—
62#14519700—24no—
63#14519698—24no—
64#14519698—22no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C2×#41×C61—22no—
72C2×#41×C61D412yes—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61D411yescharacteristic
76#127×C61D411yescharacteristic
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—21no—
80#125×C61—22no—
81#125×C61—21no—
82#125×C61—21no—
83#125×C61—21no—
84#125×C61D412yes—
85#125×C61C2312yes—
86#125×C61C2311yescharacteristic
87#124×C61C2312yes—
88#124×C61C2311yescharacteristic
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93#123×C61—21no—
94#123×C61—22no—
95#123×C61—22no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61D411yescharacteristic
103#121×C61—22no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—21no—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61D411yescharacteristic
112C4×Q8×C61D411yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C4×D4×C61—21no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61D411yescharacteristic
121#117×C61—21no—
122#117×C61—21no—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130#41×C61—44no—
131#41×C61—84no—
132C2×Q8×C61—22no—
133C2×Q8×C61C2×D411yescharacteristic
134C2×Q8×C61C2×D412yes—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D411yescharacteristic
142C2×D4×C61C2×D412yes—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×Dic61—41no—
154C2×Dic122—22no—
155C2×Dic122—44no—
156C2×Dic122—41no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D412yes—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C42×C61—22no—
190C42×C61—21no—
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2411yescharacteristic
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199#11758194—42no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—42no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61#12011yescharacteristic
207Q8×C61—41no—
208Q8×C61—42no—
209Q8×C61—82no—
210D4×C61—41no—
211D4×C61—42no—
212D4×C61—44no—
213D4×C61—42no—
214D4×C61—42no—
215D4×C61—81no—
216D4×C61—82no—
217D4×C61—82no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—82no—
230C2×C4×C61—81no—
231C2×C4×C61—82no—
232C2×C4×C61—82no—
233C2×C4×C61—82no—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×Dic61—42no—
241C2×Dic61—82no—
242Dic122—44no—
243Dic122—82no—
244#92418—42no—
245#92418—42no—
246#92418—42no—
247#19023—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—42no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—41no—
256C22×C61#45512yes—
257C22×C61C2×#12011yescharacteristic
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—42no—
268C4×C61—41no—
269C4×C61#45412yes—
270Dic61—82no—
271#3559D6111yescharacteristic
272#3239—611no—
273#3141—611no—
274C2×#491—611no—
275#1664—614no—
276#1662—614no—
277#1530—611no—
278#1528—611no—
279#1524—611no—
280C2×C61—22no—
281C2×C61—82no—
282C2×C61—81no—
283C2×C61C2×#45511yescharacteristic
284C2×C61C2×#45411yescharacteristic
285C2×C61#302211yescharacteristic
286#563C2×D6112yes—
287#562C2×D6112yes—
288#554—21no—
289#553—21no—
290#553—21no—
291#547—22no—
292#547—21no—
293#544—21no—
294#540—21no—
295#539—22no—
296#539—21no—
297#537—21no—
298#532C2×D6111yescharacteristic
299C2×C4⋊Q8—611no—
300#529C2×D6111yescharacteristic
301C2×#125C2×D6111yescharacteristic
302#501—612no—
303#491—614no—
304#491—1222no—
305#489—614no—
306#489—612no—
307#487—1221no—
308#486—1221no—
309#484—1221no—
310#481—1221no—
311#475—1221no—
312#474—1221no—
313#432—611no—
314#417—1222no—
315C2×#103—1221no—
316C2×#103—1221no—
317C4×M16—611no—
318C2×#97—611no—
319#330—614no—
320#330—612no—
321#329—612no—
322#329—614no—
323C61#1902311yescharacteristic, complemented, Sylow
