Finite Group Databasebeta

← #27151688

Subgroups — #27151688

Order 12416.

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

normal subgroups only

This group has 418 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
1#20261218C211yesmaximal, characteristic, complemented
2#20261056C212yesmaximal, complemented
3#20261056C212yesmaximal, complemented
4#20261051C212yesmaximal, complemented
5#20261046C212yesmaximal, complemented
6#20260778C211yesmaximal, characteristic, complemented
7#20260771C211yesmaximal, characteristic, complemented
8#20260740C211yesmaximal, characteristic, complemented
9C2×#15690731C211yesmaximal, characteristic, complemented
10#20260702C211yesmaximal, characteristic, complemented
11#535×C97C211yesmaximal, characteristic
12C2×#13095495C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14Q8×Dic97—21no—
15D4×Dic97—22no—
16D4×Dic97—21no—
17#15690769C2211yescharacteristic
18#15690769C2212yescomplemented
19C2×C4⋊Dic97C2211yescharacteristic
20C2×C4⋊Dic97C2212yescomplemented
21#15690743C2211yescharacteristic, complemented
22#15690743C2211yescharacteristic, complemented
23#15690743C2212yescomplemented
24#15690741C2211yescharacteristic, complemented
25#15690741C2211yescharacteristic, complemented
26#15690741C2212yescomplemented
27#15690731C2212yes—
28#15690731C2212yes—
29#15690731C2212yes—
30#15690731C2212yes—
31C22⋊#11715895C2212yescomplemented
32C22⋊#11715895C2212yescomplemented
33C2×#41×C97C2211yescharacteristic
34C22×D4×C97C2211yescharacteristic
35#127×C97—21no—
36#124×C97—21no—
37#121×C97C2212yes—
38#121×C97C2212yes—
39#120×C97—22no—
40#117×C97C2211yescharacteristic
41C22×Dic97—21no—
42C22×Dic97—22no—
43#13095495—22no—
44C2×#11715895—22no—
45C2×#11715895C2311yescharacteristic
46C2×#11715895C2311yescharacteristic
47C2×#11715895C2312yes—
48C4⋊Dic97—21no—
49C4⋊Dic97—22no—
50C4⋊Dic97C2312yes—
51C4⋊Dic97C2311yescharacteristic
52C4⋊Dic97C2311yescharacteristic
53C22⋊Dic97—21no—
54C22⋊Dic97—22no—
55C4×Dic97—21no—
56C4×Dic97—22no—
57C24×C97—21no—
58#41×C97—22no—
59#41×C97—22no—
60C2×Q8×C97D411yescharacteristic
61C2×D4×C97—22no—
62C2×D4×C97—22no—
63C2×D4×C97—21no—
64C2×D4×C97—22no—
65C2×D4×C97—22no—
66C2×D4×C97D412yes—
67C2×D4×C97D411yescharacteristic
68C2×D4×C97C2312yes—
69C22×C4×C97C2312yes—
70C22×C4×C97C2311yescharacteristic
71C4⋊C4×C97C2311yescharacteristic
72C4⋊C4×C97C2311yescharacteristic
73C22⋊C4×C97—21no—
74C22⋊C4×C97—22no—
75C22⋊C4×C97—22no—
76C42×C97—21no—
77C23×C97—22no—
78C23×C97—42no—
79C23×C97—81no—
80C23×C97—81no—
81C23×C97C2×D412yes—
82C23×C97C2×D411yescharacteristic
83Q8×C97—22no—
84D4×C97—24no—
85D4×C97—22no—
86D4×C97—22no—
87D4×C97—22no—
88D4×C97—42no—
89D4×C97—82no—
90D4×C97—82no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97C2×D412yes—
96C2×C4×C97C2×D411yescharacteristic
97C2×C4×C97#4112yes—
98C2×C4×C97#4112yes—
99C2×C4×C97C2411yescharacteristic
100C2×Dic97—21no—
101C2×Dic97—21no—
102C2×Dic97—22no—
103C2×Dic97—81no—
104C2×Dic97—82no—
105#11715895—21no—
106#11715895—21no—
107#11715895—22no—
108C22×C97—21no—
109C22×C97—21no—
110C22×C97—22no—
111C22×C97—22no—
112C22×C97—22no—
113C22×C97—42no—
114C22×C97—42no—
115C22×C97—42no—
116C22×C97—84no—
117C22×C97#12312yes—
118C22×C97C22×D411yescharacteristic
119C4×C97—22no—
120C4×C97—41no—
121C4×C97—41no—
122C4×C97#12312yes—
123C4×C97C2×#4111yescharacteristic
124C4×C97C2×#4111yescharacteristic
125Dic97—41no—
126Dic97—42no—
127Dic97—41no—
128C2×C97—22no—
