Finite Group Databasebeta

← #29667894

Subgroups — #29667894

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456967C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3#21456379C211yesmaximal, characteristic, complemented
4C2×#16840134C211yesmaximal, characteristic, complemented
5#21456213C211yesmaximal, characteristic, complemented
6#21445469C212yesmaximal, complemented
7#21445455C212yesmaximal, complemented
8#21445419C212yesmaximal, complemented
9#21445410C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3138C211yesmaximal, characteristic, complemented
12C22×#11758200—22no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840165C2212yescomplemented
16#16840158C2211yescharacteristic, complemented
17#16840138C2212yes—
18#16840136C2211yescharacteristic, complemented
19#16840134—22no—
20#16840134C2212yes—
21#16840134C2212yes—
22#16840122—22no—
23#16840113C2212yescomplemented
24#16840112C2212yescomplemented
25#16840112C2211yescharacteristic, complemented
26#16840110C2211yescharacteristic, complemented
27#16840109C2212yescomplemented
28#16840108C2212yes—
29#16840095—22no—
30#16840057—22no—
31#16839993—22no—
32#16839954—22no—
33#16839896—22no—
34C4×#11758201—22no—
35#16839815C2211yescharacteristic
36#16839764—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#533C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#490—22no—
43C61×#486C2212yes—
44C61×#486C2212yes—
45C61×#418—22no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758200—24no—
50C2×#11758200—24no—
51C2×#11758200—48no—
52C2×#11758201—22no—
53C2×#11758199—24no—
54C22×#92418D412yes—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777C2312yes—
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68#14519710—24no—
69#14519708—22no—
70C4⋊#92418D412yes—
71#14519702—24no—
72#14519701—22no—
73#14519701—24no—
74#14519699—24no—
75#14519698—24no—
76#14519697—22no—
77C22⋊#92418D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C2×M16×C61D412yes—
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61—21no—
89#127×C61—21no—
90#127×C61—21no—
91#127×C61C2312yes—
92#125×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97#117×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#102×C61—24no—
100#102×C61—24no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103C24×C61—41no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—44no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—44no—
114C2×D4×C61—82no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C2×C22⋊D61—41no—
119C22×Dic61—41no—
120#11758222—44no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×C4×D61—41no—
125C2×Dic122—22no—
126C2×Dic122—21no—
127C2×Dic122—44no—
128C2×Dic122—41no—
129M16×C61—24no—
130C2×C8×C61C2×D414yes—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61—41no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61C2411yescharacteristic
143#11758201—44no—
144#11758200—48no—
145#11758200—48no—
146#11758199—84no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—22no—
151C4⋊Dic61—21no—
152C4⋊Dic61—22no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155C2×#92418—24no—
156C2×#92418C2×D412yes—
157C2×#92418C2×D414yes—
158C2×#92418C2×D412yes—
159C23×C61—42no—
160C23×C61—82no—
161C23×C61—82no—
162C23×C61—82no—
163C23×C61#12711yescharacteristic
164C22×D61—41no—
165Q8×C61—42no—
166Q8×C61—42no—
167D4×C61—44no—
168D4×C61—44no—
169D4×C61—44no—
170D4×C61—82no—
171D4×C61—82no—
172D4×C61—82no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61#12711yescharacteristic
178C2×C4×C61#12711yescharacteristic
179C2×C4×C61#12711yescharacteristic
180C2×C4×C61#13514yes—
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C22⋊D61—82no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—82no—
188D244—42no—
189D244—42no—
190D244—42no—
191C4×D61—82no—
192Dic122—42no—
193Dic122—44no—
194Dic122—82no—
195C8×C61—24no—
196#92418—22no—
197#92418—24no—
198#92418AGL(1,ℤ/8)14yes—
199#19625—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—82no—
203C22×C61—82no—
204C22×C61—82no—
205C22×C61—164no—
206C22×C61#49512yes—
207C22×C61C2×#12711yescharacteristic
208C2×D61—81no—
209C2×D61—81no—
210C2×D61—81no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61#49512yes—
216C4×C61C2×#13512yes—
217C4×C61C2×AGL(1,ℤ/8)12yes—
218C4×C61#57712yes—
219Dic61—81no—
220Dic61—82no—
221#3230—611no—
222#3205—611no—
223#3167—611no—
224C2×#492—611no—
225#3141—611no—
226#3138D6111yescharacteristic, complemented
227#1719—612no—
228#1717—612no—
229#1708—612no—
230#1706—612no—
231C8×M16—611no—
232C2×C61—22no—
233C2×C61—162no—
234C2×C61C2×#49511yescharacteristic
235C2×C61#315111yescharacteristic
236C2×C61#315411yescharacteristic
237D61—161no—
238C22×SD16—1222no—
239#533C2×D6111yescharacteristic
