Finite Group Databasebeta

← #448×D97 #27150921

Subgroups — #448×D97 #27150921

Order 12416.

792 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×D97C211yesmaximal, characteristic, complemented
2C2×SD16×D97C212yesmaximal, complemented
3#121×D97C211yesmaximal, characteristic
4#20260876C211yesmaximal, characteristic, complemented
5#20260875C212yesmaximal, complemented
6#20260874C211yesmaximal, characteristic, complemented
7#20260873C211yesmaximal, characteristic, complemented
8#20260872C212yesmaximal, complemented
9#103×D97C212yesmaximal, complemented
10C22⋊C8×D97C211yesmaximal, characteristic, complemented
11#448×C97C211yesmaximal, characteristic, complemented
12C22×Dic194—21no—
13C22×Dic194C2211yescharacteristic, complemented
14C2×Q8×D97—28no—
15C2×Q8×D97—21no—
16C2×Q8×D97—21no—
17C2×Q8×D97C2212yes—
18C2×Q8×D97C2212yescomplemented
19C2×D4×D97—21no—
20C2×D4×D97C2211yescharacteristic
21C22×C4×D97—21no—
22C22×C4×D97C2211yescharacteristic
23C2×#13095500C2212yescomplemented
24C2×#13095499C2212yescomplemented
25C2×#13095498C2212yescomplemented
26SD16×D97—28no—
27C2×C8×D97C2212yes—
28#15690785C2211yescharacteristic
29#15690784C2211yescharacteristic
30#15690783—21no—
31#15690782—21no—
32#15690781C2211yescharacteristic
33C4⋊C4×D97C2211yescharacteristic
34C22⋊C4×D97—21no—
35#15690735C2212yescomplemented
36#15690734C2212yescomplemented
37#15690733C2212yescomplemented
38C22⋊#11715895C2211yescharacteristic, complemented
39#15690722C2212yescomplemented
40C22×Q8×C97C2211yescharacteristic, complemented
41C2×SD16×C97C2212yescomplemented
42#121×C97C2211yescharacteristic
43#103×C97C2212yescomplemented
44C22⋊C8×C97C2211yescharacteristic, complemented
45C23×D97—21no—
46C23×D97D411yescharacteristic
47C2×Dic194—28no—
48C2×Dic194—21no—
49C2×Dic194—22no—
50C2×Dic194—22no—
51C2×Dic194—24no—
52C2×Dic194C2312yes—
53Q8×D97—28no—
54Q8×D97—28no—
55Q8×D97—24no—
56Q8×D97—24no—
57Q8×D97—24no—
58Q8×D97D418yescomplemented
59C2×D388—21no—
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64D4×D97—24no—
65D4×D97—44no—
66C22×Dic97—21no—
67C22×Dic97D411yescharacteristic
68C2×C4×D97—28no—
69C2×C4×D97—22no—
70C2×C4×D97—22no—
71C2×C4×D97—24no—
72C2×C4×D97—21no—
73C2×C4×D97D411yescharacteristic
74C2×C4×D97D411yescharacteristic
75C2×C4×D97C2311yescharacteristic
76C2×C4×D97C2311yescharacteristic
77#13095500—24no—
78#13095499—24no—
79#13095498—24no—
80C2×#11715895C2312yes—
81C8×D97—24no—
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84C22⋊Dic97—21no—
85#13095488—21no—
86#13095488—21no—
87C2×Q8×C97—24no—
88C2×Q8×C97—21no—
89C2×Q8×C97C2312yes—
90C2×D4×C97—21no—
91C2×D4×C97C2311yescharacteristic
92C22×C4×C97—21no—
93C22×C4×C97C2311yescharacteristic
94SD16×C97—24no—
95C2×C8×C97C2312yes—
96C4⋊C4×C97C2311yescharacteristic
97C22⋊C4×C97—21no—
98C23×C97—21no—
99C23×C97C2×D411yescharacteristic
100C22×D97—22no—
101C22×D97—22no—
102C22×D97—21no—
103C22×D97—42no—
104C22×D97—42no—
105C22×D97—42no—
106C22×D97SD1614yescomplemented
107C22×D97C2×D411yescharacteristic
108C22×D97C2×D411yescharacteristic
109Q8×C97—24no—
