Finite Group Databasebeta

← #27151692

Subgroups — #27151692

Order 12416.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#20261302C211yesmaximal, characteristic
2#20261215C211yesmaximal, characteristic, complemented
3#20261058C212yesmaximal, complemented
4#20261057C212yesmaximal, complemented
5#20261052C212yesmaximal, complemented
6#20261047C212yesmaximal, complemented
7#20260777C211yesmaximal, characteristic, complemented
8#20260771C211yesmaximal, characteristic, complemented
9C2×#15690731C211yesmaximal, characteristic, complemented
10#20260703C211yesmaximal, characteristic, complemented
11#418×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14#15690806—21no—
15#15690797—21no—
16#15690781C2212yes—
17#15690781C2212yes—
18#15690779—22no—
19C4×Dic194—21no—
20C4×D388—21no—
21#15690770—22no—
22#15690769C2211yescharacteristic
23#15690768C2212yescomplemented
24C2×C4⋊Dic97C2211yescharacteristic
25C4⋊C4×D97C2212yescomplemented
26#15690744C2212yescomplemented
27#15690743C2211yescharacteristic, complemented
28#15690743C2211yescharacteristic, complemented
29#15690742C2212yescomplemented
30#15690741C2211yescharacteristic, complemented
31#15690741C2211yescharacteristic, complemented
32#15690731C2212yes—
33#15690731C2212yes—
34#15690722C2212yescomplemented
35#15690722C2212yescomplemented
36C22×D4×C97C2211yescharacteristic
37C2×M16×C97C2211yescharacteristic
38#117×C97C2211yescharacteristic
39#102×C97C2212yes—
40#102×C97C2212yes—
41#13095511—22no—
42#13095511—22no—
43C2×Dic194D411yescharacteristic
44C2×D388D411yescharacteristic
45C2×C22⋊D97—22no—
46C2×C22⋊D97—22no—
47C2×C22⋊D97D412yes—
48C22×Dic97—21no—
49C2×C4×D97—22no—
50C2×C4×D97C2312yes—
51C2×#11715895—22no—
52C2×#11715895C2311yescharacteristic
53C2×#11715895C2311yescharacteristic
54C4⋊Dic97—21no—
55C4⋊Dic97C2312yes—
56C4⋊Dic97C2311yescharacteristic
57C4⋊Dic97C2311yescharacteristic
58#13095490—22no—
59C22⋊Dic97—21no—
60C22⋊Dic97—22no—
61C22⋊Dic97—22no—
62#13095488—22no—
63C4×Dic97—21no—
64C4×Dic97—22no—
65C24×C97—21no—
66C2×D4×C97—21no—
67C2×D4×C97—22no—
68C2×D4×C97—22no—
69C2×D4×C97C2312yes—
70C22×C4×C97C2311yescharacteristic
71M16×C97—22no—
72C2×C8×C97C2312yes—
73C4⋊C4×C97C2311yescharacteristic
74C4⋊C4×C97C2311yescharacteristic
75C22⋊C4×C97—21no—
76C42×C97—21no—
77C23×C97—22no—
78C23×C97—42no—
79C23×C97—81no—
80C23×C97—81no—
81C23×C97C2×D411yescharacteristic
82C22×D97C2×D412yes—
83D4×C97—22no—
84D4×C97—22no—
85D4×C97—42no—
86C2×C4×C97—21no—
87C2×C4×C97—21no—
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97#4112yes—
90C2×C4×C97#4112yes—
91C2×C4×C97C2411yescharacteristic
92C22⋊D97—24no—
93C22⋊D97—82no—
94C22⋊D97—82no—
95C2×Dic97—22no—
96C2×Dic97—21no—
97C2×Dic97—21no—
98C2×Dic97—81no—
99C2×Dic97C2×D412yes—
100D388—22no—
101C4×D97—22no—
102C4×D97—22no—
103C4×D97—82no—
104Dic194—22no—
105C8×C97—22no—
106#11715895—21no—
107#11715895—21no—
108C22×C97—21no—
109C22×C97—21no—
110C22×C97—42no—
111C22×C97—42no—
112C22×C97—42no—
113C22×C97—84no—
114C22×C97#12312yes—
115C22×C97C22×D411yescharacteristic
116C2×D97—22no—
117C2×D97—22no—
118C2×D97—22no—
119C4×C97—41no—
120C4×C97—41no—
121C4×C97#12312yes—
122C4×C97C2×#4111yescharacteristic
123C4×C97C2×#4111yescharacteristic
124Dic97—22no—
125Dic97—42no—
126Dic97—41no—
127Dic97—41no—
128C2×C97—22no—
129C2×C97—82no—
130C2×C97C2×#12311yescharacteristic
131C2×C97#57611yescharacteristic
