Finite Group Databasebeta

← #27153597

Subgroups — #27153597

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D97C211yesmaximal, characteristic, complemented
2#20261386C211yesmaximal, characteristic
3#20261084C211yesmaximal, characteristic, complemented
4#20261065C211yesmaximal, characteristic, complemented
5#20261064C211yesmaximal, characteristic, complemented
6#20261018C211yesmaximal, characteristic, complemented
7#20260967C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260939C211yesmaximal, characteristic, complemented
10#20260875C211yesmaximal, characteristic, complemented
11#20260874C211yesmaximal, characteristic, complemented
12#20260820C211yesmaximal, characteristic, complemented
13C8×C22⋊D97C211yesmaximal, characteristic, complemented
14#103×D97C211yesmaximal, characteristic, complemented
15C2×#103×C97C211yesmaximal, characteristic, complemented
16C2×#13095511—21no—
17C2×Q8×D97C2211yescharacteristic
18C2×#13095500C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20C2×#13095498C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22#15690809C2211yescharacteristic
23#15690808—21no—
24#15690808C2211yescharacteristic
25#15690795—21no—
26#15690793—21no—
27#15690789—21no—
28#15690789C2211yescharacteristic
29#15690788—21no—
30#15690786—21no—
31#15690786C2211yescharacteristic
32#15690785C2211yescharacteristic
33#15690784C2211yescharacteristic
34Q8×Dic97C2211yescharacteristic
35#15690772C2211yescharacteristic
36C4×C22⋊D97—21no—
37C4×C22⋊D97C2211yescharacteristic
38C4⋊C4×D97C2211yescharacteristic
39#15690742C2211yescharacteristic
40#15690741C2211yescharacteristic
41#15690741C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690734C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690733C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690732C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C22×Q8×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C2×C4⋊C4×C97C2211yescharacteristic
55#103×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59#13095511—44no—
60C2×Dic194—21no—
61C2×Dic194—21no—
62C2×Dic194C2311yescharacteristic
63Q8×D97—22no—
64Q8×D97—41no—
65C2×D388—21no—
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97D411yescharacteristic
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97C2311yescharacteristic
73#13095500—41no—
74#13095498—22no—
75#13095498—41no—
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C8×D97—41no—
79C4⋊Dic97—21no—
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88#13095490D411yescharacteristic
89C22⋊Dic97—21no—
90C22⋊Dic97D411yescharacteristic
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488D411yescharacteristic
96C4×Dic97—21no—
97C4×Dic97C2311yescharacteristic
98C2×Q8×C97—21no—
99C2×Q8×C97—22no—
100C2×Q8×C97—41no—
101C2×Q8×C97C2311yescharacteristic
102C2×Q8×C97C2311yescharacteristic
103C22×C4×C97—21no—
104C22×C4×C97—21no—
105C22×C4×C97C2311yescharacteristic
106C2×C8×C97—41no—
107C2×C8×C97C2311yescharacteristic
108C2×C8×C97C2311yescharacteristic
109C4⋊C4×C97—21no—
110C4⋊C4×C97C2311yescharacteristic
111C4⋊C4×C97C2311yescharacteristic
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97C2×D411yescharacteristic
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—21no—
118Q8×C97—42no—
119Q8×C97#4112yescomplemented
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—42no—
127C2×C4×C97—42no—
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—42no—
131C22⋊D97SD1614yescomplemented
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138D388—21no—
139D388—21no—
140D388—42no—
141C4×D97—21no—
142C4×D97—21no—
143C4×D97—42no—
144C4×D97—42no—
145C4×D97—42no—
146Dic194—21no—
147Dic194—21no—
148Dic194—42no—
149Dic194—42no—
150C8×C97—21no—
151C8×C97—21no—
152#11715895—21no—
153#11715895—21no—
154C22×C97—21no—
155C22×C97—21no—
156C22×C97C2×SD1612yes—
157C22×C97C22×D411yescharacteristic
158C2×D97—21no—
159C2×D97—21no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97C2×SD1612yes—
164C4×C97—22no—
165C4×C97—21no—
166C4×C97—41no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97C2×#4111yescharacteristic
170C4×C97H(5,𝔽2)11yescharacteristic
171Dic97—21no—
172Dic97—21no—
173Dic97—41no—
174Dic97—41no—
175Dic97—41no—
176Dic97C2×SD1612yes—
177C2×C97—22no—