129C2×C97—42no—
130C2×C97—82no—
131C2×C97C2×#12311yescharacteristic
132C2×C97#57611yescharacteristic
133C2×C97#57711yescharacteristic
134#3096—971nomaximal, Sylow
135C97#309611yescharacteristic, complemented, Sylow
136#535D9711yescharacteristic
137#520—971no—
138#482—972no—
139#482—972no—
140#481—972no—
141#480—972no—
142#428—971no—
143#427—971no—
144#418—971no—
145C2×#102—971no—
146#408—971no—
147C2×#41C2×D9711yescharacteristic
148C22×D4C2×D9711yescharacteristic
149C2×M16—971no—
150C22×C8—971no—
151#127—21no—
152#124—21no—
153#121C2×D9712yes—
154#121C2×D9712yes—
155#120—22no—
156C4×Q8—1941no—
157C4×D4—1942no—
158C4×D4—1941no—
159#117—971no—
160#117—972no—
161#117C2×D9711yescharacteristic
162C2×C4⋊C4—971no—
163C2×C4⋊C4—972no—
164#107—971no—
165#107—971no—
166#107—972no—
167#106—971no—
168#106—971no—
169#106—972no—
170#102—972no—
171#102—972no—
172#102—972no—
173#102—972no—
174C22⋊C8—972no—
175C22⋊C8—972no—
176C24—21no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D9711yescharacteristic
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4C22⋊D9712yes—
186C2×D4C22⋊D9711yescharacteristic
187C2×D4C22×D9712yes—
188C22×C4—1942no—
189C22×C4—1941no—
190C22×C4C22×D9712yes—
191C22×C4C22×D9711yescharacteristic
192M16—1942no—
193C2×C8—971no—
194C2×C8—972no—
195C2×C8—971no—
196C2×C8—1942no—
197C4⋊C4—972no—
198C4⋊C4—971no—
199C4⋊C4—971no—
200C4⋊C4—1942no—
201C4⋊C4—1941no—
202C4⋊C4C22×D9711yescharacteristic
203C4⋊C4C22×D9711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—1942no—
208C22⋊C4—1941no—
209C42—21no—
210C42—1942no—
211C42—1941no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23C2×C22⋊D9712yes—
217C23C2×C22⋊D9711yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—42no—
224D4—82no—
225D4—82no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—7761no—
231C2×C4—7762no—
232C2×C4—1941no—
233C2×C4—1942no—
234C2×C4—1941no—
235C2×C4C2×C22⋊D9712yes—
236C2×C4C2×C22⋊D9711yescharacteristic
237C2×C4#1309551012yes—
238C2×C4#1309551012yes—
239C2×C4C23×D9711yescharacteristic
240C8—1941no—
241C8—1941no—
242C8—1942no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—84no—
252C22#1569079112yes—
253C22C22×C22⋊D9711yescharacteristic
254C4—22no—
255C4—41no—
256C4—41no—
257C4—3882no—
258C4—3881no—
259C4—3881no—
260C4#1569079112yes—
261C4C2×#1309551011yescharacteristic
262C4C2×#1309551011yescharacteristic
263C2—22no—
264C2—42no—
265C2—82no—
266C2C2×#1569079111yescharacteristic
267C2#2026153811yescharacteristic
268C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radical#535×C97C2
socleC22×C97C22×D4
hypercenter#535D97
hypocenterC97#3096
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20261218C2semidirect
2. #20261056C2semidirect
3. #20261056C2semidirect
4. #20261051C2semidirect
5. #20261046C2semidirect
6. #20260778C2semidirect
7. #20260771C2semidirect
8. #20260740C2semidirect
9. C2×#15690731C2semidirect
10. #20260702C2semidirect
18. #15690769C22semidirect
20. C2×C4⋊Dic97C22semidirect
21. #15690743C22semidirect
22. #15690743C22semidirect
23. #15690743C22semidirect
24. #15690741C22semidirect
25. #15690741C22semidirect
26. #15690741C22semidirect
31. C22⋊#11715895C22semidirect
32. C22⋊#11715895C22semidirect
135. C97#3096semidirect

Maximal subgroups

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

subgroupindex
#202612182
#202610562
#202610512
#202610462
#202607782
#202607712
#202607402
C2×#156907312
#202607022
#535×C972
#309697