324C2×#41—22no—
325C2×#41C22⋊D6112yes—
326C22×Q8—1221no—
327C2×M16—1221no—
328C22×C8—1221no—
329C4⋊Q8—21no—
330C4⋊Q8—22no—
331C4⋊Q8—612no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8C22⋊D6111yescharacteristic
336#127C22⋊D6111yescharacteristic
337#126—22no—
338#126—22no—
339#126—21no—
340#125—22no—
341#125—21no—
342#125—21no—
343#125—21no—
344#125C22⋊D6112yes—
345#125C22×D6112yes—
346#125C22×D6111yescharacteristic
347#124C22×D6111yescharacteristic
348#124C22×D6112yes—
349#123—22no—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—21no—
354#123—22no—
355#123—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8C22⋊D6111yescharacteristic
363#121—22no—
364#121—22no—
365#121—21no—
366#121—21no—
367#121—22no—
368#121—22no—
369#121—21no—
370#121—21no—
371#121C22⋊D6111yescharacteristic
372C4×Q8C22⋊D6111yescharacteristic
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22⋊D6111yescharacteristic
381#117—21no—
382#117—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22⋊D6111yescharacteristic
390C2×C4⋊C4C22⋊D6111yescharacteristic
391C2×C42C22×D6111yescharacteristic
392#107—1222no—
393#106—1222no—
394C4⋊C8—1221no—
395#103—1224no—
396#103—1222no—
397#103—1224no—
398#103—1224no—
399#102—1224no—
400#102—1222no—
401C22⋊C8—1221no—
402#97—612no—
403#97—612no—
404#97—612no—
405#97—1221no—
406C4×C8—1221no—
407#41—44no—
408#41—84no—
409C2×Q8—22no—
410C2×Q8—1222no—
411C2×Q8—2444no—
412C2×Q8—2441no—
413C2×Q8C2×C22⋊D6111yescharacteristic
414C2×Q8C2×C22⋊D6112yes—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×C22⋊D6111yescharacteristic
422C2×D4C2×C22⋊D6112yes—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—2441no—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434M16—2442no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—2442no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—2441no—
458C4⋊C4—2441no—
459C4⋊C4C2×C22⋊D6112yes—
460C4⋊C4C2×C22⋊D6112yes—
461C4⋊C4C2×C22⋊D6111yescharacteristic
462C4⋊C4C2×C22⋊D6111yescharacteristic
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C42—22no—
476C42—21no—
477C42C2×C22⋊D6111yescharacteristic
478C42C2×C22⋊D6111yescharacteristic
479C42C2×C22⋊D6111yescharacteristic
480C42C23×D6111yescharacteristic
481C23—22no—
482C23—21no—
483C23#1451974611yescharacteristic
484Q8—41no—
485Q8—42no—
486Q8—82no—
487Q8—4882no—
488Q8—2444no—
489D4—41no—
490D4—42no—
491D4—44no—
492D4—42no—
493D4—42no—
494D4—81no—
495D4—82no—
496D4—82no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—4882no—
514C2×C4—2442no—
515C2×C4#1451974611yescharacteristic
516C2×C4#1451974611yescharacteristic
517C2×C4#1451974611yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C8—2442no—
522C8—2442no—
523C8—2442no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22#1683994012yes—
533C22C2×#1451974611yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—42no—
540C4—42no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—4882no—
546C4#1683993512yes—
547C2—22no—
548C2—82no—
549C2—81no—
550C2C2×#1683994011yescharacteristic
551C2C2×#1683993511yescharacteristic
552C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3559C2
socleC22×C61C2×#120
hypercenter#3559D61
hypocenterC61#19023
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. #21457046C2semidirect
2. #21456212C2semidirect
3. C2×#16840132C2semidirect
4. #21445153C2semidirect
5. #21445142C2semidirect
6. #21444430C2semidirect
7. #21444421C2semidirect
8. #21444407C2semidirect
323. C61#19023semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214562122
C2×#168401322
#214451532
#214451422
#214444302
#214444212
#214444072
C61×#35592
#1902361