Finite Group Databasebeta

← #27151937

Subgroups — #27151937

Order 12416.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261340C211yesmaximal, characteristic, complemented
2#20261215C211yesmaximal, characteristic, complemented
3#20261054C211yesmaximal, characteristic, complemented
4#20261049C211yesmaximal, characteristic, complemented
5#20261043C211yesmaximal, characteristic, complemented
6#20261036C211yesmaximal, characteristic, complemented
7#20260936C211yesmaximal, characteristic, complemented
8#20260931C211yesmaximal, characteristic, complemented
9#20260882C211yesmaximal, characteristic, complemented
10#20260882C211yesmaximal, characteristic, complemented
11#20260848C211yesmaximal, characteristic, complemented
12#20260837C211yesmaximal, characteristic, complemented
13C4×#13095500C211yesmaximal, characteristic, complemented
14C4×D776C211yesmaximal, characteristic, complemented
15#433×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095500C2211yescharacteristic
18C2×#13095500C2211yescharacteristic
19C2×D776C2211yescharacteristic
20C2×D776C2211yescharacteristic
21#15690807C2211yescharacteristic
22#15690802C2211yescharacteristic
23#15690795—21no—
24#15690795—21no—
25#15690784C2211yescharacteristic
26#15690784C2211yescharacteristic
27#15690783—21no—
28#15690783—21no—
29C4×Dic194C2211yescharacteristic
30C4×Dic194C2211yescharacteristic
31C4×D388C2211yescharacteristic
32C4×D388C2211yescharacteristic
33C4×D388C2211yescharacteristic
34C4×D388C2211yescharacteristic
35#15690770—21no—
36#15690770—21no—
37#15690768—21no—
38C2×C4⋊Dic97C2211yescharacteristic
39C4⋊C4×D97—21no—
40#15690744C2211yescharacteristic
41#15690744C2211yescharacteristic
42#15690742C2211yescharacteristic
43#15690742C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690735C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690732C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52#117×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic
54C4⋊C8×C97C2211yescharacteristic
55C22⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—22no—
61C2×Dic194C2311yescharacteristic
62C2×D388—21no—
63C2×D388—21no—
64C2×D388C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C22×Dic97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97—21no—
76#13095500—22no—
77#13095500—22no—
78D776—22no—
79D776—22no—
80C4⋊Dic97—21no—
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88#13095490—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—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2×D411yescharacteristic
117C2×C4×C97C2×D411yescharacteristic
118C2×C4×C97C2×D411yescharacteristic
119C2×C4×C97C2411yescharacteristic
120C22⋊D97—24no—
121C22⋊D97—81no—
122C22⋊D97—81no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—81no—
129D388—21no—
130D388—21no—
131D388—21no—
132D388—21no—
133D388—21no—
134D388—81no—
135D388—81no—
136D388#4112yes—
137C4×D97—24no—
138C4×D97—81no—
139C4×D97—81no—
140C4×D97—81no—
141C4×D97—81no—
142Dic194—21no—
143Dic194#4112yes—
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—41no—
154C2×D97—41no—
155C2×D97—41no—
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—41no—
159C2×D97—41no—
160C2×D97—41no—
161C4×C97—21no—
162C4×C97—21no—
163C4×C97—21no—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97C2×#4111yescharacteristic
167C4×C97C2×#4111yescharacteristic
168Dic97—22no—
169Dic97—41no—
170Dic97—41no—
171Dic97—41no—
172Dic97—41no—
173C2×C97—41no—
174C2×C97#52511yescharacteristic
175C2×C97#57611yescharacteristic
176C2×C97#57711yescharacteristic
177D97—42no—
178D97—81no—
179D97—81no—
180#3119—971nomaximal, Sylow
181C97#311911yescharacteristic, complemented, Sylow
182#540—971no—
183#520—971no—
184#481—971no—
185#480—971no—
186#479—971no—
187#478—971no—
188#460—971no—
189#459—971no—
190#449—971no—
191#449—971no—
