Finite Group Databasebeta

← #27152191

Subgroups — #27152191

Order 12416.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15690843C211yesmaximal, characteristic, complemented
2C2×#15690841C211yesmaximal, characteristic, complemented
3C2×#15690840C211yesmaximal, characteristic, complemented
4#20261281C211yesmaximal, characteristic
5C2×#15690808C211yesmaximal, characteristic, complemented
6#20261128C212yesmaximal, complemented
7#20261126C212yesmaximal, complemented
8#20261113C212yesmaximal, complemented
9#20261106C212yesmaximal, complemented
10M16×Dic97C211yesmaximal, characteristic, complemented
11C2×#137×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×Dic194C2211yescharacteristic
14#15690843—24no—
15#15690841—24no—
16#15690840—24no—
17C2×Dic388C2212yes—
18C2×#13095501C2212yes—
19C2×#13095501C2212yes—
20C2×#13095500C2212yes—
21C2×#13095499C2212yes—
22C2×#13095498C2212yes—
23C2×#13095495C2211yescharacteristic
24#15690808—22no—
25#15690808C2212yes—
26#15690808C2212yes—
27#15690808C2211yescharacteristic
28#15690806C2211yescharacteristic
29#15690799C2211yescharacteristic
30#15690797C2211yescharacteristic
31#15690782—22no—
32C4×C22⋊D97—22no—
33C2×#13095490—22no—
34C2×C4×Dic97C2211yescharacteristic
35#15690720C2212yes—
36C8×Dic97C2212yes—
37C2×#41×C97C2211yescharacteristic
38C22×Q8×C97C2211yescharacteristic
39#137×C97—24no—
40C2×Q16×C97C2212yes—
41C2×SD16×C97C2212yes—
42C2×M16×C97C2211yescharacteristic
43#13095511—22no—
44#13095511—22no—
45C2×Dic194—22no—
46C2×Dic194—22no—
47C2×Dic194—21no—
48C2×Dic194C2311yescharacteristic
49C2×Dic194C2312yes—
50C2×D388C2311yescharacteristic
51C2×C22⋊D97—21no—
52C2×C22⋊D97—22no—
53C22×Dic97—21no—
54C22×Dic97D412yes—
55C2×C4×D97—21no—
56Dic388—24no—
57#13095501—24no—
58#13095501—24no—
59#13095500—24no—
60#13095499—24no—
61#13095498—24no—
62#13095495D414yescomplemented
63C2×#11715895C2312yes—
64#13095490—24no—
65#13095490—24no—
66#13095490—22no—
67C22⋊Dic97—22no—
68#13095488—22no—
69C4×Dic97D412yes—
70C4×Dic97C2311yescharacteristic
71C4×Dic97C2311yescharacteristic
72#41×C97—22no—
73#41×C97—22no—
74C2×Q8×C97—21no—
75C2×Q8×C97—22no—
76C2×Q8×C97—22no—
77C2×Q8×C97C2311yescharacteristic
78C2×Q8×C97C2312yes—
79C2×D4×C97—21no—
80C2×D4×C97C2311yescharacteristic
81C22×C4×C97—21no—
82C22×C4×C97—21no—
83C22×C4×C97C2311yescharacteristic
84Q16×C97—24no—
85SD16×C97—24no—
86M16×C97D414yescomplemented
87C2×C8×C97C2312yes—
88C23×C97—21no—
89C23×C97C2×D411yescharacteristic
90C22×D97—21no—
91Q8×C97—22no—
92Q8×C97—21no—
93Q8×C97—22no—
94Q8×C97—21no—
95Q8×C97—42no—
96D4×C97—21no—
97D4×C97—21no—
98D4×C97—42no—
99C2×C4×C97—21no—
100C2×C4×C97—22no—
101C2×C4×C97—42no—
102C2×C4×C97—42no—
103C2×C4×C97C2×D411yescharacteristic
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2411yescharacteristic
107C22⋊D97—42no—
108C22⋊D97—44no—
109C2×Dic97—24no—
110C2×Dic97—21no—
111C2×Dic97—22no—
112C2×Dic97—42no—
113C2×Dic97C2×D412yes—
114C2×Dic97C2×D412yes—
115D388—21no—
116D388—21no—
117C4×D97—42no—
118Dic194—22no—
119Dic194—22no—
120Dic194—21no—
121Dic194—21no—
122Dic194—42no—
123C8×C97C2×D414yes—
124#11715895C2×D414yes—
125C22×C97—22no—
126C22×C97—41no—
127C22×C97—42no—
128C22×C97#12712yes—
129C22×C97C22×D411yescharacteristic
130C2×D97—41no—
131C2×D97—42no—
132C4×C97—41no—
133C4×C97—42no—
134C4×C97#12712yes—
135C4×C97C22×D411yescharacteristic
136C4×C97C22×D411yescharacteristic
137Dic97—22no—
138Dic97—42no—
139Dic97—41no—
140Dic97#13714yescomplemented
141C2×C97—22no—
