Finite Group Databasebeta

← #27151354

Subgroups — #27151354

Order 12416.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690834C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3#20261245C211yesmaximal, characteristic, complemented
4#121×D97C211yesmaximal, characteristic, complemented
5#20260986C211yesmaximal, characteristic, complemented
6#20260981C211yesmaximal, characteristic, complemented
7#20260979C211yesmaximal, characteristic, complemented
8#20260970C211yesmaximal, characteristic, complemented
9#20260968C211yesmaximal, characteristic, complemented
10#20260965C211yesmaximal, characteristic, complemented
11#20260963C211yesmaximal, characteristic, complemented
12#20260774C211yesmaximal, characteristic, complemented
13#20260738C211yesmaximal, characteristic, complemented
14#20260733C211yesmaximal, characteristic, complemented
15#469×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97—21no—
18C2×D4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690834—24no—
21C2×#13095500C2211yescharacteristic, complemented
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095496C2211yescharacteristic, complemented
24C2×#13095495C2211yescharacteristic, complemented
25C2×#13095494C2211yescharacteristic
26C2×#13095494C2211yescharacteristic
27M16×D97—24no—
28C2×C8×D97C2211yescharacteristic, complemented
29C2×C8×D97C2211yescharacteristic, complemented
30#15690792—21no—
31#15690790C2211yescharacteristic, complemented
32#15690789C2211yescharacteristic, complemented
33#15690785C2211yescharacteristic
34#15690784C2211yescharacteristic
35#15690783—21no—
36#15690782—21no—
37#15690781C2211yescharacteristic
38#15690781C2211yescharacteristic, complemented
39#15690778—21no—
40D4×Dic97—21no—
41#15690770—21no—
42#15690768C2211yescharacteristic
43C4⋊C4×D97C2211yescharacteristic
44C22⋊C4×D97—21no—
45#15690744C2211yescharacteristic, complemented
46#15690743C2211yescharacteristic, complemented
47#15690741C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690733C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690731C2211yescharacteristic, complemented
52#15690731C2211yescharacteristic, complemented
53#15690730C2211yescharacteristic, complemented
54C2×D8×C97C2211yescharacteristic, complemented
55C2×M16×C97C2211yescharacteristic, complemented
56#121×C97C2211yescharacteristic
57#121×C97C2211yescharacteristic, complemented
58#106×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97D411yescharacteristic
63#13095510—22no—
64#13095510—22no—
65C2×Dic194C2311yescharacteristic
66C2×D388—21no—
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73D4×D97—22no—
74D4×D97—22no—
75D4×D97—44no—
76C22×Dic97—21no—
77C22×Dic97D411yescharacteristic
78C2×C4×D97—22no—
79C2×C4×D97—22no—
80C2×C4×D97—21no—
81C2×C4×D97D411yescharacteristic
82C2×C4×D97D411yescharacteristic
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2311yescharacteristic
85#13095500—22no—
86#13095499—22no—
87#13095496—22no—
88#13095495—22no—
89#13095494—22no—
90#13095494D414yescomplemented
91C2×#11715895C2311yescharacteristic
92C2×#11715895C2311yescharacteristic
93C8×D97—22no—
94C8×D97—22no—
95C4⋊Dic97—21no—
96C4⋊Dic97C2311yescharacteristic
97C4⋊Dic97C2311yescharacteristic
98#13095490—21no—
99#13095490—21no—
100C22⋊Dic97—21no—
101C22⋊Dic97—21no—
102C22⋊Dic97—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107C4×Dic97—21no—
108C2×D4×C97—21no—
109C2×D4×C97—21no—
110C2×D4×C97C2311yescharacteristic
111C2×D4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113D8×C97—22no—
114M16×C97—22no—
115C2×C8×C97C2311yescharacteristic
116C2×C8×C97C2311yescharacteristic
117C4⋊C4×C97C2311yescharacteristic
