Finite Group Databasebeta

← #27153729

Subgroups — #27153729

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261394C211yesmaximal, characteristic, complemented
2#20261389C211yesmaximal, characteristic, complemented
3#20261095C211yesmaximal, characteristic, complemented
4#20261071C211yesmaximal, characteristic, complemented
5#20261049C211yesmaximal, characteristic, complemented
6#20261031C211yesmaximal, characteristic, complemented
7#20260981C211yesmaximal, characteristic, complemented
8#20260971C211yesmaximal, characteristic, complemented
9#20260951C211yesmaximal, characteristic, complemented
10#20260882C211yesmaximal, characteristic, complemented
11#20260876C211yesmaximal, characteristic, complemented
12#20260837C211yesmaximal, characteristic, complemented
13#20260737C211yesmaximal, characteristic, complemented
14C2×#15690735C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095510—21no—
18C22×Dic194C2211yescharacteristic
19C2×#13095500C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21C2×D776C2211yescharacteristic
22#15690810C2211yescharacteristic
23#15690798C2211yescharacteristic
24#15690796—21no—
25#15690794—21no—
26#15690794—21no—
27#15690790—21no—
28#15690789C2211yescharacteristic
29#15690788—21no—
30#15690784C2211yescharacteristic
31#15690784C2211yescharacteristic
32#15690784C2211yescharacteristic
33#15690782—21no—
34C4×Dic194C2211yescharacteristic
35C4×D388C2211yescharacteristic
36#15690770—21no—
37C2×C4⋊Dic97C2211yescharacteristic
38C2×C4⋊Dic97C2211yescharacteristic
39#15690744C2211yescharacteristic
40#15690744C2211yescharacteristic
41#15690744C2211yescharacteristic
42#15690735C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690735C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52C2×M16×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57C22⋊C8×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—41no—
61#13095510—44no—
62C2×Dic194—22no—
63C2×Dic194—21no—
64C2×Dic194—21no—
65C2×Dic194—41no—
66C2×Dic194C2311yescharacteristic
67C2×Dic194C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97—21no—
74C22×Dic97—21no—
75C22×Dic97—21no—
76C22×Dic97—21no—
77C2×C4×D97—21no—
78C2×C4×D97—21no—
79#13095500—22no—
80#13095500—41no—
81D776—41no—
82C4⋊Dic97—21no—
83C4⋊Dic97—21no—
84C4⋊Dic97—21no—
85C4⋊Dic97C2311yescharacteristic
86C4⋊Dic97C2311yescharacteristic
87C4⋊Dic97C2311yescharacteristic
88C4⋊Dic97C2311yescharacteristic
89#13095490—21no—
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488—21no—
97#13095488—21no—
98C4×Dic97—21no—
99C2×D4×C97D411yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102M16×C97—41no—
103C2×C8×C97—41no—
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97D411yescharacteristic
109C22⋊C4×C97D411yescharacteristic
110C22⋊C4×C97D411yescharacteristic
111C42×C97C2311yescharacteristic
112C23×C97C2×D411yescharacteristic
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115C22×D97—21no—
116D4×C97—22no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97C2×D411yescharacteristic
122C2×C4×C97C2×D411yescharacteristic
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2411yescharacteristic
126C22⋊D97—42no—
127C22⋊D97—42no—
128C22⋊D97—42no—
129C22⋊D97—42no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—42no—
137C2×Dic97—42no—
138C2×Dic97—42no—
139D388—21no—
140D388—21no—
141D388—21no—
142D388—21no—
143C4×D97—42no—
144C4×D97—42no—
145Dic194—21no—
146Dic194—21no—
147Dic194—21no—
148Dic194—42no—
149Dic194—42no—
150Dic194#4112yescomplemented
151C8×C97—21no—
152C8×C97—21no—
153C8×C97—21no—
154C8×C97—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—21no—
159C22×C97—21no—
160C22×C97AGL(1,ℤ/8)12yes—
161C22×C97C22×D411yescharacteristic
162C2×D97—41no—
163C2×D97—41no—
164C2×D97—41no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—41no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97—21no—
171C4×C97—21no—
172C4×C97C2×#4111yescharacteristic
173C4×C97H(5,𝔽2)11yescharacteristic
174Dic97—22no—
175Dic97—41no—
176Dic97—41no—
177Dic97—41no—
178Dic97—41no—
179Dic97—41no—
180C2×C97—22no—
181C2×C97—41no—
182C2×C97#54611yescharacteristic
183C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C97#57811yescharacteristic
185D97—81no—
186D97—81no—
187#3308—971nomaximal, Sylow
