Finite Group Databasebeta

← #27151237

Subgroups — #27151237

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×#15690837C211yesmaximal, characteristic, complemented
2C2×D8×D97C211yesmaximal, characteristic, complemented
3#20261303C211yesmaximal, characteristic, complemented
4#20261207C211yesmaximal, characteristic, complemented
5#20260953C211yesmaximal, characteristic, complemented
6#20260952C211yesmaximal, characteristic, complemented
7#20260947C211yesmaximal, characteristic, complemented
8#20260931C211yesmaximal, characteristic, complemented
9#20260930C211yesmaximal, characteristic, complemented
10#20260929C211yesmaximal, characteristic, complemented
11#20260928C211yesmaximal, characteristic, complemented
12#20260760C211yesmaximal, characteristic, complemented
13#20260738C211yesmaximal, characteristic, complemented
14#20260730C211yesmaximal, characteristic, complemented
15#463×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic
19#15690837—24no—
20C2×#13095498C2211yescharacteristic, complemented
21C2×D776C2211yescharacteristic
22C2×D776C2211yescharacteristic, complemented
23C2×#13095496C2211yescharacteristic
24C2×#13095496C2211yescharacteristic
25C2×#13095496C2211yescharacteristic, complemented
26D8×D97—24no—
27C2×#13095494C2211yescharacteristic
28C2×C8×D97C2211yescharacteristic
29#15690807C2211yescharacteristic, complemented
30#15690806C2211yescharacteristic, complemented
31#15690799C2211yescharacteristic, complemented
32#15690783—21no—
33#15690783—21no—
34#15690780—21no—
35#15690780—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#15690734C2211yescharacteristic, complemented
49#15690732C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690731C2211yescharacteristic, complemented
52#15690730C2211yescharacteristic, complemented
53#15690730C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic, complemented
55C2×D8×C97C2211yescharacteristic
56#124×C97C2211yescharacteristic, complemented
57C4×D4×C97C2211yescharacteristic, complemented
58C4⋊C8×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63#13095512—22no—
64#13095512—22no—
65C2×D388—21no—
66C2×D388—21no—
67C2×D388—21no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×D388C2311yescharacteristic
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75D4×D97—22no—
76D4×D97—22no—
77D4×D97—22no—
78D4×D97D414yescomplemented
79C22×Dic97—21no—
80C2×C4×D97—21no—
81C2×C4×D97—21no—
82C2×C4×D97—21no—
83C2×C4×D97C2311yescharacteristic
84#13095498—22no—
85D776—22no—
86D776—22no—
87#13095496—22no—
88#13095496—22no—
89#13095496—22no—
90#13095494—22no—
91C2×#11715895C2311yescharacteristic
92C2×#11715895C2311yescharacteristic
93C8×D97—22no—
94C4⋊Dic97C2311yescharacteristic
95#13095490D411yescharacteristic
96#13095490D411yescharacteristic
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×Q8×C97C2311yescharacteristic
108C2×D4×C97C2311yescharacteristic
109C2×D4×C97C2311yescharacteristic
110C22×C4×C97—21no—
111SD16×C97—22no—
112D8×C97—22no—
113C2×C8×C97C2311yescharacteristic
114C2×C8×C97C2311yescharacteristic
115C4⋊C4×C97C2311yescharacteristic
116C22⋊C4×C97—21no—
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C42×C97C2311yescharacteristic
120C23×C97—21no—
121C23×C97—21no—
122C22×D97—21no—
123C22×D97—24no—
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—42no—
127C22×D97—81no—
128C22×D97—81no—
129C22×D97—81no—
130C22×D97—81no—
131C22×D97C2×D411yescharacteristic
132Q8×C97—21no—
133Q8×C97—21no—
134D4×C97—21no—
135D4×C97—21no—
136D4×C97—21no—
137D4×C97C2×D412yes—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—81no—
141C2×C4×C97C2×D411yescharacteristic
142C2×C4×C97C2×D411yescharacteristic
143C2×C4×C97C2411yescharacteristic
144C22⋊D97—24no—
145C22⋊D97—42no—
146C22⋊D97—81no—
147C22⋊D97—81no—
148C2×Dic97—21no—
149C2×Dic97—81no—
150C2×Dic97C2×D411yescharacteristic
151C2×Dic97C2×D411yescharacteristic
152C2×Dic97C2×D411yescharacteristic
153D388—21no—
154D388—21no—
155D388—21no—
156D388—21no—
157D388—21no—
158D388—42no—
159D388—81no—
160D388—81no—
161D388C2×D412yes—
162C4×D97—42no—
163C4×D97—81no—
164C4×D97—81no—
165C4×D97C2×D412yes—
166C4×D97#4112yes—
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×C97C22×D411yescharacteristic
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—22no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—41no—
