Finite Group Databasebeta

← #29733697

Subgroups — #29733697

Order 15616.

1116 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#21456279C211yesmaximal, characteristic, complemented
2#21455306C211yesmaximal, characteristic, complemented
3#21454196C211yesmaximal, characteristic, complemented
4#21446866C214yesmaximal, complemented
5#21446809C211yesmaximal, characteristic, complemented
6#21446648C211yesmaximal, characteristic, complemented
7#21446188C214yesmaximal, complemented
8C4×#14519704C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic
10C2×#14519807C2214yes—
11C2×#14519807C2212yes—
12C2×#14519777C2211yescharacteristic
13#16840157—22no—
14#16840141—21no—
15#16840134—21no—
16#16839972—21no—
17#16839972—21no—
18#16839967—21no—
19#16839967—21no—
20#16839786C2211yescharacteristic
21C2×#14519704C2212yes—
22C2×#14519704C2214yes—
23#16839770C2212yes—
24C4×#11758194C2211yescharacteristic
25#16839669—22no—
26#16839667C2214yes—
27#16839627—22no—
28C42×Dic61C2211yescharacteristic
29#16839531C2214yes—
30#16839521C2212yes—
31C61×#556—21no—
32C61×#555—21no—
33C61×#554—21no—
34C61×#553—22no—
35C61×#548—22no—
36C61×#547—21no—
37C61×#542—22no—
38C61×#537C2212yes—
39C61×#533—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520C2212yes—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122C2312yes—
46#14519807—28no—
47#14519807—44no—
48#14519807—44no—
49C2×#11758202—22no—
50C2×#11758202—24no—
51C2×#11758200—22no—
52C2×#11758200—24no—
53C2×#11758194C2312yes—
54C2×#11758194C2312yes—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61C2312yes—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66#14519704—24no—
67#14519704—44no—
68#14519700—22no—
69#14519700—22no—
70#14519688—21no—
71C4×#92418—21no—
72#14519684—28no—
73C2×#41×C61—22no—
74C2×#41×C61C2312yes—
75C2×#41×C61C2311yescharacteristic
76C22×Q8×C61D412yes—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61D411yescharacteristic
82#127×C61D411yescharacteristic
83#126×C61—24no—
84#125×C61—22no—
85#124×C61—22no—
86#124×C61—24no—
87#124×C61D412yes—
88#123×C61—24no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—22no—
92C22⋊Q8×C61—22no—
93#121×C61—22no—
94#121×C61—22no—
95#121×C61—22no—
96#121×C61—21no—
97#121×C61—21no—
98#121×C61—24no—
99C4×Q8×C61—22no—
100C4×Q8×C61—22no—
101C4×Q8×C61D411yescharacteristic
102C4×Q8×C61D411yescharacteristic
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—21no—
106C4×D4×C61—24no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C4×D4×C61D411yescharacteristic
111C4×D4×C61D411yescharacteristic
112#117×C61—22no—
113#117×C61—22no—
114#117×C61—21no—
115#117×C61C2311yescharacteristic
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61D412yes—
118C2×C4⋊C4×C61C2312yes—
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#41×C61—22no—
122#41×C61—24no—
123#41×C61—24no—
124#41×C61—22no—
125#41×C61—84no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—44no—
129C2×Q8×C61C2×D412yes—
130C2×Q8×C61C2×D411yescharacteristic
131C2×D4×C61—21no—
132C2×D4×C61—22no—
133C2×D4×C61—24no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C2×D412yes—
137C2×D4×C61C2×D411yescharacteristic
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D414yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—22no—
148C22×Dic61—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—24no—
152C2×Dic122—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C2×D411yescharacteristic
164C4⋊C4×C61C2×D411yescharacteristic
165C22⋊C4×C61—24no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C42×C61—24no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181#11758202—48no—
182#11758202—44no—
183#11758200—48no—
184#11758200—44no—
185C4⋊Dic61—41no—
186C4⋊Dic61—41no—
187C4⋊Dic61—41no—
188C4⋊Dic61—41no—
189C4×Dic61—24no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—42no—
193#11758194—22no—
194#11758194—24no—
195#11758194#4114yescomplemented
196C2×#92418—22no—
197C2×#92418—22no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61C22×D411yescharacteristic
201Q8×C61—21no—
202Q8×C61—21no—
203Q8×C61—24no—
204Q8×C61—44no—
205Q8×C61—82no—
206D4×C61—21no—
207D4×C61—21no—
208D4×C61—24no—
209D4×C61—42no—
210D4×C61—44no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—84no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—44no—
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×C61H(5,𝔽2)12yes—
236C2×Dic61—24no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—44no—
241Dic122—42no—
242Dic122—42no—
243Dic122—44no—
244#92418—24no—
245#92418—42no—
246#25615—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—42no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—42no—
253C22×C61—41no—
254C22×C61—42no—
255C22×C61C2×#12011yescharacteristic
256C22×C61#55012yes—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—42no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—44no—
263C4×C61—41no—
264C4×C61#45512yes—
265C4×C61#45612yes—
266C4×C61#53211yescharacteristic
267C4×C61#53511yescharacteristic
268C4×C61#54612yes—
269Dic61—44no—
270Dic61—82no—
271#3511D6111yescharacteristic
272#3152—611no—
273#3039—611no—
274#2862—611no—
275#2007—614no—
276#1997—611no—
277#1962—611no—
