Finite Group Databasebeta

← #27151567

Subgroups — #27151567

Order 12416.

398 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 149 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
1#20261363C211yesmaximal, characteristic, complemented
2#20261035C212yesmaximal, complemented
3#20261035C212yesmaximal, complemented
4#20261025C212yesmaximal, complemented
5#20261015C212yesmaximal, complemented
6#20260778C211yesmaximal, characteristic, complemented
7#20260771C211yesmaximal, characteristic, complemented
8#20260762C211yesmaximal, characteristic, complemented
9#20260740C211yesmaximal, characteristic
10C2×#15690731C211yesmaximal, characteristic
11#518×C97C211yesmaximal, characteristic
12C2×#13095495C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14#15690810C2212yescomplemented
15#15690809—21no—
16#15690802C2212yescomplemented
17#15690801—21no—
18#15690787—22no—
19#15690769C2211yescharacteristic
20C2×C4⋊Dic97C2211yescharacteristic
21#15690743C2211yescharacteristic, complemented
22#15690743C2212yescomplemented
23#15690743C2211yescharacteristic, complemented
24#15690741C2211yescharacteristic, complemented
25#15690741C2212yescomplemented
26#15690741C2211yescharacteristic, complemented
27C4⋊#11715895C2212yescomplemented
28C4⋊#11715895C2212yescomplemented
29#15690731C2212yes—
30#15690731C2212yes—
31#15690731C2212yes—
32#15690731C2212yes—
33C22×D4×C97C2211yescharacteristic
34C4×D4×C97—21no—
35C4×D4×C97—21no—
36C4×D4×C97C2212yes—
37C4×D4×C97C2212yes—
38#117×C97C2211yescharacteristic
39#117×C97C2211yescharacteristic
40C2×C22⋊C4×C97—22no—
41C2×Dic194—22no—
42C22×Dic97—21no—
43#13095495—22no—
44C2×#11715895—22no—
45C2×#11715895C2311yescharacteristic
46C2×#11715895C2311yescharacteristic
47C2×#11715895C2312yes—
48C4⋊Dic97—21no—
49C4⋊Dic97—22no—
50C4⋊Dic97C2312yes—
51C4⋊Dic97C2311yescharacteristic
52C4⋊Dic97C2311yescharacteristic
53#13095490—22no—
54#13095490—22no—
55C22⋊Dic97—21no—
56C4×Dic97—21no—
57C24×C97—21no—
58C2×D4×C97—21no—
59C2×D4×C97—22no—
60C2×D4×C97Q814yescomplemented
61C2×D4×C97C2312yes—
62C22×C4×C97—22no—
63C22×C4×C97—22no—
64C22×C4×C97C2311yescharacteristic
65C4⋊C4×C97D411yescharacteristic
66C4⋊C4×C97D411yescharacteristic
67C4⋊C4×C97C2311yescharacteristic
68C4⋊C4×C97C2311yescharacteristic
69C22⋊C4×C97—21no—
70C22⋊C4×C97—22no—
71C22⋊C4×C97—22no—
72C22⋊C4×C97D412yes—
73C42×C97—21no—
74C42×C97C2312yes—
75C23×C97—22no—
76C23×C97—24no—
77C23×C97—81no—
78C23×C97—81no—
79C23×C97C2×D411yescharacteristic
80D4×C97—24no—
81D4×C97—22no—
82D4×C97C2×Q814yes—
83C2×C4×C97—21no—
84C2×C4×C97—21no—
85C2×C4×C97—82no—
86C2×C4×C97—82no—
87C2×C4×C97C2×D412yes—
88C2×C4×C97C2×D412yes—
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×Q812yes—
91C2×C4×C97#4112yes—
92C2×C4×C97C2411yescharacteristic
93C2×Dic97—21no—
94C2×Dic97—21no—
95C2×Dic97—22no—
96C2×Dic97—81no—
97Dic194—82no—
98#11715895—21no—
99#11715895—21no—
100#11715895—22no—
101C22×C97—21no—
102C22×C97—21no—
103C22×C97—24no—
104C22×C97—42no—
105C22×C97—42no—
106C22×C97—84no—
107C22×C97C22⋊Q812yes—
108C22×C97C22×D411yescharacteristic
109C4×C97—22no—
110C4×C97—22no—
111C4×C97—41no—
112C4×C97—41no—
113C4×C97C22⋊Q812yes—
114C4×C97C22×Q811yescharacteristic
115C4×C97C2×#4111yescharacteristic
116Dic97—41no—
117Dic97—42no—
118Dic97—41no—
119C2×C97—22no—
120C2×C97—44no—
121C2×C97C2×C22⋊Q811yescharacteristic
122C2×C97#57611yescharacteristic