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×C97—41no—
196C4×C97C22×D411yescharacteristic
197C4×C97C2×#4111yescharacteristic
198Dic97—21no—
199Dic97—21no—
200Dic97—41no—
201Dic97#12112yes—
202C2×C97—42no—
203C2×C97—81no—
204C2×C97C2×#12111yescharacteristic
205C2×C97#57611yescharacteristic
206C2×C97#57611yescharacteristic
207D97—22no—
208D97—42no—
209D97—81no—
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#463D9711yescharacteristic, complemented
221#459—971no—
222#459—971no—
223#459—971no—
224#459—971no—
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×SD16C2×D9711yescharacteristic, complemented
234C2×D8—971no—
235C2×D8—971no—
236C2×D8—971no—
237C2×D8—971no—
238C2×D8—971no—
239C2×D8—1944no—
240C2×D8C2×D9711yescharacteristic
241C2×M16—971no—
242C22×C8—971no—
243#127—971no—
244#127—971no—
245#124—971no—
246#124C2×D9711yescharacteristic, complemented
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#103C2×D9711yescharacteristic, complemented
267#102—971no—
268#102—971no—
269#102—971no—
270#102—971no—
271#102—971no—
272#102C2×D9711yescharacteristic, complemented
273C24—1941no—
274C24—1941no—
275#41—1942no—
276#41—1942no—
277C2×Q8C22×D9711yescharacteristic
278C2×D4—974no—
279C2×D4—971no—
280C2×D4—971no—
281C2×D4—971no—
282C2×D4—1941no—
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×D4—1941no—
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—22no—
301SD16—1942no—
302D8—22no—
303D8—1942no—
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—21no—
320C22⋊C4—21no—
321C22⋊C4—971no—
322C22⋊C4—971no—
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—7761no—
335C23—7761no—
336C23—7761no—
337C23—7761no—
338C23—971no—
339C23—3882no—
340C23—1944no—
341C23—1941no—
342C23—1941no—
343C23—1941no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—7761no—
350D4—7761no—
351D4—7761no—
352D4—7761no—
353D4—972no—
354D4—3882no—
355D4—3882no—
356D4—1944no—
357D4—1941no—
358D4—1941no—
359D4—1941no—
360D4—1941no—
361D4—1941no—
362D4D4×D9712yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—81no—
366C2×C4—7761no—
367C2×C4—7761no—
368C2×C4—7761no—
369C2×C4—971no—
370C2×C4—972no—
371C2×C4—972no—
372C2×C4—971no—
373C2×C4—971no—
374C2×C4—3882no—
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—7764no—
390C22—7764no—
391C22—972no—
392C22—3881no—
393C22—3881no—
394C22—3881no—
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—41no—
408C4—972no—
409C4—3881no—
410C4—1941no—
411C4—1941no—
412C4C2×D4×D9711yescharacteristic
413C4#41×D9711yescharacteristic
414C2—42no—
415C2—81no—
416C2—7761no—
417C2—7761no—
418C2—3882no—
419C2—1942no—
420C2#121×D9711yescharacteristic
421C2#2026153711yescharacteristic
422C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#463×C97C2
socleC22×C97C22×D4
hypercenter#463D97
hypocenterC97#3041
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×D8×D97C2semidirect
3. #20261303C2semidirect
4. #20261207C2semidirect
5. #20260953C2semidirect
6. #20260952C2semidirect
7. #20260947C2semidirect
8. #20260931C2semidirect
9. #20260930C2semidirect
10. #20260929C2semidirect
11. #20260928C2semidirect
12. #20260760C2semidirect
13. #20260738C2semidirect
14. #20260730C2semidirect
15. #463×C97C2semidirect
20. C2×#13095498C22semidirect
22. C2×D776C22semidirect
25. C2×#13095496C22semidirect
29. #15690807C22semidirect
30. #15690806C22semidirect
31. #15690799C22semidirect
36. C4×D388C22semidirect
37. D4×Dic97C22semidirect
38. #15690772C22semidirect
45. C4⋊#11715895C22semidirect
46. #15690739C22semidirect
47. #15690739C22semidirect
48. #15690734C22semidirect
49. #15690732C22semidirect
50. #15690732C22semidirect
51. #15690731C22semidirect
52. #15690730C22semidirect
53. #15690730C22semidirect
54. C2×SD16×C97C22semidirect
56. #124×C97C22semidirect
57. C4×D4×C97C22semidirect
58. C4⋊C8×C97C22semidirect
59. #103×C97C22semidirect
60. #102×C97C22semidirect
78. D4×D97D4semidirect
212. C97#3041semidirect
220. #463D97semidirect
233. C2×SD16C2×D97semidirect
246. #124C2×D97semidirect
256. C4×D4C2×D97semidirect
264. C4⋊C8C2×D97semidirect
266. #103C2×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×#156908372
C2×D8×D972
#202613032
#202612072
#202609532
#202609522
#202609472
#202609312
#202609302
#202609292
#202609282
#202607602
#202607382
#202607302
#463×C972
#304197