Finite Group Databasebeta

← #27151914

Subgroups — #27151914

Order 12416.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261209C211yesmaximal, characteristic, complemented
2#20261073C211yesmaximal, characteristic, complemented
3#20261056C211yesmaximal, characteristic, complemented
4#20261022C211yesmaximal, characteristic, complemented
5#20261012C211yesmaximal, characteristic, complemented
6#20260947C211yesmaximal, characteristic, complemented
7#20260947C211yesmaximal, characteristic, complemented
8#20260885C211yesmaximal, characteristic, complemented
9#20260868C211yesmaximal, characteristic
10#20260845C211yesmaximal, characteristic, complemented
11#20260833C211yesmaximal, characteristic, complemented
12C4×#13095498C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#542×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C4×D388C2211yescharacteristic
22C4×D388C2211yescharacteristic
23C4×D388C2211yescharacteristic
24C4×D388C2211yescharacteristic
25#15690772—21no—
26#15690772—21no—
27#15690769C2211yescharacteristic
28C22⋊C4×D97—21no—
29C22⋊C4×D97—21no—
30#15690743C2211yescharacteristic, complemented
31#15690743C2211yescharacteristic, complemented
32#15690741C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35C4⋊#11715895C2211yescharacteristic, complemented
36#15690734C2211yescharacteristic, complemented
37#15690734C2211yescharacteristic, complemented
38#15690731C2211yescharacteristic, complemented
39#15690731C2211yescharacteristic, complemented
40#15690730C2211yescharacteristic
41#15690730C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic, complemented
47C4×#11715895C2211yescharacteristic, complemented
48C4⋊Q8×C97C2211yescharacteristic, complemented
49#126×C97—21no—
50#124×C97—21no—
51#124×C97C2211yescharacteristic, complemented
52#123×C97—21no—
53C22⋊Q8×C97—21no—
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic, complemented
57C4×D4×C97C2211yescharacteristic, complemented
58#117×C97C2211yescharacteristic
59C23×D97—21no—
60C2×D388—21no—
61C2×D388—22no—
62C2×D388—22no—
63C2×D388C2311yescharacteristic
64C2×D388C2311yescharacteristic
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68C2×C4×D97—21no—
69#13095498—22no—
70#13095498—22no—
71#13095496—22no—
72#13095496—22no—
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79C22⋊Dic97—21no—
80#13095488—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84C4×Dic97—21no—
85C2×Q8×C97—21no—
86C2×Q8×C97C2311yescharacteristic
87C2×D4×C97—21no—
88C2×D4×C97C2311yescharacteristic
89C22×C4×C97—21no—
90C22×C4×C97C2311yescharacteristic
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97D411yescharacteristic
95C4⋊C4×C97D411yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111C22×D97—21no—
112C22×D97—24no—
113C22×D97—81no—
114C22×D97—81no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—81no—
118D4×C97—21no—
119D4×C97—21no—
120D4×C97—81no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—81no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2×D411yescharacteristic
132C2×C4×C97C2411yescharacteristic
133C2×Dic97—21no—
134C2×Dic97—21no—
135D388—24no—
136D388—21no—
137D388—21no—
138D388#4112yes—
139D388#4112yes—
140C4×D97—81no—
141C4×D97—81no—
142C4×D97—81no—
143C4×D97—81no—
144#11715895—21no—
145#11715895—21no—
146#11715895—21no—
147#11715895—21no—
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—41no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97C22×D411yescharacteristic
155C2×D97—22no—
156C2×D97—22no—
157C2×D97—41no—
158C2×D97—41no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—84no—
162C4×C97—21no—
163C4×C97—21no—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97—41no—
170C4×C97C2×#4111yescharacteristic
171C4×C97C2×#4111yescharacteristic
172Dic97—41no—
173Dic97—41no—
174C2×C97—41no—
175C2×C97—81no—
176C2×C97#52511yescharacteristic
177C2×C97#57611yescharacteristic
178C2×C97#57711yescharacteristic
179D97—42no—
180D97—42no—
181#3118—971nomaximal, Sylow
182C97#311811yescharacteristic, complemented, Sylow
183#542D9711yescharacteristic, complemented
184#518—971no—
185#485—971no—
186#482—971no—
187#476—971no—
188#474—971no—
189#463—971no—
190#463—971no—
191#450—971no—
