Finite Group Databasebeta

← #27151234

Subgroups — #27151234

Order 12416.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690834C211yesmaximal, characteristic, complemented
2C2×D8×D97C211yesmaximal, characteristic, complemented
3#20261298C211yesmaximal, characteristic, complemented
4#20261207C211yesmaximal, characteristic, complemented
5#20260951C211yesmaximal, characteristic, complemented
6#20260950C211yesmaximal, characteristic, complemented
7#20260949C211yesmaximal, characteristic, complemented
8#20260948C211yesmaximal, characteristic, complemented
9#20260929C211yesmaximal, characteristic, complemented
10#20260928C211yesmaximal, characteristic, complemented
11#20260927C211yesmaximal, characteristic, complemented
12#20260760C211yesmaximal, characteristic, complemented
13#20260738C211yesmaximal, characteristic, complemented
14#20260728C211yesmaximal, characteristic, complemented
15#459×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic
19#15690834—24no—
20C2×#13095500C2211yescharacteristic, complemented
21C2×#13095499C2211yescharacteristic, complemented
22C2×D776C2211yescharacteristic
23C2×#13095496C2211yescharacteristic
24C2×#13095496C2211yescharacteristic
25C2×#13095496C2211yescharacteristic, complemented
26D8×D97—24no—
27C2×#13095494C2211yescharacteristic
28C2×C8×D97C2211yescharacteristic
29#15690806C2211yescharacteristic, complemented
30#15690798C2211yescharacteristic, complemented
31#15690797C2211yescharacteristic, complemented
32#15690782—21no—
33#15690782—21no—
34#15690778—21no—
35#15690778—21no—
36C4×D388C2211yescharacteristic, complemented
37D4×Dic97C2211yescharacteristic, complemented
38#15690772C2211yescharacteristic, complemented
39C4×C22⋊D97—21no—
40#15690770—21no—
41#15690768C2211yescharacteristic
42#15690767C2211yescharacteristic
43C2×#13095488—21no—
44C22⋊C4×D97—21no—
45C4⋊#11715895C2211yescharacteristic, complemented
46#15690739C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690733C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690731C2211yescharacteristic, complemented
52#15690731C2211yescharacteristic, complemented
53#15690730C2211yescharacteristic, complemented
54C2×D8×C97C2211yescharacteristic
55C2×D8×C97C2211yescharacteristic, complemented
56#127×C97C2211yescharacteristic, complemented
57C4×D4×C97C2211yescharacteristic, complemented
58C4⋊C8×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63#13095510—22no—
64#13095510—22no—
65C2×Dic194C2311yescharacteristic
66C2×D388C2311yescharacteristic
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—22no—
76D4×D97D414yescomplemented
77C22×Dic97—21no—
78C22×Dic97—21no—
79C2×C4×D97—21no—
80C2×C4×D97—21no—
81C2×C4×D97C2311yescharacteristic
82#13095500—22no—
83#13095499—22no—
84D776—22no—
85#13095496—22no—
86#13095496—22no—
87#13095496—22no—
88#13095494—22no—
89C2×#11715895C2311yescharacteristic
90C2×#11715895C2311yescharacteristic
91C8×D97—22no—
92C4⋊Dic97C2311yescharacteristic
93#13095490D411yescharacteristic
94#13095490D411yescharacteristic
95C22⋊Dic97—21no—
96C22⋊Dic97—21no—
97C22⋊Dic97—21no—
98#13095488—21no—
99#13095488—21no—
100#13095488—21no—
101#13095488—21no—
102#13095488—21no—
103#13095488D411yescharacteristic
104#13095488D411yescharacteristic
105C4×Dic97—21no—
106C4×Dic97C2311yescharacteristic
107C2×D4×C97—21no—
108C2×D4×C97—21no—
109C2×D4×C97C2311yescharacteristic
110C2×D4×C97C2311yescharacteristic
111C2×D4×C97C2311yescharacteristic
112C22×C4×C97—21no—
113D8×C97—22no—
114D8×C97—22no—
115C2×C8×C97C2311yescharacteristic
116C2×C8×C97C2311yescharacteristic
117C4⋊C4×C97C2311yescharacteristic