110Q8×C97—22no—
111Q8×C97C2×D414yescomplemented
112D4×C97—22no—
113D4×C97—42no—
114C2×C4×C97—24no—
115C2×C4×C97—22no—
116C2×C4×C97—22no—
117C2×C4×C97—21no—
118C2×C4×C97C2×D411yescharacteristic
119C2×C4×C97C2411yescharacteristic
120C22⋊D97—42no—
121C22⋊D97—42no—
122C22⋊D97—42no—
123C2×Dic97—24no—
124C2×Dic97—22no—
125C2×Dic97—22no—
126C2×Dic97—21no—
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129D388—22no—
130D388—42no—
131C4×D97—28no—
132C4×D97—28no—
133C4×D97—22no—
134C4×D97—22no—
135C4×D97—22no—
136C4×D97—42no—
137C4×D97C2×D414yes—
138Dic194—28no—
139Dic194—24no—
140Dic194—22no—
141Dic194—28no—
142Dic194C2×D414yescomplemented
143C8×C97—22no—
144#11715895—22no—
145C22×C97—21no—
146C22×C97—21no—
147C22×C97—41no—
148C22×C97—41no—
149C22×C97—41no—
150C22×C97C2×SD1612yescomplemented
151C22×C97C22×D411yescharacteristic
152C2×D97—21no—
153C2×D97—21no—
154C2×D97—28no—
155C2×D97—41no—
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—84no—
159C2×D97#12012yes—
160C2×D97C2×SD1612yes—
161C2×D97C2×SD1612yescomplemented
162C2×D97#13712yes—
163C4×C97—24no—
164C4×C97—21no—
165C4×C97—41no—
166C4×C97C22×D412yes—
167Dic97—21no—
168Dic97—24no—
169Dic97—41no—
170Dic97#12012yes—
171C2×C97—22no—
172C2×C97—81no—
173C2×C97C2×#12011yescharacteristic
174C2×C97C22×SD1611yescharacteristic
175C2×C97C2×#13711yescharacteristic
176D97—22no—
177D97—81no—
178D97#44814yesdirect product factor
179C2×#448—971nomaximal, Sylow
180C97C2×#44811yescharacteristic, complemented, Sylow
181C23×Q8—971no—
182C22×SD16—972no—
183C2×#121—971no—
184#448—971no—
185#448—972no—
186#448—971no—
187#448—972no—
188#448—971no—
189#448D9711yescharacteristic, direct product factor
190C2×#103—972no—
191C2×C22⋊C8—971no—
192C22×Q8—972no—
193C22×Q8—972no—
194C22×Q8—971no—
195C22×Q8—1941no—
196C22×Q8—1941no—
197C22×Q8—1941no—
198C22×Q8—1948no—
199C22×Q8C2×D9711yescharacteristic, complemented
200C22×D4—971no—
201C22×D4—1941no—
202C23×C4—971no—
203C23×C4—1941no—
204C2×SD16—972no—
205C2×SD16—972no—
206C2×SD16—972no—
207C2×SD16—1948no—
208C2×SD16C2×D9712yescomplemented
209C22×C8—972no—
210#121—971no—
211#121—971no—
212#121—971no—
213#121—1941no—
214#121—1941no—
215#121C2×D9711yescharacteristic
216C2×C4⋊C4—971no—
217C2×C22⋊C4—1941no—
218#103—972no—
219#103—972no—
220#103—972no—
221#103C2×D9712yescomplemented
222C22⋊C8—971no—
223C22⋊C8—972no—
224C22⋊C8C2×D9711yescharacteristic, complemented
225C24—971no—
226C24—1941no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—978no—
230C2×Q8—972no—
231C2×Q8—1942no—
232C2×Q8—1942no—
233C2×Q8—1944no—
234C2×Q8—1948no—
235C2×Q8—1944no—
236C2×Q8—1944no—
237C2×Q8—1948no—
238C2×Q8—1948no—
239C2×Q8—1944no—
240C2×Q8—1941no—
241C2×Q8C22×D9712yes—
242C2×D4—21no—
243C2×D4—971no—
244C2×D4—3884no—
245C2×D4—1941no—
246C2×D4—1944no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4—1941no—
250C2×D4C22×D9711yescharacteristic
251C22×C4—21no—
252C22×C4—971no—
253C22×C4—971no—
254C22×C4—971no—
255C22×C4—971no—
256C22×C4—971no—
257C22×C4—1944no—
258C22×C4—1941no—
259C22×C4—1942no—
260C22×C4—1942no—