192#447—971no—
193#442—971no—
194#440—971no—
195C4×SD16—971no—
196C4×D8—971no—
197#433—971no—
198C22×D4—971no—
199C2×SD16—971no—
200C2×SD16—971no—
201C2×D8—971no—
202C2×D8—971no—
203C4⋊Q8C2×D9711yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D9711yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D9711yescharacteristic
210#121C2×D9711yescharacteristic
211C4×Q8C2×D9711yescharacteristic, complemented
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—1941no—
217C4×D4—1941no—
218C4×D4C2×D9711yescharacteristic, complemented
219#117—971no—
220#117C2×D9711yescharacteristic
221C2×C22⋊C4—1941no—
222C2×C22⋊C4—1941no—
223#107—971no—
224#107—971no—
225#106—971no—
226#106—971no—
227C4⋊C8—971no—
228C4⋊C8—971no—
229#103—971no—
230#103—971no—
231#102—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235#102—971no—
236#102—971no—
237C22⋊C8—971no—
238C22⋊C8—971no—
239#97—971no—
240C4×C8—971no—
241C24—1941no—
242C2×Q8—21no—
243C2×Q8C22×D9711yescharacteristic
244C2×D4—21no—
245C2×D4—971no—
246C2×D4—971no—
247C2×D4—1942no—
248C2×D4—1942no—
249C2×D4—1941no—
250C2×D4C22×D9711yescharacteristic
251C22×C4—21no—
252C22×C4—1941no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4C22×D9711yescharacteristic
257SD16—1942no—
258SD16—1942no—
259D8—1942no—
260D8—1942no—
261C2×C8—971no—
262C2×C8—971no—
263C2×C8—971no—
264C2×C8—971no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4C22⋊D9711yescharacteristic
271C4⋊C4C22⋊D9711yescharacteristic
272C4⋊C4C22×D9711yescharacteristic
273C4⋊C4C22×D9711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4C22⋊D9711yescharacteristic
285C22⋊C4C22⋊D9711yescharacteristic
286C42—21no—
287C42—1941no—
288C42C22×D9711yescharacteristic
289C42C22×D9711yescharacteristic
290C23—21no—
291C23—7761no—
292C23—7761no—
293C23—1944no—
294C23—1941no—
295C23—1941no—
296C23C2×C22⋊D9711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—972no—
304D4—972no—
305D4—1944no—
306D4—1941no—
307D4—1941no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—7761no—
315C2×C4—7761no—
316C2×C4—7761no—
317C2×C4—7761no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4C2×C22⋊D9711yescharacteristic
321C2×C4C2×C22⋊D9711yescharacteristic
322C2×C4C2×C22⋊D9711yescharacteristic
323C2×C4C2×C22⋊D9711yescharacteristic
324C2×C4C2×C22⋊D9711yescharacteristic
325C2×C4C23×D9711yescharacteristic
326C8—1941no—
327C8—1941no—
328C8—1941no—
329C8—1941no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—7764no—
337C22—3881no—
338C22—3881no—
339C22—3881no—
340C22—3881no—
341C22—1942no—
342C22—1942no—
343C22C22×C22⋊D9711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—3881no—
353C4—3881no—
354C4#41×D9711yescharacteristic
355C4#41×D9711yescharacteristic
356C2—41no—
357C2—81no—
358C2—3882no—
359C2—3882no—
360C2#2026124211yescharacteristic
361C2#2026153811yescharacteristic
362C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#542×C97C2
socleC22×C97C22×D4
prime radical#20260868C2
hypercenter#542D97
hypocenterC97#3118
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20261209C2semidirect
2. #20261073C2semidirect
3. #20261056C2semidirect
4. #20261022C2semidirect
5. #20261012C2semidirect
6. #20260947C2semidirect
7. #20260947C2semidirect
8. #20260885C2semidirect
10. #20260845C2semidirect
11. #20260833C2semidirect
12. C4×#13095498C2semidirect
13. C4×#13095496C2semidirect
14. #20260796C2semidirect
15. #542×C97C2semidirect
30. #15690743C22semidirect
31. #15690743C22semidirect
32. #15690741C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. C4⋊#11715895C22semidirect
36. #15690734C22semidirect
37. #15690734C22semidirect
38. #15690731C22semidirect
39. #15690731C22semidirect
46. #15690721C22semidirect
47. C4×#11715895C22semidirect
48. C4⋊Q8×C97C22semidirect
51. #124×C97C22semidirect
56. C4×Q8×C97C22semidirect
57. C4×D4×C97C22semidirect
182. C97#3118semidirect
183. #542D97semidirect
203. C4⋊Q8C2×D97semidirect
206. #124C2×D97semidirect
211. C4×Q8C2×D97semidirect
218. C4×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612092
#202610732
#202610562
#202610222
#202610122
#202609472
#202608852
#202608682
#202608452
#202608332
C4×#130954982
C4×#130954962
#202607962
#542×C972
#311897