Finite Group Databasebeta

← #27151411

Subgroups — #27151411

Order 12416.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2C2×#15690818C211yesmaximal, characteristic, complemented
3#20261315C211yesmaximal, characteristic, complemented
4#20261301C211yesmaximal, characteristic, complemented
5#20260988C211yesmaximal, characteristic, complemented
6#20260987C211yesmaximal, characteristic, complemented
7#20260984C211yesmaximal, characteristic, complemented
8#20260972C211yesmaximal, characteristic, complemented
9#20260969C211yesmaximal, characteristic, complemented
10#20260966C211yesmaximal, characteristic, complemented
11#20260961C211yesmaximal, characteristic, complemented
12#20260776C211yesmaximal, characteristic, complemented
13#20260730C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#468×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095511C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic
19#15690837—24no—
20C2×#13095498C2211yescharacteristic
21C2×D776C2211yescharacteristic
22C2×#13095496C2211yescharacteristic
23#15690818—24no—
24C2×#13095494C2211yescharacteristic
25C2×#13095494C2211yescharacteristic, complemented
26C22×#11715895C2211yescharacteristic, complemented
27C2×C8×D97C2211yescharacteristic
28C2×C8×D97C2211yescharacteristic, complemented
29#15690806—21no—
30#15690796—21no—
31#15690788—21no—
32#15690786C2211yescharacteristic, complemented
33#15690785—21no—
34#15690785C2211yescharacteristic
35#15690785C2211yescharacteristic, complemented
36#15690784C2211yescharacteristic
37#15690784C2211yescharacteristic, complemented
38#15690783—21no—
39#15690783—21no—
40#15690781C2211yescharacteristic
41#15690780—21no—
42#15690778—21no—
43#15690768C2211yescharacteristic
44C4⋊C4×D97C2211yescharacteristic
45#15690743C2211yescharacteristic, complemented
46#15690742C2211yescharacteristic, complemented
47#15690741C2211yescharacteristic, complemented
48#15690734C2211yescharacteristic, complemented
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic, complemented
51#15690731C2211yescharacteristic
52#15690730C2211yescharacteristic
53#15690730C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic
55C2×M16×C97C2211yescharacteristic, complemented
56C22⋊Q8×C97C2211yescharacteristic, complemented
57#121×C97C2211yescharacteristic
58#107×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic
61C23×D97—21no—
62#13095512—22no—
63#13095512—22no—
64#13095511—22no—
65#13095511—22no—
66C2×Dic194D411yescharacteristic
67C2×D388—21no—
68C2×D388—21no—
69C2×D388—21no—
70C2×D388D411yescharacteristic
71C2×D388C2311yescharacteristic
72C2×D388C2311yescharacteristic
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76C2×C22⋊D97—21no—
77C2×C22⋊D97D411yescharacteristic
78C2×C22⋊D97D411yescharacteristic
79D4×D97—22no—
80D4×D97—22no—
81C2×C4×D97—21no—
82C2×C4×D97—21no—
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2311yescharacteristic
85#13095498—22no—
86D776—22no—
87#13095496—22no—
88#13095494—22no—
89#13095494D414yescomplemented
90C2×#11715895—22no—
91C2×#11715895C2311yescharacteristic
92C2×#11715895C2311yescharacteristic
93C8×D97—22no—
94C8×D97—22no—
95C4⋊Dic97C2311yescharacteristic
96C4⋊Dic97C2311yescharacteristic
97#13095490—21no—
98#13095490—21no—
99C22⋊Dic97—21no—
100#13095488—21no—
101#13095488—21no—
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106C4×Dic97—21no—
107C2×Q8×C97C2311yescharacteristic
108C2×D4×C97—21no—
109C2×D4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111SD16×C97—22no—