261C22×C4—1948no—
262C22×C4—1941no—
263C22×C4C22×D9711yescharacteristic
264SD16—24no—
265SD16—1944no—
266SD16—1944no—
267SD16—1944no—
268C2×C8—972no—
269C2×C8—1944no—
270C2×C8C22×D9712yes—
271C4⋊C4—971no—
272C4⋊C4—1941no—
273C4⋊C4C22×D9711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—1941no—
276C22⋊C4—1941no—
277C22⋊C4—1941no—
278C23—21no—
279C23—971no—
280C23—974no—
281C23—971no—
282C23—3882no—
283C23—3882no—
284C23—3882no—
285C23—1942no—
286C23—1942no—
287C23—1941no—
288C23D4×D9711yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—974no—
292Q8—1948no—
293Q8—1948no—
294Q8—1944no—
295Q8—1942no—
296Q8D4×D9714yescomplemented
297D4—22no—
298D4—42no—
299D4—3882no—
300D4—3882no—
301D4—3882no—
302D4—3882no—
303D4—1942no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—971no—
309C2×C4—974no—
310C2×C4—971no—
311C2×C4—3882no—
312C2×C4—1942no—
313C2×C4—1942no—
314C2×C4—1948no—
315C2×C4—1948no—
316C2×C4—1942no—
317C2×C4—1941no—
318C2×C4—1944no—
319C2×C4—1942no—
320C2×C4—1942no—
321C2×C4D4×D9711yescharacteristic
322C2×C4C23×D9711yescharacteristic
323C8—22no—
324C8—1942no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—7764no—
331C22—972no—
332C22—972no—
333C22—972no—
334C22—972no—
335C22—3881no—
336C22—3881no—
337C22—3881no—
338C22—1948no—
339C22—1941no—
340C22—1941no—
341C22SD16×D9712yescomplemented
342C22C2×D4×D9711yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—972no—
347C4—3881no—
348C4—1941no—
349C4—1944no—
350C4C2×D4×D9712yes—
351C2—22no—
352C2—81no—
353C2—7761no—
354C2—974no—
355C2—1942no—
356C2#120×D9711yescharacteristic
357C2C2×SD16×D9711yescharacteristic
358C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#448×C97C2
socleC22×C97C22×D4
prime radical#121×D97C2
hypercenter#448D97
hypocenterC97C2×#448
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D97C2semidirect
2. C2×SD16×D97C2semidirect
4. #20260876C2semidirect
5. #20260875C2semidirect
6. #20260874C2semidirect
7. #20260873C2semidirect
8. #20260872C2semidirect
9. #103×D97C2semidirect
10. C22⋊C8×D97C2semidirect
11. #448×C97C2semidirect
13. C22×Dic194C22semidirect
18. C2×Q8×D97C22semidirect
23. C2×#13095500C22semidirect
24. C2×#13095499C22semidirect
25. C2×#13095498C22semidirect
35. #15690735C22semidirect
36. #15690734C22semidirect
37. #15690733C22semidirect
38. C22⋊#11715895C22semidirect
39. #15690722C22semidirect
40. C22×Q8×C97C22semidirect
41. C2×SD16×C97C22semidirect
43. #103×C97C22semidirect
44. C22⋊C8×C97C22semidirect
58. Q8×D97D4semidirect
106. C22×D97SD16semidirect
111. Q8×C97C2×D4semidirect
142. Dic194C2×D4semidirect
150. C22×C97C2×SD16semidirect
161. C2×D97C2×SD16semidirect
178. D97#448direct (N × H)
180. C97C2×#448semidirect
189. #448D97direct (N × H)
199. C22×Q8C2×D97semidirect
208. C2×SD16C2×D97semidirect
221. #103C2×D97semidirect
224. C22⋊C8C2×D97semidirect
296. Q8D4×D97semidirect
341. C22SD16×D97semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D972
C2×SD16×D972
#121×D972
#202608762
#202608752
#202608742
#202608732
#202608722
#103×D972
C22⋊C8×D972
#448×C972
C2×#44897