188C97#330811yescharacteristic, complemented, Sylow
189#548—971no—
190#547—971no—
191#489—971no—
192#484—971no—
193#480—971no—
194#477—971no—
195#468—971no—
196#466—971no—
197#463—971no—
198#449—971no—
199#448—971no—
200#440—971no—
201#435D9711yescharacteristic, complemented
202#417—971no—
203C2×#103—971no—
204C2×#41—971no—
205C2×#41—1941no—
206C22×Q8—971no—
207C2×SD16—971no—
208C2×SD16—971no—
209C2×D8—971no—
210C2×M16C2×D9711yescharacteristic
211C22×C8C2×D9711yescharacteristic
212C4⋊Q8—971no—
213#124—971no—
214#124—1941no—
215#123—1941no—
216#123—1941no—
217C22⋊Q8—971no—
218C22⋊Q8—1941no—
219C22⋊Q8—1941no—
220#121—971no—
221#121—971no—
222#121—971no—
223#121—1941no—
224C4×Q8—971no—
225C4×D4—971no—
226C4×D4—1941no—
227C4×D4C2×D9711yescharacteristic
228C2×C4⋊C4—971no—
229C2×C4⋊C4—971no—
230#107—971no—
231#107—971no—
232#107—971no—
233C4⋊C8C2×D9711yescharacteristic
234#103—971no—
235#103—971no—
236#103—971no—
237#103—971no—
238#103—971no—
239#103—971no—
240#102—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244C22⋊C8C2×D9711yescharacteristic
245C22⋊C8C2×D9711yescharacteristic
246#97C2×D9711yescharacteristic
247#41—3881no—
248#41—3884no—
249#41—1942no—
250C2×Q8—971no—
251C2×Q8—971no—
252C2×Q8—3881no—
253C2×Q8—1942no—
254C2×Q8—1941no—
255C2×Q8—1941no—
256C2×D4—971no—
257C2×D4—971no—
258C2×D4—1941no—
259C2×D4—1941no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4D38811yescharacteristic
263C22×C4—1941no—
264C22×C4—1941no—
265C22×C4—1941no—
266C22×C4—1941no—
267C22×C4—1941no—
268C22×C4C22×D9711yescharacteristic
269C22×C4C22×D9711yescharacteristic
270SD16—3881no—
271SD16—1942no—
272D8—3881no—
273M16—41no—
274C2×C8—41no—
275C2×C8C22×D9711yescharacteristic
276C2×C8C22×D9711yescharacteristic
277C2×C8C22×D9711yescharacteristic
278C2×C8C22×D9711yescharacteristic
279C4⋊C4—971no—
280C4⋊C4—971no—
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—1941no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4D38811yescharacteristic
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4D38811yescharacteristic
297C22⋊C4D38811yescharacteristic
298C42—1941no—
299C42C22×D9711yescharacteristic
300C23—1941no—
301C23—1941no—
302C23C2×D38811yescharacteristic
303C23C2×D38811yescharacteristic
304Q8—972no—
305Q8—3882no—
306Q8—3882no—
307Q8—1941no—
308Q8—1941no—
309Q8—1941no—
310D4—22no—
311D4—3882no—
312D4—3882no—
313D4—3882no—
314D4—3882no—
315D4—1941no—
316D4—1941no—
317D4—1941no—
318D4—1941no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3882no—
324C2×C4—3882no—
325C2×C4—3882no—
326C2×C4—3882no—
327C2×C4—3882no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4—1941no—
333C2×C4—1941no—
334C2×C4C2×D38811yescharacteristic
335C2×C4C2×D38811yescharacteristic
336C2×C4C2×D38811yescharacteristic
337C2×C4C2×D38811yescharacteristic
338C2×C4C23×D9711yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—3881no—
349C22—3881no—
350C22—3881no—
351C22—3881no—
352C22—3881no—
353C22—3881no—
354C22#1569083812yes—
355C22C22×D38811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—3881no—
361C4—3881no—
362C4—3881no—
363C4—3881no—
364C4—3881no—
365C4—1942no—
366C4#41×D9711yescharacteristic
367C4#1569085511yescharacteristic
368C2—22no—
369C2—41no—
370C2—7761no—
371C2—7761no—
372C2#2026138011yescharacteristic
373C2C2×#1569083811yescharacteristic
374C2#2026155211yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#435×C97C2
socleC22×C97C22×D4
hypercenter#435D97
hypocenterC97#3308
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #20261394C2semidirect
2. #20261389C2semidirect
3. #20261095C2semidirect
4. #20261071C2semidirect
5. #20261049C2semidirect
6. #20261031C2semidirect
7. #20260981C2semidirect
8. #20260971C2semidirect
9. #20260951C2semidirect
10. #20260882C2semidirect
11. #20260876C2semidirect
12. #20260837C2semidirect
13. #20260737C2semidirect
14. C2×#15690735C2semidirect
15. #435×C97C2semidirect
150. Dic194#41semidirect
188. C97#3308semidirect
201. #435D97semidirect

Maximal subgroups

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

subgroupindex
#202613942
#202613892
#202610952
#202610712
#202610492
#202610312
#202609812
#202609712
#202609512
#202608822
#202608762
#202608372
#202607372
C2×#156907352
#435×C972
#330897