118C4⋊C4×C97C2311yescharacteristic
119C22⋊C4×C97—21no—
120C22⋊C4×C97—21no—
121C23×C97—21no—
122C23×C97—21no—
123C23×C97C2×D411yescharacteristic
124C22×D97—22no—
125C22×D97—22no—
126C22×D97—21no—
127C22×D97—41no—
128C22×D97—42no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97C2×D411yescharacteristic
132C22×D97C2×D411yescharacteristic
133D4×C97—21no—
134D4×C97—21no—
135D4×C97—21no—
136D4×C97—21no—
137D4×C97—42no—
138D4×C97—42no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97C2×D411yescharacteristic
144C2×C4×C97C2411yescharacteristic
145C22⋊D97—42no—
146C22⋊D97—42no—
147C22⋊D97—42no—
148C22⋊D97—42no—
149C22⋊D97—42no—
150C2×Dic97—22no—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—42no—
155C2×Dic97C2×D411yescharacteristic
156C2×Dic97C2×D411yescharacteristic
157D388—21no—
158D388—21no—
159D388—42no—
160C4×D97—22no—
161C4×D97—21no—
162C4×D97—21no—
163C4×D97—22no—
164C4×D97—42no—
165C4×D97C2×D412yes—
166C4×D97#4112yes—
167Dic194—21no—
168Dic194—21no—
169C8×C97—21no—
170C8×C97C2×D412yescomplemented
171#11715895—21no—
172#11715895C2×D412yescomplemented
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97C22×D411yescharacteristic
183C2×D97—21no—
184C2×D97—21no—
185C2×D97—21no—
186C2×D97—21no—
187C2×D97—44no—
188C2×D97—41no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—84no—
192C2×D97#12112yes—
193C2×D97AGL(1,ℤ/8)12yescomplemented
194C4×C97—21no—
195C4×C97—41no—
196C4×C97—41no—
197C4×C97C22×D411yescharacteristic
198C4×C97C2×#4111yescharacteristic
199Dic97—21no—
200Dic97—41no—
201Dic97—41no—
202Dic97—41no—
203Dic97#12112yes—
204C2×C97—41no—
205C2×C97—81no—
206C2×C97—81no—
207C2×C97C2×#12111yescharacteristic
208C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C97#57511yescharacteristic
210D97—22no—
211D97—22no—
212D97—81no—
213#3055—971nomaximal, Sylow
214C97#305511yescharacteristic, complemented, Sylow
215C2×AGL(1,ℤ/8)—971no—
216C22×M16—971no—
217#525—971no—
218C2×#121—971no—
219#469—971no—
220#469D9711yescharacteristic, complemented
221#468—971no—
222#468—971no—
223#466—971no—
224#466—971no—
225#465—971no—
226#465—971no—
227#428—971no—
228#418—971no—
229#417—971no—
230C2×#41—971no—
231C22×D4—971no—
232C22×D4—1941no—
233C23×C4—971no—
234AGL(1,ℤ/8)—1944no—
235C2×SD16—971no—
236C2×SD16—971no—
237C2×D8—971no—
238C2×D8C2×D9711yescharacteristic, complemented
239C2×M16—971no—
240C2×M16—971no—
241C2×M16—971no—
242C2×M16—1944no—
243C2×M16C2×D9711yescharacteristic, complemented
244C22×C8—971no—
245C22×C8—971no—
246#123—1941no—
247C22⋊Q8—971no—
248C22⋊Q8—971no—
249#121—971no—
250#121—971no—
251#121—971no—
252#121—971no—
253#121—1941no—
254#121—1941no—
255#121C2×D9711yescharacteristic
256#121C2×D9711yescharacteristic, complemented
257#120—1941no—
258C4×D4—1941no—
259C4×D4—1941no—
260#117—971no—
261C2×C4⋊C4—971no—
262C2×C22⋊C4—1941no—
263#107—971no—
264#107—971no—
265#106—971no—
266#106C2×D9711yescharacteristic, complemented
267#103—971no—
268#103—971no—
269#102—971no—
270#102—971no—
271#102—971no—
272#102—971no—
273#102C2×D9711yescharacteristic, complemented
274#102C2×D9711yescharacteristic, complemented
275C24—971no—
276C24—1941no—
277#41—1942no—
278#41—1942no—
279C2×Q8—971no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—971no—
283C2×D4—3884no—
284C2×D4—1941no—
285C2×D4—1942no—
286C2×D4—1942no—
287C2×D4—1941no—
288C2×D4—1941no—
289C2×D4—1941no—
290C2×D4—1941no—
291C2×D4—1941no—
292C2×D4C22×D9711yescharacteristic
293C2×D4C22×D9711yescharacteristic
294C22×C4—971no—
295C22×C4—971no—
296C22×C4—971no—
297C22×C4—971no—
298C22×C4—971no—
299C22×C4—1942no—
300C22×C4—1942no—
301C22×C4—1941no—
302C22×C4—1941no—
303C22×C4C22×D9711yescharacteristic