192#442—971no—
193#440—971no—
194C4×SD16—971no—
195C4×D8—971no—
196#433D9711yescharacteristic, complemented
197C2×#41—971no—
198C2×SD16—971no—
199C2×SD16—971no—
200C2×D8—971no—
201C2×D8—971no—
202#127—971no—
203#125—971no—
204#123—1941no—
205#123—1941no—
206#121—971no—
207#121—971no—
208#121—1941no—
209#121—1941no—
210C4×Q8—971no—
211C4×Q8—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—1941no—
217C4×D4—1941no—
218#117—1941no—
219#117C2×D9711yescharacteristic
220C2×C4⋊C4—971no—
221C2×C4⋊C4—1941no—
222#107—971no—
223#107—971no—
224#106—971no—
225#106—971no—
226C4⋊C8C2×D9711yescharacteristic
227C4⋊C8C2×D9711yescharacteristic
228#103—971no—
229#103—971no—
230#102—971no—
231#102—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235#102—971no—
236C22⋊C8C2×D9711yescharacteristic
237C22⋊C8C2×D9711yescharacteristic
238#97C2×D9711yescharacteristic
239C4×C8C2×D9711yescharacteristic
240#41—1942no—
241#41—1942no—
242C2×Q8—971no—
243C2×D4—971no—
244C2×D4—971no—
245C2×D4—971no—
246C2×D4—1941no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4—1941no—
250C2×D4—1941no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4—1941no—
257C22×C4C22×D9711yescharacteristic
258SD16—1942no—
259SD16—1942no—
260D8—1942no—
261D8—1942no—
262C2×C8C22×D9711yescharacteristic
263C2×C8C22×D9711yescharacteristic
264C2×C8C22×D9711yescharacteristic
265C2×C8C22×D9711yescharacteristic
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4—1941no—
274C4⋊C4—1941no—
275C4⋊C4D38811yescharacteristic
276C4⋊C4D38811yescharacteristic
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4D38811yescharacteristic
284C22⋊C4D38811yescharacteristic
285C42—1941no—
286C42—1941no—
287C42C22×D9711yescharacteristic
288C42C22×D9711yescharacteristic
289C23—1941no—
290C23—1941no—
291C23—1941no—
292C23C2×D38811yescharacteristic
293Q8—972no—
294Q8—1941no—
295D4—7761no—
296D4—7761no—
297D4—7761no—
298D4—7761no—
299D4—972no—
300D4—1944no—
301D4—1941no—
302D4—1941no—
303D4—1941no—
304D4—1941no—
305D4—1941no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—7761no—
309C2×C4—7761no—
310C2×C4—7761no—
311C2×C4—7761no—
312C2×C4—7761no—
313C2×C4—1944no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4C2×D38811yescharacteristic
320C2×C4C2×D38811yescharacteristic
321C2×C4C2×D38811yescharacteristic
322C2×C4C2×D38811yescharacteristic
323C2×C4C2×D38811yescharacteristic
324C2×C4C23×D9711yescharacteristic
325C8—21no—
326C8—21no—
327C8—21no—
328C8—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—3881no—
333C22—3881no—
334C22—3881no—
335C22—3881no—
336C22—3881no—
337C22—3881no—
338C22—3881no—
339C22—3881no—
340C22—1942no—
341C22C22×D38811yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—3881no—
348C4—3881no—
349C4—3881no—
350C4—3881no—
351C4—1942no—
352C4#41×D9711yescharacteristic
353C4#41×D9711yescharacteristic
354C2—41no—
355C2—7761no—
356C2—7761no—
357C2—3882no—
358C2#2026124711yescharacteristic
359C2#2026154011yescharacteristic
360C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#433×C97C2
socleC22×C97C22×D4
hypercenter#433D97
hypocenterC97#3119
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. #20261340C2semidirect
2. #20261215C2semidirect
3. #20261054C2semidirect
4. #20261049C2semidirect
5. #20261043C2semidirect
6. #20261036C2semidirect
7. #20260936C2semidirect
8. #20260931C2semidirect
9. #20260882C2semidirect
10. #20260882C2semidirect
11. #20260848C2semidirect
12. #20260837C2semidirect
13. C4×#13095500C2semidirect
14. C4×D776C2semidirect
15. #433×C97C2semidirect
181. C97#3119semidirect
196. #433D97semidirect

Maximal subgroups

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

subgroupindex
#202613402
#202612152
#202610542
#202610492
#202610432
#202610362
#202609362
#202609312
#202608822
#202608482
#202608372
C4×#130955002
C4×D7762
#433×C972
#311997