Finite Group Databasebeta

← #27151425

Subgroups — #27151425

Order 12416.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#15690841C211yesmaximal, characteristic, complemented
2C2×#15690816C211yesmaximal, characteristic, complemented
3#20261316C211yesmaximal, characteristic
4#20261309C211yesmaximal, characteristic, complemented
5#20260992C212yesmaximal, complemented
6#20260984C212yesmaximal, complemented
7#20260975C212yesmaximal, complemented
8#20260967C212yesmaximal, complemented
9#20260775C211yesmaximal, characteristic, complemented
10#20260732C211yesmaximal, characteristic, complemented
11C2×#103×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C2×#13095511C2211yescharacteristic
14#15690841—24no—
15C2×#13095501C2212yes—
16C2×#13095498C2212yes—
17#15690816—24no—
18C2×#13095495C2211yescharacteristic
19C2×#13095494C2212yes—
20C2×C8×D97C2212yes—
21#15690808—21no—
22#15690799—21no—
23#15690793—22no—
24#15690790—21no—
25#15690790C2212yes—
26#15690786C2212yes—
27#15690786C2212yes—
28#15690785—21no—
29#15690785C2212yes—
30#15690784—22no—
31#15690769C2211yescharacteristic
32#15690743C2212yes—
33#15690741C2212yes—
34#15690735C2211yescharacteristic
35#15690735C2211yescharacteristic
36#15690732C2211yescharacteristic
37#15690732C2211yescharacteristic
38C22×Q8×C97C2211yescharacteristic
39C22×C8×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#103×C97C2212yes—
42#103×C97C2212yes—
43#13095511—22no—
44#13095511—22no—
45#13095511—22no—
46#13095511—22no—
47C2×Dic194D411yescharacteristic
48C2×Dic194C2311yescharacteristic
49C2×D388—22no—
50C2×D388D411yescharacteristic
51C2×D388C2311yescharacteristic
52C2×C22⋊D97—21no—
53C2×C22⋊D97D412yes—
54C2×C4×D97—21no—
55C2×C4×D97C2312yes—
56#13095501—24no—
57#13095498—24no—
58#13095495D414yescomplemented
59#13095494—24no—
60C2×#11715895C2312yes—
61C8×D97—24no—
62C4⋊Dic97—22no—
63C4⋊Dic97C2311yescharacteristic
64C4⋊Dic97C2311yescharacteristic
65#13095490—22no—
66#13095490—22no—
67C22⋊Dic97—21no—
68#13095488—22no—
69#13095488—22no—
70#13095488—22no—
71#13095488—22no—
72C4×Dic97—21no—
73C2×Q8×C97—21no—
74C2×Q8×C97—22no—
75C2×Q8×C97—22no—
76C2×Q8×C97C2312yes—
77C22×C4×C97—21no—
78C22×C4×C97—21no—
79C22×C4×C97C2311yescharacteristic
80C2×C8×C97—22no—
81C2×C8×C97C2311yescharacteristic
82C2×C8×C97C2311yescharacteristic
83C4⋊C4×C97—21no—
84C4⋊C4×C97C2312yes—
85C23×C97C2×D411yescharacteristic
86C22×D97—21no—
87C22×D97C2×D412yes—
88Q8×C97—22no—
89Q8×C97—22no—
90Q8×C97—42no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97—81no—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2×D412yes—
97C2×C4×C97#4112yes—
98C2×C4×C97C2411yescharacteristic
99C22⋊D97—24no—
100C22⋊D97—42no—
101C2×Dic97—21no—
102C2×Dic97—21no—
103C2×Dic97—21no—
104C2×Dic97C2×D412yes—
105D388—22no—
106D388—21no—
107D388—21no—
108D388—82no—
109C4×D97—22no—
110C4×D97—22no—
111C4×D97—42no—
112Dic194—22no—
113Dic194—21no—
114Dic194—21no—
115C8×C97—21no—
116C8×C97—21no—
117#11715895C2×D414yes—
118C22×C97—21no—
119C22×C97—21no—
120C22×C97#12112yes—
121C22×C97C22×D411yescharacteristic
122C2×D97—22no—
123C2×D97—22no—
124C2×D97—22no—
125C2×D97—41no—
126C2×D97—41no—
127C2×D97—41no—
128C4×C97—42no—
129C4×C97—42no—
130C4×C97#12112yes—
131C4×C97C22×D411yescharacteristic
132C4×C97C2×#4111yescharacteristic
133Dic97—22no—
134Dic97—41no—
135Dic97—41no—
136Dic97—41no—
137C2×C97—22no—
138C2×C97C2×#12111yescharacteristic
139C2×C97#57711yescharacteristic
