Finite Group Databasebeta

← #29734449

Subgroups — #29734449

Order 15616.

1156 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#21456366C211yesmaximal, characteristic, complemented
2#21455634C211yesmaximal, characteristic, complemented
3#21454175C211yesmaximal, characteristic, complemented
4#21447026C214yesmaximal, complemented
5#21446797C211yesmaximal, characteristic, complemented
6#21446660C211yesmaximal, characteristic, complemented
7#21446029C214yesmaximal, complemented
8C4×#14519704C211yesmaximal, characteristic, complemented
9C61×#3508C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11C2×#14519764—22no—
12#16840172—22no—
13#16840165—21no—
14#16840158—21no—
15#16840057—21no—
16#16840052—21no—
17#16840047—21no—
18#16840042—21no—
19#16839805C2212yes—
20#16839805C2214yes—
21#16839786C2211yescharacteristic
22C2×#14519704C2212yes—
23#16839763C2212yes—
24C2×#14519694C2214yes—
25C4×#11758194C2211yescharacteristic
26#16839665—22no—
27#16839665C2214yes—
28C42×Dic61C2211yescharacteristic
29#16839521C2214yes—
30#16839521C2212yes—
31C61×#559—21no—
32C61×#559—21no—
33C61×#550—21no—
34C61×#550—21no—
35C61×#546—22no—
36C61×#546—22no—
37C61×#540—22no—
38C61×#535C2212yes—
39C61×#533C2211yescharacteristic
40C61×#532—21no—
41C61×#532—21no—
42C2×#127×C61C2211yescharacteristic
43C61×#518C2212yes—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46C2×#11758194C2312yes—
47C2×#11758194C2312yes—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519764—48no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×C22⋊Dic61—28no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61C2312yes—
59#14519710—22no—
60#14519710—24no—
61#14519708—22no—
62#14519708—24no—
63C4⋊#92418—21no—
64C4⋊#92418—21no—
65#14519704—24no—
66#14519701—22no—
67#14519698—22no—
68#14519694—28no—
69#14519688—21no—
70C4×#92418—21no—
71C2×#41×C61—22no—
72C2×#41×C61—22no—
73C2×#41×C61C2311yescharacteristic
74C22×D4×C61D412yes—
75C22×D4×C61D412yes—
76C22×D4×C61C2312yes—
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61D411yescharacteristic
79#127×C61—22no—
80#127×C61—21no—
81#127×C61—21no—
82#127×C61—21no—
83#127×C61—21no—
84#125×C61D412yes—
85#124×C61—22no—
86#124×C61—22no—
87#124×C61—22no—
88#123×C61—24no—
89#123×C61—24no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92#121×C61—22no—
93#121×C61—22no—
94#121×C61—24no—
95#121×C61—24no—
96#121×C61—22no—
97#121×C61—22no—
98#120×C61—24no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101C4×Q8×C61D411yescharacteristic
102C4×Q8×C61D411yescharacteristic
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61D411yescharacteristic
112C4×D4×C61D411yescharacteristic
113#117×C61—21no—
114#117×C61C2312yes—
115#117×C61C2311yescharacteristic
116C2×C22⋊C4×C61—24no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C24×C61C2×D414yes—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—84no—
123#41×C61—84no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61C2×D411yescharacteristic
127C2×D4×C61—21no—
128C2×D4×C61—24no—
129C2×D4×C61—24no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—24no—
137C2×D4×C61—24no—
138C2×D4×C61—44no—
139C2×D4×C61—44no—
140C2×D4×C61C2×D411yescharacteristic
141C22×C4×C61—24no—
142C22×C4×C61—24no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—22no—
150C22×Dic61—22no—
151C2×Dic122—44no—
152C2×Dic122—41no—
153C2×Dic122—41no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D411yescharacteristic
161C4⋊C4×C61C2×D411yescharacteristic
