Finite Group Databasebeta

← #27142265

Subgroups — #27142265

Order 12416.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#20261299C211yesmaximal, characteristic
2#20261209C211yesmaximal, characteristic, complemented
3#20260752C211yesmaximal, characteristic, complemented
4C2×#15690726C211yesmaximal, characteristic, complemented
5#20260492C211yesmaximal, characteristic, complemented
6#20260491C211yesmaximal, characteristic, complemented
7#421×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690806C412yescomplemented
11#15690799C412yescomplemented
12#15690784—22no—
13#15690778—22no—
14C4×D388—22no—
15#15690772—22no—
16#15690769C2211yescharacteristic
17#15690769C2211yescharacteristic
18C22⋊C4×D97—22no—
19#15690737—22no—
20#15690726—22no—
21C2×#41×C97C2211yescharacteristic
22C2×M16×C97C2211yescharacteristic
23#117×C97C2211yescharacteristic
24#104×C97—22no—
25C23×D97D412yes—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388C2×C412yes—
31C2×D388D412yes—
32C2×C22⋊D97—24no—
33C2×C4×D97—24no—
34#13095495—21no—
35#13095495—21no—
36C2×#11715895—21no—
37C4⋊Dic97—21no—
38C4⋊Dic97—21no—
39C22⋊Dic97—21no—
40C22⋊Dic97—21no—
41#13095488—24no—
42#13095488—24no—
43C4×Dic97C2×C412yescomplemented
44C4×Dic97D412yescomplemented
45#41×C97—21no—
46#41×C97—21no—
47#41×C97—21no—
48#41×C97—21no—
49C2×Q8×C97C2×C411yescharacteristic, complemented
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic, complemented
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—21no—
55M16×C97—21no—
56C2×C8×C97—21no—
57C4⋊C4×C97—21no—
58C22⋊C4×C97—21no—
59C42×C97D412yescomplemented
60C23×C97—21no—
61C23×C97C2×D411yescharacteristic
62C22×D97—22no—
63C22×D97—22no—
64C22×D97—82no—
65C22×D97C22⋊C414yes—
66Q8×C97—21no—
67Q8×C97—21no—
68D4×C97—21no—
69D4×C97—21no—
70D4×C97—41no—
71D4×C97—41no—
72C2×C4×C97—21no—
73C2×C4×C97—22no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97C22×C411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97#4111yescharacteristic
81C2×C4×C97#4111yescharacteristic
82C22⋊D97—84no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85D388—22no—
86D388—22no—
87D388—22no—
88D388—22no—
89C4×D97—84no—
90C8×C97—41no—
91#11715895—41no—
92C22×C97—41no—
93C22×C97—42no—
94C22×C97—41no—
95C22×C97C2×C22⋊C411yescharacteristic
96C22×C97#12011yescharacteristic
97C22×C97#12311yescharacteristic
98C2×D97—24no—
99C2×D97—44no—
100C2×D97—44no—
101C2×D97—82no—
102C4×C97—42no—
103C4×C97—41no—
104C4×C97C4×D411yescharacteristic
105C4×C97C4×D411yescharacteristic
106C4×C97#12111yescharacteristic
107C4×C97#12111yescharacteristic
108Dic97—42no—
109Dic97—42no—
110C2×C97—21no—
111C2×C97—21no—
112C2×C97—21no—
113C2×C97—81no—
114C2×C97#38611yescharacteristic
115D97—44no—
116#1911—971nomaximal, Sylow
117C97#191111yescharacteristic, complemented, Sylow
118#535—971no—
119#518—971no—
120#421—971no—
121#421D9711yescharacteristic, complemented
122C2×#100—971no—
123H(3,ℤ/4)—971no—
124H(3,ℤ/4)—971no—
125C2×#41C2×D9711yescharacteristic
126C22×D4—971no—
127C2×M16—971no—
128C2×M16C2×D9711yescharacteristic
129#127—972no—
130#124—972no—
131#121—1942no—
132#120—1942no—
133C4×D4—1942no—
134C4×D4—1942no—
135#117—971no—
136#117—971no—
137#117C2×D9711yescharacteristic
138C2×C22⋊C4—1942no—
139#104—22no—
140#104—1942no—
141#100—1942no—
142C24—972no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C4×D9711yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—972no—
150C2×D4—972no—
151C2×D4—1944no—
152C2×D4—1941no—
153C2×D4—1941no—
154C2×D4—1941no—
155C2×D4—1941no—
156C2×D4C4×D9711yescharacteristic, complemented
157C22×C4—21no—
158C22×C4—1944no—
159C22×C4C22×D9711yescharacteristic
160M16—21no—
161M16—21no—
162M16—1941no—
163M16—1941no—
164C2×C8—21no—
165C2×C8—1941no—
166C4⋊C4—21no—
167C4⋊C4—1941no—
168C4⋊C4—1941no—
169C22⋊C4—21no—
170C22⋊C4—1944no—
171C22⋊C4—1944no—
172C22⋊C4—1941no—
173C22⋊C4—1941no—
174C42—972no—
175C42—972no—
176C42C22⋊D9712yescomplemented
177C23—21no—
178C23—7762no—
179C23—974no—
180C23—1942no—
181C23—1942no—
182C23D4×D9711yescharacteristic
183Q8—21no—
184Q8—21no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—7764no—
190D4—1942no—
191D4—1942no—
192D4—1942no—
193D4—1942no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—7764no—
199C2×C4—1942no—
200C2×C4—1942no—
201C2×C4C2×C4×D9711yescharacteristic
202C2×C4D4×D9711yescharacteristic
203C2×C4D4×D9711yescharacteristic
204C2×C4C2×C22⋊D9711yescharacteristic
205C2×C4#1309551011yescharacteristic
206C2×C4#1309551111yescharacteristic
207C8—41no—
208C8—3881no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—7762no—
213C22—3884no—
214C22—3884no—
215C22—1944no—
216C22C22⋊C4×D9711yescharacteristic
217C22#1569077811yescharacteristic
218C22#1569079211yescharacteristic
219C4—42no—
220C4—41no—
221C4—3882no—
222C4—3882no—
223C4#1569077011yescharacteristic
224C4C4×C22⋊D9711yescharacteristic
225C4#1569078111yescharacteristic
226C4#1569078311yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—3884no—
232C2#2026061111yescharacteristic
subgroupG/H
centerC2#20260611
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#421×C97C2
socleC2×C97#386
prime radical#20261299C2
cosocleC22×C4×C97C23
hypercenter#421D97
hypocenterC97#1911
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20261209C2semidirect
3. #20260752C2semidirect
4. C2×#15690726C2semidirect
5. #20260492C2semidirect
6. #20260491C2semidirect
7. #421×C97C2semidirect
10. #15690806C4semidirect
11. #15690799C4semidirect
43. C4×Dic97C2×C4semidirect
44. C4×Dic97D4semidirect
49. C2×Q8×C97C2×C4semidirect
51. C2×D4×C97C2×C4semidirect
59. C42×C97D4semidirect
117. C97#1911semidirect
121. #421D97semidirect
147. C2×Q8C4×D97semidirect
156. C2×D4C4×D97semidirect
176. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612992
#202612092
#202607522
C2×#156907262
#202604922
#202604912
#421×C972
#191197