278#1882—614no—
279C4×#104—611no—
280C2×C61—22no—
281C2×C61—81no—
282C2×C61—82no—
283C2×C61#243111yescharacteristic
284C2×C61C2×#45511yescharacteristic
285C2×C61C2×#45611yescharacteristic
286C2×#137—614no—
287C2×#137—612no—
288#556—21no—
289#555—21no—
290#554—21no—
291#553—22no—
292#548—22no—
293#547—21no—
294#542—22no—
295#537C2×D6112yes—
296#533—21no—
297#532—21no—
298#532C2×D6111yescharacteristic
299C2×C4⋊Q8—611no—
300C2×C4⋊Q8C2×D6111yescharacteristic
301#520C2×D6112yes—
302C4×#41C2×D6111yescharacteristic
303#497—1222no—
304#494—1221no—
305#492—1221no—
306#462—1221no—
307#462—1221no—
308#461—1221no—
309#461—1221no—
310#423—611no—
311C2×#104—612no—
312C2×#104—614no—
313#419—612no—
314C4×M16—611no—
315#391—614no—
316#391—1222no—
317#382—1222no—
318C43—611no—
319#343—614no—
320#339—612no—
321C61#2561511yescharacteristic, complemented, Sylow
322C2×#41—22no—
323C2×#41C22×D6111yescharacteristic
324C2×#41C22×D6112yes—
325C22×Q8—612no—
326C22×Q8C22⋊D6112yes—
327#137—1228no—
328#137—2444no—
329#137—2444no—
330C2×Q16—1222no—
331C2×Q16—1224no—
332C2×SD16—1222no—
333C2×SD16—1224no—
334C2×M16—612no—
335C2×M16—612no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8—1221no—
341C4⋊Q8—1221no—
342C4⋊Q8—1221no—
343C4⋊Q8—1221no—
344C4⋊Q8C22⋊D6111yescharacteristic
345#127C22⋊D6111yescharacteristic
346#126—24no—
347#125—22no—
348#124—22no—
349#124—24no—
350#124C22⋊D6112yes—
351#123—24no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356#121—22no—
357#121—22no—
358#121—22no—
359#121—21no—
360#121—21no—
361#121—24no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8C22⋊D6111yescharacteristic
365C4×Q8C22⋊D6111yescharacteristic
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—24no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4C22⋊D6111yescharacteristic
374C4×D4C22⋊D6111yescharacteristic
375#117—22no—
376#117—22no—
377#117—21no—
378#117C22×D6111yescharacteristic
379C2×C4⋊C4—22no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C22⋊D6112yes—
383C2×C4⋊C4C22×D6112yes—
384C2×C42—21no—
385C2×C42—612no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391#104—1224no—
392#104—2444no—
393#103—1222no—
394#103—1222no—
395#97—1221no—
396C4×C8—1221no—
397#95—1228no—
398#41—22no—
399#41—24no—
400#41—24no—
401#41—22no—
402#41—84no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1224no—
409C2×Q8—1224no—
410C2×Q8C2×C22⋊D6112yes—
411C2×Q8C2×C22⋊D6111yescharacteristic
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4C2×C22⋊D6112yes—
418C2×D4C2×C22⋊D6111yescharacteristic
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4C2×C22⋊D6111yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6114yes—
429C22×C4C23×D6111yescharacteristic
430Q16—2448no—
431Q16—2444no—
432SD16—2448no—
433SD16—2444no—
434M16—614no—
435M16—1222no—
436M16—1224no—
437C2×C8—1222no—
438C2×C8—1222no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—2441no—
450C4⋊C4—2441no—
451C4⋊C4—2441no—
452C4⋊C4—2441no—
453C4⋊C4C2×C22⋊D6111yescharacteristic
454C4⋊C4C2×C22⋊D6111yescharacteristic
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C42—24no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—1224no—
468C42—1222no—
469C42—1222no—
470C42—2442no—
471C42C2×C22⋊D6111yescharacteristic
472C42C2×C22⋊D6111yescharacteristic
473C42C2×C22⋊D6111yescharacteristic
474C42C2×C22⋊D6111yescharacteristic
475C23—21no—
476C23—22no—
477C23C22×C22⋊D6111yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—24no—
481Q8—44no—
482Q8—82no—
483Q8—2442no—
484Q8—2442no—
485Q8—2444no—
486D4—21no—
487D4—21no—
488D4—24no—
489D4—42no—
490D4—44no—
491D4—42no—
492D4—42no—
493D4—84no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—84no—
508C2×C4—1224no—
509C2×C4—2442no—
510C2×C4—2444no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4#1451974611yescharacteristic
514C2×C4#1451974611yescharacteristic
515C2×C4#1451974611yescharacteristic
516C2×C4#1451974611yescharacteristic
517C2×C4C22×C22⋊D6111yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4#41×D6112yes—
520C2×C4#1451982312yes—
521C8—1224no—
522C8—2442no—
523C22—21no—
524C22—21no—
525C22—41no—
526C22—42no—
527C22—41no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22C2×#1451974611yescharacteristic
532C22#1684043112yes—
533C4—21no—
534C4—21no—
535C4—41no—
536C4—44no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—4882no—
541C4—2444no—
542C4#1683994012yes—
543C4#1683994512yes—
544C4#1684030811yescharacteristic
545C4#1684032611yescharacteristic
546C4#1684040012yes—
547C2—22no—
548C2—81no—
549C2—82no—
550C2#2144943411yescharacteristic
551C2C2×#1683994011yescharacteristic
552C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#25615
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21456279C2semidirect
2. #21455306C2semidirect
3. #21454196C2semidirect
4. #21446866C2semidirect
5. #21446809C2semidirect
6. #21446648C2semidirect
7. #21446188C2semidirect
8. C4×#14519704C2semidirect
195. #11758194#41semidirect
321. C61#25615semidirect

Maximal subgroups

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

subgroupindex
#214562792
#214553062
#214541962
#214468662
#214468092
#214466482
#214461882
C4×#145197042
C61×#35112
#2561561