162C4⋊C4×C61C2×D412yes—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—22no—
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C22⋊Dic61—416no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—41no—
184C4⋊Dic61—41no—
185C4×Dic61—24no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—42no—
189#11758194—22no—
190#11758194—24no—
191#11758194#4114yescomplemented
192C2×#92418—22no—
193C2×#92418—22no—
194C23×C61—24no—
195C23×C61—21no—
196C23×C61—28no—
197C23×C61—44no—
198C23×C61—44no—
199C23×C61C22×D411yescharacteristic
200Q8×C61—21no—
201Q8×C61—21no—
202Q8×C61—42no—
203Q8×C61—42no—
204D4×C61—21no—
205D4×C61—21no—
206D4×C61—42no—
207D4×C61—42no—
208D4×C61—42no—
209D4×C61—44no—
210D4×C61—44no—
211D4×C61—44no—
212D4×C61—82no—
213D4×C61—82no—
214D4×C61—84no—
215D4×C61—84no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—84no—
227C2×C4×C61—84no—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4112yes—
235C2×C4×C612−1+412yes—
236C2×Dic61—24no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—44no—
240C2×Dic61—42no—
241Dic122—82no—
242Dic122—82no—
243#92418—24no—
244#92418—42no—
245#25687—611nomaximal, Sylow
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—41no—
249C22×C61—44no—
250C22×C61—41no—
251C22×C61—44no—
252C22×C61—41no—
253C22×C61—44no—
254C22×C61—88no—
255C22×C61C2×#12011yescharacteristic
256C22×C61#54712yes—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—42no—
262C4×C61—41no—
263C4×C61—42no—
264C4×C61#45312yes—
265C4×C61#45512yes—
266C4×C61#53211yescharacteristic
267C4×C61#53611yescharacteristic
268C4×C61#54812yes—
269Dic61—44no—
270Dic61—82no—
271#3508D6111yescharacteristic
272#3165—611no—
273#3079—611no—
274#2858—611no—
275#2037—614no—
276#1994—611no—
277#1965—611no—
278#1856—614no—
279C4×#104—611no—
280C2×C61—22no—
281C2×C61—81no—
282C2×C61—84no—
283C2×C61#243211yescharacteristic
284C2×C61C2×#45311yescharacteristic
285C2×C61C2×#45511yescharacteristic
286#559—21no—
287#559—21no—
288#550—21no—
289#550—21no—
290#546—22no—
291#546—22no—
292#540—22no—
293#535C2×D6112yes—
294#533C2×D6111yescharacteristic
295#532—21no—
296#532—21no—
297C2×C4⋊Q8—611no—
298C2×#127C2×D6111yescharacteristic
299C2×#124—1222no—
300#518C2×D6112yes—
301C4×#41C2×D6111yescharacteristic
302#501—1222no—
303#500—1221no—
304#498—1221no—
305#477—1221no—
306#476—1221no—
307#475—1221no—
308#474—1221no—
309#428—612no—
310#428—614no—
311#423—611no—
312C2×#104—612no—
313#417—612no—
314C2×#100—614no—
315C4×M16—611no—
316#390—614no—
317#390—1222no—
318C43—611no—
319#339—612no—
320#339—614no—
321C61#2568711yescharacteristic, complemented, Sylow
322C2×#41—22no—
323C2×#41—22no—
324C2×#41C22×D6111yescharacteristic
325C22×Q8—1221no—
326C22×D4C22⋊D6112yes—
327C22×D4C22⋊D6112yes—
328C22×D4C22×D6112yes—
329C2×M16—612no—
330C2×M16—612no—
331C4⋊Q8—1221no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8C22⋊D6111yescharacteristic
336C4⋊Q8C22⋊D6111yescharacteristic
337#127—22no—
338#127—21no—
339#127—21no—
340#127—21no—
341#127—21no—
342#125C22⋊D6112yes—
343#124—22no—
344#124—22no—
345#124—22no—
346#124—2448no—
347#123—24no—
348#123—24no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351#121—22no—
352#121—22no—
353#121—24no—
354#121—24no—
355#121—22no—
356#121—22no—
357#120—24no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8C22⋊D6111yescharacteristic
361C4×Q8C22⋊D6111yescharacteristic
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4C22⋊D6111yescharacteristic
