Finite Group Databasebeta

← #27151929

Subgroups — #27151929

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#20261215C211yesmaximal, characteristic, complemented
2#20261051C211yesmaximal, characteristic, complemented
3#20261046C211yesmaximal, characteristic, complemented
4#20261039C211yesmaximal, characteristic, complemented
5#20261032C211yesmaximal, characteristic, complemented
6#20260932C211yesmaximal, characteristic, complemented
7#20260927C211yesmaximal, characteristic, complemented
8#20260878C211yesmaximal, characteristic, complemented
9#20260878C211yesmaximal, characteristic, complemented
10#20260845C211yesmaximal, characteristic, complemented
11#20260835C211yesmaximal, characteristic, complemented
12C4×#13095499C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#540×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095499C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C4×Dic194C2211yescharacteristic
22C4×Dic194C2211yescharacteristic
23C4×D388C2211yescharacteristic
24C4×D388C2211yescharacteristic
25#15690770—21no—
26#15690770—21no—
27#15690768—21no—
28C2×C4⋊Dic97C2211yescharacteristic
29C4⋊C4×D97—21no—
30#15690743C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690733C2211yescharacteristic
37#15690733C2211yescharacteristic
38#15690731C2211yescharacteristic
39#15690731C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48#127×C97C2211yescharacteristic
49#125×C97C2211yescharacteristic
50#123×C97—21no—
51#123×C97—21no—
52#121×C97—21no—
53#121×C97—21no—
54#121×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58#117×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—22no—
61C2×Dic194C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C22×Dic97—21no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68#13095499—22no—
69#13095499—22no—
70#13095496—22no—
71#13095496—22no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095488—21no—
82#13095488—21no—
83C4×Dic97—21no—
84C4×Dic97—21no—
85C2×D4×C97—21no—
86C2×D4×C97—21no—
87C2×D4×C97—21no—
88C2×D4×C97—21no—
89C2×D4×C97C2311yescharacteristic
90C2×D4×C97C2311yescharacteristic
91C22×C4×C97—21no—
92C22×C4×C97—21no—
93C22×C4×C97C2311yescharacteristic
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97D411yescharacteristic
97C4⋊C4×C97D411yescharacteristic
98C4⋊C4×C97C2311yescharacteristic
99C4⋊C4×C97C2311yescharacteristic
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97—21no—
104C22⋊C4×C97D411yescharacteristic
105C22⋊C4×C97D411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97—21no—
110C23×C97C2×D411yescharacteristic
111C22×D97—21no—
112D4×C97—21no—
113D4×C97—21no—
114D4×C97—21no—
115D4×C97—21no—
116D4×C97—81no—
117D4×C97—81no—
118D4×C97—81no—
119D4×C97—81no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—81no—
125C2×C4×C97—81no—
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—24no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—81no—
137D388—21no—
138D388#4112yes—
139C4×D97—24no—
140C4×D97—81no—
141C4×D97—81no—
142Dic194—21no—
143Dic194#4112yes—
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×C97—41no—
155C22×C97—41no—
156C22×C97—41no—
157C22×C97C22×D411yescharacteristic
158C2×D97—22no—
159C2×D97—41no—
160C2×D97—41no—
161C4×C97—21no—
162C4×C97—21no—
163C4×C97—21no—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97—41no—
167C4×C97—41no—
168C4×C97C2×#4111yescharacteristic
169C4×C97C2×#4111yescharacteristic
170Dic97—22no—
171Dic97—41no—
172Dic97—41no—
173C2×C97—41no—
174C2×C97—81no—
175C2×C97—81no—
176C2×C97#52511yescharacteristic
177C2×C97#57611yescharacteristic
178C2×C97#57711yescharacteristic
179D97—42no—
180#3119—971nomaximal, Sylow
181C97#311911yescharacteristic, complemented, Sylow
182#540D9711yescharacteristic, complemented
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#433—971no—
197C2×#41—971no—
198C2×SD16—971no—
199C2×SD16—971no—
200C2×D8—971no—
201C2×D8—971no—
202#127C2×D9711yescharacteristic
203#125C2×D9711yescharacteristic
204#123—21no—
205#123—21no—
206#121—21no—
207#121—21no—
208#121C2×D9711yescharacteristic
209#121C2×D9711yescharacteristic
210C4×Q8—971no—
211C4×Q8—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—1941no—
215C4×D4—1941no—
216C4×D4C2×D9711yescharacteristic
217C4×D4C2×D9711yescharacteristic
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⋊C8—971no—
227C4⋊C8—971no—
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⋊C8—971no—
237C22⋊C8—971no—
238#97—971no—
239C4×C8—971no—
240#41—1942no—
241#41—1942no—
242C2×Q8—971no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—971no—
248C2×D4—1941no—
249C2×D4C22×D9711yescharacteristic
250C2×D4C22×D9711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4—1941no—
257C22×C4C22×D9711yescharacteristic
258SD16—1942no—
259SD16—1942no—
260D8—1942no—
261D8—1942no—
262C2×C8—971no—
263C2×C8—971no—
264C2×C8—971no—
265C2×C8—971no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4C22⋊D9711yescharacteristic
274C4⋊C4C22⋊D9711yescharacteristic
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4C22⋊D9711yescharacteristic
284C22⋊C4C22⋊D9711yescharacteristic
285C42—1941no—
286C42—1941no—
287C42C22×D9711yescharacteristic
288C42C22×D9711yescharacteristic
289C23—21no—
290C23—21no—
291C23—1941no—
292C23C2×C22⋊D9711yescharacteristic
293Q8—972no—
294Q8—1941no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—972no—
304D4—1944no—
305D4—1941no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—7761no—
313C2×C4—7761no—
314C2×C4—7761no—
315C2×C4—1944no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4C2×C22⋊D9711yescharacteristic
320C2×C4C2×C22⋊D9711yescharacteristic
321C2×C4C2×C22⋊D9711yescharacteristic
322C2×C4C2×C22⋊D9711yescharacteristic
323C2×C4C2×C22⋊D9711yescharacteristic
324C2×C4C23×D9711yescharacteristic
325C8—1941no—
326C8—1941no—
327C8—1941no—
328C8—1941no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3881no—
339C22—3881no—
340C22—1942no—
341C22C22×C22⋊D9711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—3881no—
350C4—3881no—
351C4—1942no—
352C4#41×D9711yescharacteristic
353C4#41×D9711yescharacteristic
354C2—41no—
355C2—81no—
356C2—81no—
357C2—3882no—
358C2#2026124211yescharacteristic
359C2#2026153811yescharacteristic
360C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#540×C97C2
socleC22×C97C22×D4
hypercenter#540D97
hypocenterC97#3119
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20261215C2semidirect
2. #20261051C2semidirect
3. #20261046C2semidirect
4. #20261039C2semidirect
5. #20261032C2semidirect
6. #20260932C2semidirect
7. #20260927C2semidirect
8. #20260878C2semidirect
9. #20260878C2semidirect
10. #20260845C2semidirect
11. #20260835C2semidirect
12. C4×#13095499C2semidirect
13. C4×#13095496C2semidirect
14. #20260796C2semidirect
15. #540×C97C2semidirect
181. C97#3119semidirect
182. #540D97semidirect

Maximal subgroups

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

subgroupindex
#202612152
#202610512
#202610462
#202610392
#202610322
#202609322
#202609272
#202608782
#202608452
#202608352
C4×#130954992
C4×#130954962
#202607962
#540×C972
#311997