Finite Group Databasebeta

← #27153610

Subgroups — #27153610

Order 12416.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261440C211yesmaximal, characteristic
2#20261398C211yesmaximal, characteristic
3#20261084C211yesmaximal, characteristic, complemented
4#20261072C211yesmaximal, characteristic, complemented
5#20261029C211yesmaximal, characteristic, complemented
6#20260977C211yesmaximal, characteristic, complemented
7#20260974C211yesmaximal, characteristic, complemented
8#20260944C211yesmaximal, characteristic, complemented
9#20260893C211yesmaximal, characteristic, complemented
10#20260892C211yesmaximal, characteristic, complemented
11#20260829C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic
13C2×#15690733C211yesmaximal, characteristic, complemented
14#103×D97C211yesmaximal, characteristic, complemented
15#484×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18C2×Dic388C2211yescharacteristic
19C2×Dic388C2211yescharacteristic
20C2×#13095501C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23#15690809—21no—
24#15690809C2211yescharacteristic
25#15690808C2211yescharacteristic
26#15690800—21no—
27#15690790C2211yescharacteristic
28#15690790C2211yescharacteristic
29#15690789—21no—
30#15690789C2211yescharacteristic
31#15690788—21no—
32#15690788—21no—
33#15690787—21no—
34#15690775C2211yescharacteristic
35#15690775C2211yescharacteristic
36C4×C22⋊D97—21no—
37C4×C22⋊D97C2211yescharacteristic
38#15690770—21no—
39C4⋊C4×D97C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690735C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690734C2211yescharacteristic
45#15690733C2211yescharacteristic
46#15690733C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690733C2211yescharacteristic
49#15690733C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C22⋊Q8×C97C2211yescharacteristic
53C2×C4⋊C4×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59C2×Dic194—22no—
60C2×Dic194—21no—
61C2×Dic194—21no—
62C2×Dic194—21no—
63C2×Dic194—41no—
64C2×Dic194C2311yescharacteristic
65C2×Dic194C2311yescharacteristic
66C2×Dic194C2311yescharacteristic
67Q8×D97—22no—
68Q8×D97—41no—
69C2×C22⋊D97D411yescharacteristic
70C22×Dic97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97C2311yescharacteristic
74Dic388—22no—
75Dic388—41no—
76#13095501—41no—
77C2×#11715895—41no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C8×D97—41no—
81C4⋊Dic97—21no—
82C4⋊Dic97—21no—
83C4⋊Dic97C2311yescharacteristic
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88#13095490—21no—
89#13095490—21no—
90#13095490—21no—
91#13095490D411yescharacteristic
92C22⋊Dic97—21no—
93C22⋊Dic97D411yescharacteristic
94#13095488—21no—
95#13095488—21no—
96#13095488D411yescharacteristic
97C4×Dic97—21no—
98C4×Dic97—21no—
99C4×Dic97C2311yescharacteristic
100C2×Q8×C97C2311yescharacteristic
101C22×C4×C97—21no—
102C22×C4×C97C2311yescharacteristic
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97—21no—
106C4⋊C4×C97—21no—
107C4⋊C4×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C23×C97C2×D411yescharacteristic
112C22×D97C2×D411yescharacteristic
113Q8×C97—21no—
114Q8×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—81no—
122C2×C4×C97C2×D411yescharacteristic
123C2×C4×C97C2411yescharacteristic
124C22⋊D97Q1614yescomplemented
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—42no—
130C2×Dic97C2×D411yescharacteristic
131C2×Dic97C2×D411yescharacteristic
132C2×Dic97C2×D411yescharacteristic
133C4×D97—21no—
134C4×D97—21no—
135C4×D97—42no—
136C4×D97—81no—
137Dic194—21no—
138Dic194—21no—
139Dic194—21no—
140Dic194—21no—
141Dic194—21no—
142Dic194—42no—
143Dic194—42no—
144Dic194—81no—
145Dic194#4112yes—
146C8×C97—21no—
147C8×C97—21no—
148#11715895—21no—
149#11715895—21no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C97C2×Q1612yes—
153C22×C97C22×D411yescharacteristic
154C2×D97—21no—
155C2×D97—21no—
156C2×D97C2×Q1612yes—
157C4×C97—21no—
158C4×C97—41no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97C2×#4111yescharacteristic
163C4×C97H(5,𝔽2)11yescharacteristic
164Dic97—22no—
165Dic97—21no—
166Dic97—21no—
167Dic97—41no—
168Dic97—41no—