123C2×C97#57711yescharacteristic
124#3081—971nomaximal, Sylow
125C97#308111yescharacteristic, complemented, Sylow
126#544—971no—
127#518D9711yescharacteristic
128#478—972no—
129#478—972no—
130#476—972no—
131#474—972no—
132#428—971no—
133#427—971no—
134#424—971no—
135#418—971no—
136C2×#102—971no—
137C22×D4C2×D9711yescharacteristic
138C2×M16—971no—
139C22×C8—971no—
140C4⋊Q8—972no—
141C4⋊Q8—1941no—
142#125—972no—
143#125—1941no—
144C22⋊Q8—1942no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4C2×D9712yes—
148C4×D4C2×D9712yes—
149#117—971no—
150#117C2×D9711yescharacteristic
151#117C2×D9711yescharacteristic
152C2×C4⋊C4—971no—
153C2×C22⋊C4—22no—
154#107—971no—
155#107—971no—
156#107—972no—
157#106—971no—
158#106—971no—
159#106—972no—
160C4⋊C8—972no—
161C4⋊C8—972no—
162#102—972no—
163#102—972no—
164#102—972no—
165#102—972no—
166C24—21no—
167C2×Q8—1942no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic19414yescomplemented
171C2×D4C22×D9712yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—1941no—
175C22×C4C22×D9711yescharacteristic
176M16—1942no—
177C2×C8—971no—
178C2×C8—972no—
179C2×C8—971no—
180C2×C8—1942no—
181C4⋊C4—972no—
182C4⋊C4—971no—
183C4⋊C4—971no—
184C4⋊C4—1942no—
185C4⋊C4—1942no—
186C4⋊C4—1941no—
187C4⋊C4—1942no—
188C4⋊C4C22⋊D9711yescharacteristic
189C4⋊C4C22⋊D9711yescharacteristic
190C4⋊C4C22×D9711yescharacteristic
191C4⋊C4C22×D9711yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1941no—
196C22⋊C4C22⋊D9712yes—
197C42—21no—
198C42—1941no—
199C42C22×D9712yes—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23C2×C22⋊D9711yescharacteristic
205Q8—7762no—
206D4—24no—
207D4—22no—
208D4C2×Dic19414yes—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—7761no—
214C2×C4—1941no—
215C2×C4—1941no—
216C2×C4—1942no—
217C2×C4C2×C22⋊D9712yes—
218C2×C4C2×C22⋊D9712yes—
219C2×C4C2×C22⋊D9711yescharacteristic
220C2×C4C2×Dic19412yes—
221C2×C4#1309551112yes—
222C2×C4C23×D9711yescharacteristic
223C8—1941no—
224C8—1941no—
225C8—1942no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1569078912yes—
233C22C22×C22⋊D9711yescharacteristic
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—3882no—
239C4—3881no—
240C4—3881no—
241C4#1569078912yes—
242C4C22×Dic19411yescharacteristic
243C4C2×#1309551111yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1569078911yescharacteristic
247C2#2026153811yescharacteristic
248C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#518×C97C2
socleC22×C97C22×D4
prime radicalC22×D4×C97C22
hypercenter#518D97
hypocenterC97#3081
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20261363C2semidirect
2. #20261035C2semidirect
3. #20261035C2semidirect
4. #20261025C2semidirect
5. #20261015C2semidirect
6. #20260778C2semidirect
7. #20260771C2semidirect
8. #20260762C2semidirect
14. #15690810C22semidirect
16. #15690802C22semidirect
21. #15690743C22semidirect
22. #15690743C22semidirect
23. #15690743C22semidirect
24. #15690741C22semidirect
25. #15690741C22semidirect
26. #15690741C22semidirect
27. C4⋊#11715895C22semidirect
28. C4⋊#11715895C22semidirect
60. C2×D4×C97Q8semidirect
125. C97#3081semidirect
170. C2×D4Dic194semidirect

Maximal subgroups

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

subgroupindex
#202613632
#202610352
#202610252
#202610152
#202607782
#202607712
#202607622
#202607402
C2×#156907312
#518×C972
#308197