240#532—611no—
241C2×C4⋊Q8—611no—
242C2×#127C2×D6111yescharacteristic
243#500—612no—
244#498—611no—
245#493—612no—
246#492—611no—
247#492—612no—
248#492—612no—
249#492—1222no—
250#490—22no—
251#489—1222no—
252#487—611no—
253#487—612no—
254#487—612no—
255#486—611no—
256#486—612no—
257#486—612no—
258#486C2×D6112yes—
259#486C2×D6112yes—
260#484—1222no—
261#477—1222no—
262#465—1222no—
263#459—1222no—
264#447—1222no—
265C4×SD16—1222no—
266#431—611no—
267#418—22no—
268#417—1222no—
269C4×M16C2×D6111yescharacteristic
270C2×C4×C8—611no—
271#322—612no—
272C82—612no—
273C61#1962511yescharacteristic, complemented, Sylow
274C2×#41—1221no—
275C22×Q8—1221no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4—21no—
279C2×SD16—1224no—
280C2×SD16—1224no—
281C2×SD16—1222no—
282C2×SD16—2448no—
283C2×D8—1224no—
284C2×M16C22⋊D6112yes—
285C22×C8—612no—
286C4⋊Q8—612no—
287C4⋊Q8—611no—
288C4⋊Q8—1221no—
289C4⋊Q8—1221no—
290C4⋊Q8—1221no—
291C4⋊Q8C22×D6111yescharacteristic
292C4⋊Q8C22×D6111yescharacteristic
293#127—21no—
294#127—21no—
295#127—21no—
296#127—611no—
297#127C22×D6112yes—
298#125—21no—
299#124—1221no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302#121—1221no—
303#121—1221no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×D4—1221no—
307C4×D4—1221no—
308#117—21no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C42C22×D6111yescharacteristic
312#107—1224no—
313#106—1222no—
314C4⋊C8—612no—
315#103—1222no—
316#103—1224no—
317#103—1224no—
318#102—24no—
319#102—24no—
320#102—1222no—
321#102—1224no—
322#102—1224no—
323C22⋊C8—612no—
324#97C22⋊D6112yes—
325C4×C8—612no—
326C4×C8—611no—
327C4×C8—612no—
328C4×C8—611no—
329C4×C8C22×D6112yes—
330C24—41no—
331#41—2444no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—1222no—
335C2×Q8—1221no—
336C2×Q8—2444no—
337C2×Q8—2441no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—44no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—44no—
346C2×D4—82no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—2441no—
351C22×C4—2441no—
352C22×C4—2441no—
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356SD16—2448no—
357SD16—2444no—
358SD16—2448no—
359D8—4884no—
360M16—24no—
361C2×C8—612no—
362C2×C8—612no—
363C2×C8—614no—
364C2×C8—1224no—
365C2×C8C2×C22⋊D6114yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1222no—
375C4⋊C4—1222no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C22⋊C4—41no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—41no—
382C42D4×D6111yescharacteristic
383C42D4×D6111yescharacteristic
384C42C2×C22⋊D6111yescharacteristic
385C42C23×D6111yescharacteristic
386C23—42no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—2441no—
391C23#1451977311yescharacteristic
392Q8—42no—
393Q8—42no—
394Q8—4882no—
395Q8—2442no—
396Q8—2444no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—4882no—
404D4—2442no—
405D4—2442no—
406D4—2442no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—4882no—
412C2×C4—4882no—
413C2×C4—2441no—
414C2×C4—2442no—
415C2×C4#1451977311yescharacteristic
416C2×C4#1451977311yescharacteristic
417C2×C4#1451977311yescharacteristic
418C2×C4#1451979714yes—
419C2×C4C2×D4×D6111yescharacteristic
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4C22×C22⋊D6111yescharacteristic
422C8—24no—
423C8—614no—
424C8—1222no—
425C8—1224no—
426C22—21no—
427C22—21no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—164no—
432C22—4881no—
433C22—4881no—
434C22—4881no—
435C22#1684014412yes—
436C22C2×#1451977311yescharacteristic
437C4—21no—
438C4—21no—
439C4—81no—
440C4—81no—
441C4—4881no—
442C4—4882no—
443C4#1684014412yes—
444C4C2×#1451979712yes—
445C4AGL(1,ℤ/8)×D6112yes—
446C4#1684057212yes—
447C2—22no—
448C2—162no—
449C2—9761no—
450C2C2×#1684014411yescharacteristic
451C2#2145626711yescharacteristic
452C2#2145629311yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3138C2
socleC22×C61C2×#127
prime radical#21456737C2
hypercenter#3138D61
hypocenterC61#19625
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
3. #21456379C2semidirect
4. C2×#16840134C2semidirect
5. #21456213C2semidirect
6. #21445469C2semidirect
7. #21445455C2semidirect
8. #21445419C2semidirect
9. #21445410C2semidirect
10. M16×#92418C2semidirect
11. C61×#3138C2semidirect
15. #16840165C22semidirect
16. #16840158C22semidirect
18. #16840136C22semidirect
23. #16840113C22semidirect
24. #16840112C22semidirect
25. #16840112C22semidirect
26. #16840110C22semidirect
27. #16840109C22semidirect
38. #16839483C22semidirect
39. C8×#92418C22semidirect
226. #3138D61semidirect
273. C61#19625semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214567372
#214563792
C2×#168401342
#214562132
#214454692
#214454552
#214454192
#214454102
M16×#924182
C61×#31382
#1962561