178C2×C97#54611yescharacteristic
179C2×C97C22×SD1611yescharacteristic
180C2×C97#57811yescharacteristic
181D97—22no—
182D97—81no—
183#3298—971nomaximal, Sylow
184C97#329811yescharacteristic, complemented, Sylow
185D4×Q8—971no—
186#547—971no—
187#487—971no—
188#483—971no—
189#483—971no—
190#475—971no—
191#465—971no—
192#465—971no—
193#461—971no—
194#448—971no—
195#448—971no—
196C4×SD16—971no—
197C8×D4—971no—
198C2×#103—971no—
199C2×#103D9711yescharacteristic, complemented
200C2×#41—1941no—
201C22×Q8—971no—
202C22×Q8C2×D9711yescharacteristic
203C2×SD16—971no—
204C2×SD16—971no—
205C2×SD16—971no—
206C22×C8—971no—
207C22×C8C2×D9711yescharacteristic
208C4⋊Q8—971no—
209C4⋊Q8—971no—
210C4⋊Q8—1941no—
211#123—1941no—
212#123—1941no—
213C22⋊Q8—971no—
214C22⋊Q8—971no—
215C22⋊Q8—1941no—
216C22⋊Q8—1941no—
217C22⋊Q8—1941no—
218#121—971no—
219#121—971no—
220C4×Q8—971no—
221C4×D4—971no—
222C4×D4—971no—
223C4×D4—1941no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4C2×D9711yescharacteristic
226#106—971no—
227#106—971no—
228#106—971no—
229C4⋊C8—971no—
230#103—971no—
231#103—971no—
232#103—971no—
233#103—971no—
234#103—971no—
235#103C2×D9711yescharacteristic
236#103C2×D9711yescharacteristic
237#103C2×D9711yescharacteristic
238#103C2×D9711yescharacteristic
239#102—971no—
240C22⋊C8—971no—
241C22⋊C8—971no—
242C4×C8—971no—
243#41—3884no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—971no—
248C2×Q8—3881no—
249C2×Q8—1941no—
250C2×Q8—1941no—
251C2×Q8—1942no—
252C2×Q8C22×D9711yescharacteristic
253C2×Q8C22×D9711yescharacteristic
254C2×D4—971no—
255C2×D4—971no—
256C2×D4—1941no—
257C2×D4—1941no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—971no—
261C22×C4—1941no—
262C22×C4—1941no—
263C22×C4—1941no—
264C22×C4C22×D9711yescharacteristic
265SD16—3881no—
266SD16—3881no—
267SD16—1942no—
268C2×C8—41no—
269C2×C8—971no—
270C2×C8—971no—
271C2×C8—3881no—
272C2×C8C22×D9711yescharacteristic
273C2×C8C22×D9711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—971no—
276C4⋊C4—971no—
277C4⋊C4—971no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4—1941no—
283C4⋊C4—1941no—
284C4⋊C4—1941no—
285C4⋊C4C22×D9711yescharacteristic
286C4⋊C4C22×D9711yescharacteristic
287C22⋊C4—971no—
288C22⋊C4—971no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C42—971no—
295C42—1941no—
296C23—971no—
297C23—1941no—
298C23D4×D9711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—3882no—
304Q8—3882no—
305Q8—1941no—
306Q8—1941no—
307Q8#1309551112yescomplemented
308D4—974no—
309D4—3882no—
310D4—3882no—
311D4—1941no—
312D4—1941no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—971no—
322C2×C4—971no—
323C2×C4—971no—
324C2×C4—3882no—
325C2×C4—3882no—
326C2×C4—3882no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4D4×D9711yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—21no—
335C8—21no—
336C8—1941no—
337C8—1941no—
338C22—21no—
339C22—21no—
340C22—972no—
341C22—3881no—
342C22—3881no—
343C22—3881no—
344C22—1941no—
345C22—1941no—
346C22SD16×D9712yes—
347C22C2×D4×D9711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—972no—
354C4—3881no—
355C4—3881no—
356C4—3881no—
357C4—1941no—
358C4—1941no—
359C4C2×#1309551111yescharacteristic
360C4#1569085511yescharacteristic
361C2—22no—
362C2—7761no—
363C2—1942no—
364C2#2026137511yescharacteristic
365C2C2×SD16×D9711yescharacteristic
366C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#103×C97C2
socleC22×C97C22×D4
prime radical#20261386C2
hypercenterC2×#103D97
hypocenterC97#3298
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D97C2semidirect
3. #20261084C2semidirect
4. #20261065C2semidirect
5. #20261064C2semidirect
6. #20261018C2semidirect
7. #20260967C2semidirect
8. #20260964C2semidirect
9. #20260939C2semidirect
10. #20260875C2semidirect
11. #20260874C2semidirect
12. #20260820C2semidirect
13. C8×C22⋊D97C2semidirect
14. #103×D97C2semidirect
15. C2×#103×C97C2semidirect
119. Q8×C97#41semidirect
131. C22⋊D97SD16semidirect
184. C97#3298semidirect
199. C2×#103D97semidirect
307. Q8#13095511semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D972
#202613862
#202610842
#202610652
#202610642
#202610182
#202609672
#202609642
#202609392
#202608752
#202608742
#202608202
C8×C22⋊D972
#103×D972
C2×#103×C972
#329897