112M16×C97—22no—
113C2×C8×C97C2311yescharacteristic
114C2×C8×C97C2311yescharacteristic
115C4⋊C4×C97—21no—
116C4⋊C4×C97C2311yescharacteristic
117C4⋊C4×C97C2311yescharacteristic
118C22⋊C4×C97—21no—
119C22⋊C4×C97—21no—
120C23×C97—21no—
121C23×C97C2×D411yescharacteristic
122C22×D97—21no—
123C22×D97—21no—
124C22×D97—42no—
125C22×D97—81no—
126C22×D97—81no—
127C22×D97C2×D411yescharacteristic
128C22×D97C2×D411yescharacteristic
129Q8×C97—21no—
130Q8×C97—21no—
131D4×C97—21no—
132D4×C97—21no—
133D4×C97—81no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2411yescharacteristic
141C22⋊D97—22no—
142C22⋊D97—22no—
143C22⋊D97—42no—
144C22⋊D97—81no—
145C22⋊D97—81no—
146C22⋊D97—81no—
147C2×Dic97—21no—
148C2×Dic97—21no—
149C2×Dic97C2×D411yescharacteristic
150C2×Dic97C2×D411yescharacteristic
151D388—22no—
152D388—21no—
153D388—21no—
154D388—21no—
155D388—21no—
156D388—42no—
157D388—81no—
158D388—81no—
159C4×D97—21no—
160C4×D97—21no—
161C4×D97—42no—
162C4×D97—81no—
163C4×D97C2×D412yes—
164C4×D97#4112yes—
165Dic194—22no—
166C8×C97—21no—
167C8×C97C2×D412yes—
168#11715895—21no—
169#11715895C2×D412yes—
170C22×C97—21no—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97—41no—
174C22×C97—41no—
175C22×C97—41no—
176C22×C97C22×D411yescharacteristic
177C2×D97—21no—
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—21no—
181C2×D97—21no—
182C2×D97—41no—
183C2×D97—41no—
184C2×D97—41no—
185C2×D97—41no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—84no—
189C2×D97#12112yes—
190C4×C97—21no—
191C4×C97—41no—
192C4×C97—41no—
193C4×C97—41no—
194C4×C97C22×D411yescharacteristic
195C4×C97C2×#4111yescharacteristic
196Dic97—21no—
197Dic97—41no—
198Dic97—41no—
199Dic97#12112yes—
200C2×C97—41no—
201C2×C97—81no—
202C2×C97C2×#12111yescharacteristic
203C2×C97#57611yescharacteristic
204C2×C97#57711yescharacteristic
205D97—22no—
206D97—41no—
207D97—81no—
208D97—81no—
209#3062—971nomaximal, Sylow
210C97#306211yescharacteristic, complemented, Sylow
211C2×AGL(1,ℤ/8)—971no—
212C2×#131—971no—
213#537—971no—
214#535—971no—
215#469—971no—
216#469—971no—
217#468—971no—
218#468D9711yescharacteristic, complemented
219#466—971no—
220#466—971no—
221#465—971no—
222#465—971no—
223#428—971no—
224#416—971no—
225C2×#102—971no—
226C2×#41—971no—
227C2×#41—971no—
228C22×D4—971no—
229AGL(1,ℤ/8)—1944no—
230C2×SD16—971no—
231C2×SD16C2×D9711yescharacteristic
232C2×D8—971no—
233C2×D8—971no—
234#131—1944no—
235C2×M16—971no—
236C2×M16—971no—
237C2×M16C2×D9711yescharacteristic, complemented
238C22×C8—971no—
239C22×C8—971no—
240C22×C8—971no—
241#127—1941no—
242#124—1941no—
243C22⋊Q8—971no—
244C22⋊Q8—1941no—
245C22⋊Q8C2×D9711yescharacteristic, complemented
246#121—971no—
247#121—971no—
248#121—971no—
249#121—971no—
250#121—971no—
251#121—1941no—
252#121—1941no—
253#121—1941no—
254#121C2×D9711yescharacteristic
255#120—1941no—
256#120—1941no—
257#117—971no—
258C2×C4⋊C4—971no—
259#107—971no—
260#107C2×D9711yescharacteristic, complemented
261#106—971no—
262#106—971no—
263#103—971no—
264#103C2×D9711yescharacteristic, complemented
265#102—971no—
266#102—971no—
267#102—971no—
268#102—971no—
269#102—971no—
270#102C2×D9711yescharacteristic
271C24—1941no—
272#41—1942no—
273#41—1942no—
274#41—1942no—
275#41—1942no—
276C2×Q8—971no—
277C2×Q8C22×D9711yescharacteristic
278C2×D4—21no—
279C2×D4—971no—
280C2×D4—971no—
281C2×D4—971no—
282C2×D4—971no—
283C2×D4—971no—
284C2×D4—1941no—
285C2×D4—1941no—
286C2×D4—1941no—