304SD16—1942no—
305SD16—1942no—
306D8—22no—
307D8—1942no—
308M16—22no—
309M16—974no—
310M16—1942no—
311M16—1942no—
312C2×C8—971no—
313C2×C8—971no—
314C2×C8—1942no—
315C2×C8—1942no—
316C2×C8C22×D9711yescharacteristic
317C2×C8C22×D9711yescharacteristic
318C4⋊C4—971no—
319C4⋊C4—971no—
320C4⋊C4—1941no—
321C4⋊C4—1941no—
322C4⋊C4—1941no—
323C4⋊C4C22×D9711yescharacteristic
324C4⋊C4C22×D9711yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C22⋊C4—1941no—
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C42—1941no—
335C23—21no—
336C23—21no—
337C23—971no—
338C23—971no—
339C23—3881no—
340C23—3882no—
341C23—3882no—
342C23—3882no—
343C23—1942no—
344C23—1942no—
345C23—1941no—
346C23D4×D9711yescharacteristic
347Q8—1941no—
348Q8—1941no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—42no—
354D4—42no—
355D4—3882no—
356D4—3882no—
357D4—3882no—
358D4—3882no—
359D4—3882no—
360D4—3882no—
361D4—1941no—
362D4—1941no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—971no—
368C2×C4—972no—
369C2×C4—972no—
370C2×C4—971no—
371C2×C4—3882no—
372C2×C4—3882no—
373C2×C4—1942no—
374C2×C4—1942no—
375C2×C4—1942no—
376C2×C4—1941no—
377C2×C4—1941no—
378C2×C4—1941no—
379C2×C4—1941no—
380C2×C4—1941no—
381C2×C4D4×D9711yescharacteristic
382C2×C4C23×D9711yescharacteristic
383C8—21no—
384C8—972no—
385C8—1941no—
386C8D4×D9712yescomplemented
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—7764no—
397C22—972no—
398C22—972no—
399C22—3884no—
400C22—3881no—
401C22—3881no—
402C22—3881no—
403C22—1941no—
404C22—1941no—
405C22—1941no—
406C22—1941no—
407C22C2×D4×D9711yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—972no—
412C4—3881no—
413C4—3881no—
414C4—3881no—
415C4—1941no—
416C4C2×D4×D9711yescharacteristic
417C4#41×D9711yescharacteristic
418C2—41no—
419C2—81no—
420C2—81no—
421C2—7761no—
422C2—1942no—
423C2—1942no—
424C2#121×D9711yescharacteristic
425C2AGL(1,ℤ/8)×D9711yescharacteristic
426C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#469×C97C2
socleC22×C97C22×D4
hypercenter#469D97
hypocenterC97#3055
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. C2×M16×D97C2semidirect
3. #20261245C2semidirect
4. #121×D97C2semidirect
5. #20260986C2semidirect
6. #20260981C2semidirect
7. #20260979C2semidirect
8. #20260970C2semidirect
9. #20260968C2semidirect
10. #20260965C2semidirect
11. #20260963C2semidirect
12. #20260774C2semidirect
13. #20260738C2semidirect
14. #20260733C2semidirect
15. #469×C97C2semidirect
21. C2×#13095500C22semidirect
22. C2×#13095499C22semidirect
23. C2×#13095496C22semidirect
24. C2×#13095495C22semidirect
28. C2×C8×D97C22semidirect
29. C2×C8×D97C22semidirect
31. #15690790C22semidirect
32. #15690789C22semidirect
38. #15690781C22semidirect
45. #15690744C22semidirect
46. #15690743C22semidirect
47. #15690741C22semidirect
48. #15690735C22semidirect
49. #15690733C22semidirect
50. #15690732C22semidirect
51. #15690731C22semidirect
52. #15690731C22semidirect
53. #15690730C22semidirect
54. C2×D8×C97C22semidirect
55. C2×M16×C97C22semidirect
57. #121×C97C22semidirect
58. #106×C97C22semidirect
59. #102×C97C22semidirect
60. #102×C97C22semidirect
90. #13095494D4semidirect
170. C8×C97C2×D4semidirect
172. #11715895C2×D4semidirect
193. C2×D97AGL(1,ℤ/8)semidirect
214. C97#3055semidirect
220. #469D97semidirect
238. C2×D8C2×D97semidirect
243. C2×M16C2×D97semidirect
256. #121C2×D97semidirect
266. #106C2×D97semidirect
273. #102C2×D97semidirect
274. #102C2×D97semidirect
386. C8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
C2×M16×D972
#202612452
#121×D972
#202609862
#202609812
#202609792
#202609702
#202609682
#202609652
#202609632
#202607742
#202607382
#202607332
#469×C972
#305597