132C2×C97#57711yescharacteristic
133D97—42no—
134#3096—971nomaximal, Sylow
135C97#309611yescharacteristic, complemented, Sylow
136#535—971no—
137#520—971no—
138#482—972no—
139#482—972no—
140#481—972no—
141#480—972no—
142#428—971no—
143#427—971no—
144#418D9711yescharacteristic, complemented
145C2×#102—971no—
146#408—971no—
147C2×#41—971no—
148C22×D4C2×D9711yescharacteristic
149C2×M16C2×D9711yescharacteristic
150C22×C8—971no—
151#127—1941no—
152#124—1941no—
153#121—972no—
154#121—972no—
155#120—1942no—
156C4×Q8—1941no—
157C4×D4—1941no—
158C4×D4—1942no—
159#117—971no—
160#117—972no—
161#117C2×D9711yescharacteristic
162C2×C4⋊C4—971no—
163C2×C4⋊C4—972no—
164#107—972no—
165#107—971no—
166#107—971no—
167#106—972no—
168#106—971no—
169#106—971no—
170#102—972no—
171#102—972no—
172#102C2×D9712yes—
173#102C2×D9712yes—
174C22⋊C8—972no—
175C22⋊C8—972no—
176C24—21no—
177#41—1942no—
178#41—1942no—
179C2×Q8—971no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—972no—
184C2×D4—971no—
185C2×D4—1942no—
186C2×D4—1942no—
187C2×D4C22×D9712yes—
188C22×C4—972no—
189C22×C4—1942no—
190C22×C4—1941no—
191C22×C4C22×D9711yescharacteristic
192M16—22no—
193C2×C8—971no—
194C2×C8—971no—
195C2×C8—1942no—
196C2×C8C22×D9712yes—
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—1942no—
206C22⋊C4—1942no—
207C22⋊C4—1942no—
208C22⋊C4—1941no—
209C42—21no—
210C42—1942no—
211C42—1941no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—972no—
217C23D4×D9711yescharacteristic
218Q8—1942no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—7762no—
223D4—7762no—
224D4—1944no—
225D4—1942no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—7762no—
229C2×C4—7761no—
230C2×C4—972no—
231C2×C4—1941no—
232C2×C4—1941no—
233C2×C4—1942no—
234C2×C4—1942no—
235C2×C4—1942no—
236C2×C4D4×D9711yescharacteristic
237C2×C4#1309551012yes—
238C2×C4#1309551112yes—
239C2×C4C23×D9711yescharacteristic
240C8—22no—
241C8—1941no—
242C8—1941no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—1942no—
250C22—1942no—
251C22—1942no—
252C22#1569079212yes—
253C22C2×D4×D9711yescharacteristic
254C4—41no—
255C4—41no—
256C4—3882no—
257C4—3881no—
258C4—3881no—
259C4—1942no—
260C4#1569079212yes—
261C4C2×#1309551011yescharacteristic
262C4C2×#1309551111yescharacteristic
263C2—22no—
264C2—82no—
265C2—3882no—
266C2C2×#1569079211yescharacteristic
267C2#2026153911yescharacteristic
268C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#418×C97C2
socleC22×C97C22×D4
prime radical#20261302C2
hypercenter#418D97
hypocenterC97#3096
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #20261215C2semidirect
3. #20261058C2semidirect
4. #20261057C2semidirect
5. #20261052C2semidirect
6. #20261047C2semidirect
7. #20260777C2semidirect
8. #20260771C2semidirect
9. C2×#15690731C2semidirect
10. #20260703C2semidirect
11. #418×C97C2semidirect
23. #15690768C22semidirect
25. C4⋊C4×D97C22semidirect
26. #15690744C22semidirect
27. #15690743C22semidirect
28. #15690743C22semidirect
29. #15690742C22semidirect
30. #15690741C22semidirect
31. #15690741C22semidirect
34. #15690722C22semidirect
35. #15690722C22semidirect
135. C97#3096semidirect
144. #418D97semidirect

Maximal subgroups

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

subgroupindex
#202613022
#202612152
#202610582
#202610572
#202610522
#202610472
#202607772
#202607712
C2×#156907312
#202607032
#418×C972
#309697