287C2×D4—1942no—
288C2×D4—1942no—
289C2×D4—1941no—
290C2×D4—1941no—
291C2×D4—1941no—
292C2×D4—1941no—
293C2×D4C22×D9711yescharacteristic
294C22×C4—971no—
295C22×C4—971no—
296C22×C4—1941no—
297C22×C4—1941no—
298C22×C4C22×D9711yescharacteristic
299SD16—22no—
300SD16—1942no—
301D8—1942no—
302D8—1942no—
303M16—22no—
304M16—974no—
305M16—1942no—
306C2×C8—971no—
307C2×C8—971no—
308C2×C8—1942no—
309C2×C8—1942no—
310C2×C8—1942no—
311C2×C8C22×D9711yescharacteristic
312C2×C8C22×D9711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—971no—
315C4⋊C4—971no—
316C4⋊C4—1941no—
317C4⋊C4—1941no—
318C4⋊C4C22×D9711yescharacteristic
319C4⋊C4C22×D9711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C22⋊C4—1941no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C42—1941no—
330C23—21no—
331C23—7761no—
332C23—7761no—
333C23—971no—
334C23—971no—
335C23—3882no—
336C23—1941no—
337C23—1941no—
338C23D4×D9711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—1942no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—7761no—
346D4—7761no—
347D4—7761no—
348D4—7761no—
349D4—7761no—
350D4—3882no—
351D4—3882no—
352D4—1942no—
353D4—1942no—
354D4—1942no—
355D4—1941no—
356D4—1941no—
357D4—1941no—
358D4—1941no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—7761no—
365C2×C4—971no—
366C2×C4—972no—
367C2×C4—972no—
368C2×C4—971no—
369C2×C4—3882no—
370C2×C4—1941no—
371C2×C4—1941no—
372C2×C4—1941no—
373C2×C4—1941no—
374C2×C4D4×D9711yescharacteristic
375C2×C4C23×D9711yescharacteristic
376C8—21no—
377C8—972no—
378C8—1941no—
379C8D4×D9712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—7764no—
387C22—972no—
388C22—3881no—
389C22—3881no—
390C22—3881no—
391C22—3881no—
392C22—3881no—
393C22—3881no—
394C22—1941no—
395C22—1941no—
396C22—1941no—
397C22—1941no—
398C22—1941no—
399C22C2×D4×D9711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—972no—
405C4—3881no—
406C4—3881no—
407C4—1941no—
408C4C2×D4×D9711yescharacteristic
409C4#41×D9711yescharacteristic
410C2—41no—
411C2—81no—
412C2—7761no—
413C2—7761no—
414C2—3881no—
415C2—1942no—
416C2#121×D9711yescharacteristic
417C2#2026153911yescharacteristic
418C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#468×C97C2
socleC22×C97C22×D4
hypercenter#468D97
hypocenterC97#3062
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690818C2semidirect
3. #20261315C2semidirect
4. #20261301C2semidirect
5. #20260988C2semidirect
6. #20260987C2semidirect
7. #20260984C2semidirect
8. #20260972C2semidirect
9. #20260969C2semidirect
10. #20260966C2semidirect
11. #20260961C2semidirect
12. #20260776C2semidirect
13. #20260730C2semidirect
14. #102×D97C2semidirect
15. #468×C97C2semidirect
25. C2×#13095494C22semidirect
26. C22×#11715895C22semidirect
28. C2×C8×D97C22semidirect
32. #15690786C22semidirect
35. #15690785C22semidirect
37. #15690784C22semidirect
45. #15690743C22semidirect
46. #15690742C22semidirect
47. #15690741C22semidirect
48. #15690734C22semidirect
50. #15690732C22semidirect
53. #15690730C22semidirect
55. C2×M16×C97C22semidirect
56. C22⋊Q8×C97C22semidirect
58. #107×C97C22semidirect
59. #103×C97C22semidirect
89. #13095494D4semidirect
210. C97#3062semidirect
218. #468D97semidirect
237. C2×M16C2×D97semidirect
245. C22⋊Q8C2×D97semidirect
260. #107C2×D97semidirect
264. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908182
#202613152
#202613012
#202609882
#202609872
#202609842
#202609722
#202609692
#202609662
#202609612
#202607762
#202607302
#102×D972
#468×C972
#306297