118C22⋊C4×C97—21no—
119C42×C97C2311yescharacteristic
120C23×C97—21no—
121C23×C97—21no—
122C23×C97—21no—
123C22×D97—21no—
124C22×D97—24no—
125C22×D97—21no—
126C22×D97—42no—
127C22×D97—81no—
128C22×D97—81no—
129C22×D97—81no—
130C22×D97—81no—
131C22×D97C2×D411yescharacteristic
132D4×C97—21no—
133D4×C97—21no—
134D4×C97—21no—
135D4×C97—21no—
136D4×C97—21no—
137D4×C97—81no—
138D4×C97—81no—
139D4×C97C2×D412yes—
140C2×C4×C97—21no—
141C2×C4×C97—81no—
142C2×C4×C97C2×D411yescharacteristic
143C2×C4×C97C2×D411yescharacteristic
144C2×C4×C97C2411yescharacteristic
145C22⋊D97—24no—
146C22⋊D97—42no—
147C22⋊D97—42no—
148C22⋊D97—81no—
149C22⋊D97—81no—
150C2×Dic97—21no—
151C2×Dic97—21no—
152C2×Dic97—42no—
153C2×Dic97—81no—
154C2×Dic97C2×D411yescharacteristic
155C2×Dic97C2×D411yescharacteristic
156C2×Dic97C2×D411yescharacteristic
157D388—21no—
158D388—21no—
159D388—21no—
160D388C2×D412yes—
161C4×D97—81no—
162C4×D97—81no—
163C4×D97C2×D412yes—
164C4×D97#4112yes—
165Dic194—21no—
166Dic194—21no—
167C8×C97—21no—
168C8×C97—21no—
169#11715895—21no—
170#11715895—21no—
171C22×C97—22no—
172C22×C97—41no—
173C22×C97—41no—
174C22×C97—41no—
175C22×C97—41no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97C22×D411yescharacteristic
181C2×D97—21no—
182C2×D97—21no—
183C2×D97—22no—
184C2×D97—41no—
185C2×D97—41no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—41no—
189C2×D97—84no—
190C2×D97—84no—
191C2×D97#12112yes—
192C4×C97—21no—
193C4×C97—21no—
194C4×C97—41no—
195C4×C97C22×D411yescharacteristic
196C4×C97C2×#4111yescharacteristic
197Dic97—21no—
198Dic97—21no—
199Dic97—41no—
200Dic97—41no—
201Dic97#12112yes—
202C2×C97—42no—
203C2×C97—81no—
204C2×C97—81no—
205C2×C97C2×#12111yescharacteristic
206C2×C97#57611yescharacteristic
207C2×C97#57611yescharacteristic
208D97—22no—
209D97—42no—
210D97—81no—
211#3041—971nomaximal, Sylow
212C97#304111yescharacteristic, complemented, Sylow
213C2×AGL(1,ℤ/8)—971no—
214C22×D8—971no—
215#535—971no—
216#518—971no—
217#463—971no—
218#463—971no—
219#463—971no—
220#463—971no—
221#459—971no—
222#459—971no—
223#459—971no—
224#459D9711yescharacteristic, complemented
225#424—971no—
226#418—971no—
227#416—971no—
228C2×#41—971no—
229C22×D4—971no—
230C22×D4—971no—
231AGL(1,ℤ/8)—1944no—
232C2×SD16—971no—
233C2×SD16—971no—
234C2×D8—971no—
235C2×D8—971no—
236C2×D8—971no—
237C2×D8—971no—
238C2×D8—1944no—
239C2×D8C2×D9711yescharacteristic
240C2×D8C2×D9711yescharacteristic, complemented
241C2×M16—971no—
242C22×C8—971no—
243#127—971no—
244#127C2×D9711yescharacteristic, complemented
245#124—971no—
246#124—971no—
247#121—1941no—
248#121—1941no—
249#120—1941no—
250#120—1941no—
251C4×D4—971no—
252C4×D4—971no—
253C4×D4—971no—
254C4×D4—1941no—
255C4×D4—1941no—
256C4×D4C2×D9711yescharacteristic, complemented
257#117—971no—
258#117—971no—
259C2×C22⋊C4—1941no—
260C2×C22⋊C4—1941no—
261C4⋊C8—971no—
262C4⋊C8—971no—
263C4⋊C8—971no—
264C4⋊C8C2×D9711yescharacteristic, complemented
265#103—971no—
266#103—971no—
267#102—971no—
268#102—971no—
269#102—971no—
270#102—971no—
271#102C2×D9711yescharacteristic, complemented
272#102C2×D9711yescharacteristic, complemented
273C24—1941no—
274C24—1941no—
275#41—1942no—
276#41—1942no—
277C2×Q8—971no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—974no—
281C2×D4—971no—
282C2×D4—971no—
283C2×D4—1942no—
284C2×D4—1942no—
285C2×D4—1942no—
286C2×D4—1941no—
287C2×D4—1941no—
288C2×D4—1941no—
289C2×D4—1941no—
290C2×D4—1941no—
291C2×D4C22×D9711yescharacteristic
292C2×D4C22×D9711yescharacteristic
293C2×D4C22×D9711yescharacteristic
294C22×C4—21no—
295C22×C4—971no—
296C22×C4—1941no—
