Finite Group Databasebeta

← #27151922

Subgroups — #27151922

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#20261356C211yesmaximal, characteristic, complemented
2#20261209C211yesmaximal, characteristic, complemented
3#20261077C211yesmaximal, characteristic, complemented
4#20261060C211yesmaximal, characteristic, complemented
5#20261026C211yesmaximal, characteristic, complemented
6#20261016C211yesmaximal, characteristic, complemented
7#20260951C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260888C211yesmaximal, characteristic, complemented
10#20260871C211yesmaximal, characteristic
11#20260848C211yesmaximal, characteristic, complemented
12#20260837C211yesmaximal, characteristic, complemented
13C4×#13095500C211yesmaximal, characteristic, complemented
14C4×D776C211yesmaximal, characteristic, complemented
15#433×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C2×#13095500C2211yescharacteristic
18C2×#13095500C2211yescharacteristic
19C2×D776C2211yescharacteristic
20C2×D776C2211yescharacteristic
21#15690810C2211yescharacteristic, complemented
22#15690805—21no—
23#15690798C2211yescharacteristic, complemented
24#15690796—21no—
25#15690792—21no—
26#15690789C2211yescharacteristic
27#15690788—21no—
28#15690784C2211yescharacteristic
29C4×Dic194C2211yescharacteristic, complemented
30C4×D388C2211yescharacteristic
31C4×D388C2211yescharacteristic
32C4×D388C2211yescharacteristic
33C4×D388C2211yescharacteristic
34C4×D388C2211yescharacteristic, complemented
35#15690772—21no—
36#15690772—21no—
37#15690769C2211yescharacteristic
38C22⋊C4×D97—21no—
39C22⋊C4×D97—21no—
40#15690744C2211yescharacteristic, complemented
41#15690744C2211yescharacteristic, complemented
42#15690742C2211yescharacteristic, complemented
43#15690742C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690735C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic
47#15690732C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic, complemented
51#15690732C2211yescharacteristic, complemented
52#117×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic, complemented
54C4⋊C8×C97C2211yescharacteristic, complemented
55C22⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic, complemented
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62C2×D388—21no—
63C2×D388—22no—
64C2×D388—22no—
65C2×D388C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74#13095500—22no—
75#13095500—22no—
76D776—22no—
77D776—22no—
78C4⋊Dic97—21no—
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84#13095490—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C22×C4×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97D411yescharacteristic
103C4⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C22⋊C4×C97D411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97C2×D411yescharacteristic
109C22×D97—21no—
110C22×D97—21no—
111C22×D97—24no—
112C22×D97—21no—
113C22×D97—81no—
114C22×D97—81no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97C2×D411yescharacteristic
118C2×C4×C97C2×D411yescharacteristic
119C2×C4×C97C2×D411yescharacteristic
120C2×C4×C97C2×D411yescharacteristic
121C2×C4×C97C2×D411yescharacteristic
122C2×C4×C97C2411yescharacteristic
123C22⋊D97—81no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129D388—24no—
130D388—21no—
131D388—21no—
132D388—21no—
133D388—21no—
134D388#4112yes—
135D388#4112yes—
136C4×D97—81no—
137C4×D97—81no—
138C4×D97—81no—
139C4×D97—81no—
140C4×D97—81no—
141Dic194—21no—
142Dic194—21no—
143Dic194—81no—
144C8×C97—21no—
145C8×C97—21no—
146C8×C97—21no—
147C8×C97—21no—
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97C22×D411yescharacteristic
152C2×D97—22no—
153C2×D97—22no—
154C2×D97—41no—
155C2×D97—41no—
156C2×D97—41no—
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×C97C2×#4111yescharacteristic
168C4×C97C2×#4111yescharacteristic
169Dic97—41no—
170Dic97—41no—
171Dic97—41no—
172Dic97—41no—
173Dic97—41no—
174C2×C97—41no—
175C2×C97#52511yescharacteristic
176C2×C97#57611yescharacteristic
177C2×C97#57711yescharacteristic
178D97—42no—
179D97—42no—
180D97—81no—
181#3118—971nomaximal, Sylow
182C97#311811yescharacteristic, complemented, Sylow