169Dic97—41no—
170Dic97C2×Q1612yes—
171C2×C97—22no—
172C2×C97#54611yescharacteristic
173C2×C97C22×Q1611yescharacteristic
174C2×C97#57711yescharacteristic
175D97—22no—
176#3299—971nomaximal, Sylow
177C97#329911yescharacteristic, complemented, Sylow
178#554—971no—
179D4×Q8—971no—
180#487—971no—
181#484—971no—
182#484D9711yescharacteristic, complemented
183#477—971no—
184#467—971no—
185#467—971no—
186#462—971no—
187#451—971no—
188#451—971no—
189C4×Q16—971no—
190C8×D4—971no—
191C2×#103—971no—
192C2×#103—971no—
193C22×Q8—971no—
194C22×Q8—971no—
195C2×Q16—971no—
196C2×Q16—971no—
197C2×Q16—971no—
198C22×C8—971no—
199C22×C8—971no—
200C4⋊Q8—971no—
201C4⋊Q8—971no—
202C4⋊Q8—1941no—
203#125—1941no—
204C22⋊Q8—971no—
205C22⋊Q8—971no—
206C22⋊Q8—971no—
207C22⋊Q8—1941no—
208C22⋊Q8—1941no—
209C22⋊Q8—1941no—
210C22⋊Q8—1941no—
211C22⋊Q8C2×D9711yescharacteristic
212C4×Q8—971no—
213C4×Q8—971no—
214C4×D4—971no—
215C4×D4—1941no—
216C4×D4—1941no—
217C2×C4⋊C4—971no—
218C2×C4⋊C4C2×D9711yescharacteristic
219#107—971no—
220#107C2×D9711yescharacteristic
221#107C2×D9711yescharacteristic
222C4⋊C8—971no—
223#103—971no—
224#103—971no—
225#103—971no—
226#103—971no—
227#103—971no—
228#103—971no—
229#103—971no—
230#103—971no—
231#103C2×D9711yescharacteristic
232#103C2×D9711yescharacteristic
233C22⋊C8—971no—
234C22⋊C8C2×D9711yescharacteristic
235C4×C8—971no—
236C2×Q8—971no—
237C2×Q8—971no—
238C2×Q8—971no—
239C2×Q8—3881no—
240C2×Q8—3881no—
241C2×Q8—1941no—
242C2×Q8—1941no—
243C2×Q8—1942no—
244C2×Q8—1942no—
245C2×Q8—1941no—
246C2×Q8C22×D9711yescharacteristic
247C2×D4—971no—
248C22×C4—21no—
249C22×C4—971no—
250C22×C4—1941no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4C22×D9711yescharacteristic
254Q16—3881no—
255Q16—3881no—
256Q16—1942no—
257C2×C8—971no—
258C2×C8—971no—
259C2×C8—3881no—
260C2×C8—3881no—
261C2×C8C22×D9711yescharacteristic
262C2×C8C22×D9711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—1941no—
268C4⋊C4—1941no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4—1941no—
274C4⋊C4—1941no—
275C4⋊C4—1941no—
276C4⋊C4C22×D9711yescharacteristic
277C4⋊C4C22×D9711yescharacteristic
278C4⋊C4C22×D9711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—971no—
281C22⋊C4—971no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C42—971no—
286C42—1941no—
287C42—1941no—
288C23—971no—
289C23D4×D9711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—7761no—
293Q8—972no—
294Q8—3882no—
295Q8—3882no—
296Q8—1941no—
297Q8—1941no—
298Q8—1941no—
299Q8—1941no—
300Q8—1941no—
301D4—974no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—81no—
309C2×C4—7761no—
310C2×C4—971no—
311C2×C4—971no—
312C2×C4—971no—
313C2×C4—3882no—
314C2×C4—3882no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4D4×D9711yescharacteristic
322C2×C4C23×D9711yescharacteristic
323C8—21no—
324C8—21no—
325C8—1941no—
326C8—1941no—
327C22—21no—
328C22—21no—
329C22—972no—
330C22—1941no—
331C22—1941no—
332C22Q16×D9712yes—
333C22C2×D4×D9711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—972no—
340C4—3881no—
341C4—3881no—
342C4—3881no—
343C4—1941no—
344C4—1942no—
345C4—1941no—
346C4#41×D9711yescharacteristic
347C4#1569085611yescharacteristic
348C2—22no—
349C2—1942no—
350C2#2026137711yescharacteristic
351C2C2×Q16×D9711yescharacteristic
352C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#484×C97C2
socleC22×C97C22×D4
prime radicalC2×C22⋊D97D4
hypercenter#484D97
hypocenterC97#3299
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #20261084C2semidirect
4. #20261072C2semidirect
5. #20261029C2semidirect
6. #20260977C2semidirect
7. #20260974C2semidirect
8. #20260944C2semidirect
9. #20260893C2semidirect
10. #20260892C2semidirect
11. #20260829C2semidirect
13. C2×#15690733C2semidirect
14. #103×D97C2semidirect
15. #484×C97C2semidirect
124. C22⋊D97Q16semidirect
177. C97#3299semidirect
182. #484D97semidirect

Maximal subgroups

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

subgroupindex
#202614402
#202613982
#202610842
#202610722
#202610292
#202609772
#202609742
#202609442
#202608932
#202608922
#202608292
#202607982
C2×#156907332
#103×D972
#484×C972
#329997