297C22×C4—1941no—
298C22×C4—1941no—
299C22×C4—1941no—
300SD16—1942no—
301SD16—1942no—
302D8—22no—
303D8—22no—
304D8—1942no—
305D8—1942no—
306D8—1942no—
307D8—1942no—
308M16—1942no—
309C2×C8—971no—
310C2×C8—971no—
311C2×C8—1942no—
312C2×C8C22×D9711yescharacteristic
313C2×C8C22×D9711yescharacteristic
314C4⋊C4—971no—
315C4⋊C4—971no—
316C4⋊C4—971no—
317C4⋊C4C22×D9711yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—971no—
320C22⋊C4—971no—
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C22⋊C4—1941no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C42—971no—
330C42—1941no—
331C42C22×D9711yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—7761no—
336C23—7761no—
337C23—7761no—
338C23—7761no—
339C23—971no—
340C23—3882no—
341C23—1944no—
342C23—1941no—
343C23—1941no—
344Q8—1941no—
345Q8—1941no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—7761no—
354D4—7761no—
355D4—972no—
356D4—3882no—
357D4—3882no—
358D4—1944no—
359D4—1941no—
360D4—1941no—
361D4—1941no—
362D4D4×D9712yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—7761no—
366C2×C4—7761no—
367C2×C4—7761no—
368C2×C4—971no—
369C2×C4—972no—
370C2×C4—972no—
371C2×C4—971no—
372C2×C4—971no—
373C2×C4—3882no—
374C2×C4—1941no—
375C2×C4—1941no—
376C2×C4D4×D9711yescharacteristic
377C2×C4D4×D9711yescharacteristic
378C2×C4C23×D9711yescharacteristic
379C8—21no—
380C8—21no—
381C8—1941no—
382C8—1941no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—7764no—
393C22—7764no—
394C22—972no—
395C22—3881no—
396C22—3881no—
397C22—3881no—
398C22—3881no—
399C22—3881no—
400C22—1942no—
401C22—1941no—
402C22—1941no—
403C22C2×D4×D9711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—972no—
408C4—3881no—
409C4—3881no—
410C4—1941no—
411C4—1941no—
412C4C2×D4×D9711yescharacteristic
413C4#41×D9711yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—7761no—
418C2—3882no—
419C2—1942no—
420C2#121×D9711yescharacteristic
421C2#2026153611yescharacteristic
422C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#459×C97C2
socleC22×C97C22×D4
hypercenter#459D97
hypocenterC97#3041
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. C2×D8×D97C2semidirect
3. #20261298C2semidirect
4. #20261207C2semidirect
5. #20260951C2semidirect
6. #20260950C2semidirect
7. #20260949C2semidirect
8. #20260948C2semidirect
9. #20260929C2semidirect
10. #20260928C2semidirect
11. #20260927C2semidirect
12. #20260760C2semidirect
13. #20260738C2semidirect
14. #20260728C2semidirect
15. #459×C97C2semidirect
20. C2×#13095500C22semidirect
21. C2×#13095499C22semidirect
25. C2×#13095496C22semidirect
29. #15690806C22semidirect
30. #15690798C22semidirect
31. #15690797C22semidirect
36. C4×D388C22semidirect
37. D4×Dic97C22semidirect
38. #15690772C22semidirect
45. C4⋊#11715895C22semidirect
46. #15690739C22semidirect
47. #15690739C22semidirect
48. #15690735C22semidirect
49. #15690733C22semidirect
50. #15690732C22semidirect
51. #15690731C22semidirect
52. #15690731C22semidirect
53. #15690730C22semidirect
55. C2×D8×C97C22semidirect
56. #127×C97C22semidirect
57. C4×D4×C97C22semidirect
58. C4⋊C8×C97C22semidirect
59. #102×C97C22semidirect
60. #102×C97C22semidirect
76. D4×D97D4semidirect
212. C97#3041semidirect
224. #459D97semidirect
240. C2×D8C2×D97semidirect
244. #127C2×D97semidirect
256. C4×D4C2×D97semidirect
264. C4⋊C8C2×D97semidirect
271. #102C2×D97semidirect
272. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
C2×D8×D972
#202612982
#202612072
#202609512
#202609502
#202609492
#202609482
#202609292
#202609282
#202609272
#202607602
#202607382
#202607282
#459×C972
#304197