140C2×C97#57811yescharacteristic
141D97—42no—
142D97—81no—
143#3063—971nomaximal, Sylow
144C97#306311yescharacteristic, complemented, Sylow
145C2×#137—971no—
146C2×#131—971no—
147#537—971no—
148#536—971no—
149#470—972no—
150#468—972no—
151#467—972no—
152#465—972no—
153#428—971no—
154#416—971no—
155C2×#103D9711yescharacteristic, complemented
156C2×#41—971no—
157C2×#41—971no—
158C22×Q8C2×D9711yescharacteristic
159#137—1944no—
160C2×Q16—972no—
161C2×SD16—972no—
162#131—1944no—
163C2×M16—971no—
164C2×M16—972no—
165C22×C8—972no—
166C22×C8C2×D9711yescharacteristic
167C4⋊Q8—1941no—
168#124—1941no—
169#123—1942no—
170C22⋊Q8—972no—
171C22⋊Q8—972no—
172C22⋊Q8—972no—
173C22⋊Q8—1941no—
174#121—972no—
175#121—1941no—
176#121—1942no—
177#117—971no—
178C2×C4⋊C4C2×D9711yescharacteristic
179#107—972no—
180#106—972no—
181#103—971no—
182#103—971no—
183#103C2×D9712yes—
184#103C2×D9712yes—
185#102—971no—
186#102—971no—
187#41—1942no—
188#41—1942no—
189#41—1942no—
190#41—1942no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—971no—
195C2×Q8—971no—
196C2×Q8C22×D9712yes—
197C2×D4—972no—
198C2×D4—971no—
199C2×D4—971no—
200C2×D4—1942no—
201C2×D4—1941no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—972no—
205C22×C4—1941no—
206C22×C4C22×D9711yescharacteristic
207Q16—1944no—
208SD16—1944no—
209M16—974no—
210M16—1944no—
211C2×C8—22no—
212C2×C8—972no—
213C2×C8—1944no—
214C2×C8C22×D9711yescharacteristic
215C2×C8C22×D9711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—971no—
218C4⋊C4—971no—
219C4⋊C4—1942no—
220C4⋊C4—1942no—
221C4⋊C4—1942no—
222C4⋊C4C22×D9712yes—
223C22⋊C4—1942no—
224C22⋊C4—1942no—
225C22⋊C4—1942no—
226C22⋊C4—1942no—
227C22⋊C4—1941no—
228C42—1941no—
229C23—972no—
230C23—1941no—
231C23D4×D9711yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—42no—
235Q8—1941no—
236Q8—1941no—
237Q8—1942no—
238D4—7762no—
239D4—3882no—
240D4—1944no—
241D4—1942no—
242D4—1941no—
243D4—1941no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—972no—
249C2×C4—3882no—
250C2×C4—1941no—
251C2×C4—1941no—
252C2×C4—1941no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4D4×D9711yescharacteristic
256C2×C4D4×D9712yes—
257C2×C4#1309551112yes—
258C2×C4C23×D9711yescharacteristic
259C8—21no—
260C8—21no—
261C8—974no—
262C22—21no—
263C22—21no—
264C22—3881no—
265C22—3881no—
266C22—3881no—
267C22—1942no—
268C22—1942no—
269C22—1942no—
270C22#1569078312yes—
271C22C2×D4×D9711yescharacteristic
272C4—42no—
273C4—42no—
274C4—3881no—
275C4—3881no—
276C4—3881no—
277C4—1942no—
278C4#1569078312yes—
279C4C2×D4×D9711yescharacteristic
280C4C2×#1309551111yescharacteristic
281C2—22no—
282C2—7761no—
283C2—3882no—
284C2C2×#1569078311yescharacteristic
285C2#2026154111yescharacteristic
286C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#103×C97C2
socleC22×C97C22×D4
prime radical#20261316C2
hypercenterC2×#103D97
hypocenterC97#3063
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. C2×#15690841C2semidirect
2. C2×#15690816C2semidirect
4. #20261309C2semidirect
5. #20260992C2semidirect
6. #20260984C2semidirect
7. #20260975C2semidirect
8. #20260967C2semidirect
9. #20260775C2semidirect
10. #20260732C2semidirect
11. C2×#103×C97C2semidirect
58. #13095495D4semidirect
144. C97#3063semidirect
155. C2×#103D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156908162
#202613162
#202613092
#202609922
#202609842
#202609752
#202609672
#202607752
#202607322
C2×#103×C972
#306397