183#542—971no—
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#433D9711yescharacteristic, complemented
198C22×D4—971no—
199C2×SD16—971no—
200C2×SD16—971no—
201C2×D8—971no—
202C2×D8—971no—
203C4⋊Q8—971no—
204#126—1941no—
205#124—971no—
206#124—1941no—
207#123—1941no—
208C22⋊Q8—971no—
209C22⋊Q8—1941no—
210#121—971no—
211C4×Q8—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—971no—
217C4×D4—1941no—
218C4×D4—1941no—
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⋊C8C2×D9711yescharacteristic, complemented
228C4⋊C8C2×D9711yescharacteristic, complemented
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⋊C8C2×D9711yescharacteristic
238C22⋊C8C2×D9711yescharacteristic
239#97C2×D9711yescharacteristic, complemented
240C4×C8C2×D9711yescharacteristic, complemented
241C24—1941no—
242C2×Q8—971no—
243C2×Q8—1941no—
244C2×D4—971no—
245C2×D4—971no—
246C2×D4—971no—
247C2×D4—1942no—
248C2×D4—1942no—
249C2×D4—1941no—
250C2×D4—1941no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4C22×D9711yescharacteristic
257SD16—1942no—
258SD16—1942no—
259D8—1942no—
260D8—1942no—
261C2×C8C22×D9711yescharacteristic
262C2×C8C22×D9711yescharacteristic
263C2×C8C22×D9711yescharacteristic
264C2×C8C22×D9711yescharacteristic
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4D38811yescharacteristic
273C4⋊C4D38811yescharacteristic
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C22⋊C4—1941no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4D38811yescharacteristic
285C22⋊C4D38811yescharacteristic
286C42—1941no—
287C42—1941no—
288C42C22×D9711yescharacteristic
289C42C22×D9711yescharacteristic
290C23—7761no—
291C23—7761no—
292C23—1944no—
293C23—1941no—
294C23—1941no—
295C23—1941no—
296C23C2×D38811yescharacteristic
297Q8—7761no—
298Q8—1941no—
299Q8—1941no—
300D4—7761no—
301D4—972no—
302D4—972no—
303D4—1944no—
304D4—1941no—
305D4—1941no—
306D4—1941no—
307D4—1941no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—7761no—
311C2×C4—7761no—
312C2×C4—7761no—
313C2×C4—7761no—
314C2×C4—7761no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4C2×D38811yescharacteristic
321C2×C4C2×D38811yescharacteristic
322C2×C4C2×D38811yescharacteristic
323C2×C4C2×D38811yescharacteristic
324C2×C4C2×D38811yescharacteristic
325C2×C4C23×D9711yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—7764no—
334C22—3881no—
335C22—3881no—
336C22—3881no—
337C22—3881no—
338C22—3881no—
339C22—3881no—
340C22—3881no—
341C22—1942no—
342C22—1942no—
343C22C22×D38811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—3881no—
350C4—3881no—
351C4—3881no—
352C4—3881no—
353C4—3881no—
354C4#41×D9711yescharacteristic
355C4#41×D9711yescharacteristic
356C2—41no—
357C2—7761no—
358C2—3882no—
359C2—3882no—
360C2#2026124711yescharacteristic
361C2#2026154011yescharacteristic
362C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#433×C97C2
socleC22×C97C22×D4
prime radical#20260871C2
hypercenter#433D97
hypocenterC97#3118
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #20261356C2semidirect
2. #20261209C2semidirect
3. #20261077C2semidirect
4. #20261060C2semidirect
5. #20261026C2semidirect
6. #20261016C2semidirect
7. #20260951C2semidirect
8. #20260951C2semidirect
9. #20260888C2semidirect
11. #20260848C2semidirect
12. #20260837C2semidirect
13. C4×#13095500C2semidirect
14. C4×D776C2semidirect
15. #433×C97C2semidirect
21. #15690810C22semidirect
23. #15690798C22semidirect
29. C4×Dic194C22semidirect
34. C4×D388C22semidirect
40. #15690744C22semidirect
41. #15690744C22semidirect
42. #15690742C22semidirect
43. #15690742C22semidirect
44. #15690735C22semidirect
45. #15690735C22semidirect
50. #15690732C22semidirect
51. #15690732C22semidirect
53. C4⋊C8×C97C22semidirect
54. C4⋊C8×C97C22semidirect
57. #97×C97C22semidirect
58. C4×C8×C97C22semidirect
182. C97#3118semidirect
197. #433D97semidirect
227. C4⋊C8C2×D97semidirect
228. C4⋊C8C2×D97semidirect
239. #97C2×D97semidirect
240. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202612092
#202610772
#202610602
#202610262
#202610162
#202609512
#202608882
#202608712
#202608482
#202608372
C4×#130955002
C4×D7762
#433×C972
#311897