142C2×C97—81no—
143C2×C97C2×#12711yescharacteristic
144C2×C97C2×#13712yes—
145D97—81no—
146#3152—971nomaximal, Sylow
147C97#315211yescharacteristic, complemented, Sylow
148C2×#137—971no—
149C2×#137—971no—
150C2×#137—971no—
151C2×#137D9711yescharacteristic, complemented
152#532—971no—
153C2×C4⋊Q8—971no—
154#497—972no—
155#497—972no—
156#494—972no—
157#492—972no—
158C4×M16—971no—
159C2×#41—971no—
160C2×#41C2×D9711yescharacteristic
161C22×Q8—971no—
162C22×Q8C2×D9711yescharacteristic
163#137—24no—
164#137—1944no—
165#137—1944no—
166#137—1944no—
167C2×Q16—972no—
168C2×Q16—972no—
169C2×Q16—972no—
170C2×Q16C2×D9712yes—
171C2×SD16—972no—
172C2×SD16—972no—
173C2×SD16—972no—
174C2×SD16C2×D9712yes—
175C2×M16—971no—
176C2×M16C2×D9711yescharacteristic
177C4⋊Q8—972no—
178C4⋊Q8—972no—
179C4⋊Q8—971no—
180C4⋊Q8—1942no—
181#127—971no—
182#124—971no—
183#124—971no—
184#121—1942no—
185C4×D4—1942no—
186C2×C4⋊C4—1942no—
187C2×C42—971no—
188#97—972no—
189C4×C8—972no—
190#41—22no—
191#41—22no—
192#41—1942no—
193#41—1942no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—971no—
198C2×Q8—972no—
199C2×Q8—1942no—
200C2×Q8—1942no—
201C2×Q8—1941no—
202C2×Q8C22×D9711yescharacteristic
203C2×Q8C22×D9712yes—
204C2×D4—21no—
205C2×D4—971no—
206C2×D4—1942no—
207C2×D4—1941no—
208C2×D4C22×D9711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—972no—
212C22×C4—1941no—
213C22×C4—1941no—
214C22×C4C22×D9711yescharacteristic
215Q16—24no—
216Q16—1944no—
217Q16—1944no—
218Q16—1944no—
219SD16—24no—
220SD16—1944no—
221SD16—1944no—
222SD16—1944no—
223M16—974no—
224M16C22⋊D9714yescomplemented
225C2×C8—972no—
226C2×C8C22×D9712yes—
227C4⋊C4—1944no—
228C4⋊C4—1944no—
229C4⋊C4—1942no—
230C22⋊C4—1942no—
231C22⋊C4—1942no—
232C42—972no—
233C42—971no—
234C42—971no—
235C23—21no—
236C23—1941no—
237C23D4×D9711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—3882no—
244Q8—1942no—
245Q8—1941no—
246Q8—1942no—
247Q8—1941no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—3884no—
252D4—3882no—
253D4—1941no—
254D4—1941no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—972no—
260C2×C4—972no—
261C2×C4—3882no—
262C2×C4—3882no—
263C2×C4—1944no—
264C2×C4—1941no—
265C2×C4—1942no—
266C2×C4D4×D9711yescharacteristic
267C2×C4D4×D9712yes—
268C2×C4C2×C22⋊D9712yes—
269C2×C4C23×D9711yescharacteristic
270C8—974no—
271C8C2×C22⋊D9714yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—3882no—
276C22—3881no—
277C22#1569080612yes—
278C22C2×D4×D9711yescharacteristic
279C4—41no—
280C4—42no—
281C4—974no—
282C4—3881no—
283C4—3882no—
284C4—1942no—
285C4#1569080612yes—
286C4C2×D4×D9711yescharacteristic
287C4C22×C22⋊D9711yescharacteristic
288C2—22no—
289C2—81no—
290C2—7761no—
291C2C2×#1569080611yescharacteristic
292C2#137×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radical#20261281C2
hypercenterC2×#137D97
hypocenterC97#3152
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#15690843C2semidirect
2. C2×#15690841C2semidirect
3. C2×#15690840C2semidirect
5. C2×#15690808C2semidirect
6. #20261128C2semidirect
7. #20261126C2semidirect
8. #20261113C2semidirect
9. #20261106C2semidirect
10. M16×Dic97C2semidirect
11. C2×#137×C97C2semidirect
62. #13095495D4semidirect
86. M16×C97D4semidirect
140. Dic97#137semidirect
147. C97#3152semidirect
151. C2×#137D97semidirect
224. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C2×#156908412
C2×#156908402
#202612812
C2×#156908082
#202611282
#202611262
#202611132
#202611062
M16×Dic972
C2×#137×C972
#315297