371C4×D4C22⋊D6111yescharacteristic
372#117—21no—
373#117C22×D6111yescharacteristic
374#117C22×D6112yes—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1221no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—1228no—
379C2×C42—21no—
380C2×C42—612no—
381C2×C42—1221no—
382C2×C42—1221no—
383C2×C42C22×D6111yescharacteristic
384#107—1222no—
385#107—1224no—
386#106—1222no—
387#106—1224no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390#104—1224no—
391#103—1222no—
392#102—1222no—
393#100—1228no—
394#97—1221no—
395C4×C8—1221no—
396C24C2×C22⋊D6114yes—
397#41—22no—
398#41—22no—
399#41—84no—
400#41—84no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—2444no—
404C2×Q8—2441no—
405C2×Q8—2441no—
406C2×Q8C2×C22⋊D6111yescharacteristic
407C2×D4—21no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4C2×C22⋊D6111yescharacteristic
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C23×D6111yescharacteristic
431M16—614no—
432M16—1222no—
433M16—1224no—
434C2×C8—1222no—
435C2×C8—1222no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—2441no—
445C4⋊C4—2441no—
446C4⋊C4C2×C22⋊D6111yescharacteristic
447C4⋊C4C2×C22⋊D6112yes—
448C4⋊C4C2×C22⋊D6111yescharacteristic
449C4⋊C4C2×C22⋊D6112yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—21no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—24416no—
459C42—21no—
460C42—22no—
461C42—21no—
462C42—21no—
463C42—1224no—
464C42—1222no—
465C42—1222no—
466C42—2442no—
467C42C2×C22⋊D6111yescharacteristic
468C42C2×C22⋊D6111yescharacteristic
469C42C2×C22⋊D6111yescharacteristic
470C42C2×C22⋊D6111yescharacteristic
471C23—24no—
472C23—21no—
473C23—28no—
474C23—44no—
475C23—44no—
476C23C22×C22⋊D6111yescharacteristic
477Q8—21no—
478Q8—21no—
479Q8—42no—
480Q8—42no—
481Q8—4882no—
482Q8—4882no—
483D4—21no—
484D4—21no—
485D4—42no—
486D4—44no—
487D4—42no—
488D4—42no—
489D4—44no—
490D4—44no—
491D4—82no—
492D4—82no—
493D4—84no—
494D4—84no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—1224no—
508C2×C4—2444no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4#1451974611yescharacteristic
513C2×C4#1451974611yescharacteristic
514C2×C4#1451974611yescharacteristic
515C2×C4#1451974611yescharacteristic
516C2×C4C22×C22⋊D6111yescharacteristic
517C2×C4C22×C22⋊D6111yescharacteristic
518C2×C4#41×D6112yes—
519C2×C4#1451982412yes—
520C8—1224no—
521C8—2442no—
522C22—21no—
523C22—21no—
524C22—41no—
525C22—44no—
526C22—41no—
527C22—44no—
528C22—41no—
529C22—44no—
530C22—88no—
531C22C2×#1451974611yescharacteristic
532C22#1684041112yes—
533C4—21no—
534C4—21no—
535C4—41no—
536C4—42no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—4882no—
541C4—2444no—
542C4#1683993012yes—
543C4#1683994012yes—
544C4#1684030811yescharacteristic
545C4#1684033311yescharacteristic
546C4#1684041912yes—
547C2—22no—
548C2—84no—
549C2—81no—
550C2#2144944311yescharacteristic
551C2C2×#1683993011yescharacteristic
552C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#25687
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21456366C2semidirect
2. #21455634C2semidirect
3. #21454175C2semidirect
4. #21447026C2semidirect
5. #21446797C2semidirect
6. #21446660C2semidirect
7. #21446029C2semidirect
8. C4×#14519704C2semidirect
191. #11758194#41semidirect
321. C61#25687semidirect

Maximal subgroups

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

subgroupindex
#214563662
#214556342
#214541752
#214470262
#214467972
#214466602
#214460292
C4×#145197042
C61×#35082
#2568761