Finite Group Databasebeta

← #27140332

Subgroups — #27140332

Order 12416.

264 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 61 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#124×D97C211yesmaximal, characteristic
2#20260790C211yesmaximal, characteristic, complemented
3#20260687C211yesmaximal, characteristic, complemented
4#20260475C211yesmaximal, characteristic, complemented
5#20260474C211yesmaximal, characteristic, complemented
6#20260474C211yesmaximal, characteristic, complemented
7#329×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97C411yescharacteristic
9C2×D4×D97C411yescharacteristic
10C2×#13095494—21no—
11#15690799C411yescharacteristic
12#15690798C2211yescharacteristic
13#15690797C411yescharacteristic
14#15690796—21no—
15#15690796—21no—
16C22⋊C4×D97—21no—
17C22⋊C4×D97—21no—
18C42×D97C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690721C2211yescharacteristic, complemented
22#15690721C2211yescharacteristic, complemented
23C8×Dic97—21no—
24#124×C97C2211yescharacteristic
25#97×C97C2211yescharacteristic, complemented
26#97×C97C2211yescharacteristic, complemented
27C23×D97—21no—
28C2×Dic194—21no—
29C2×Dic194C2×C411yescharacteristic
30Q8×D97—42no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33D4×D97—42no—
34C2×C4×D97—21no—
35C2×C4×D97C2×C411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38#13095494—24no—
39C2×#11715895—21no—
40C2×#11715895—21no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43#13095488—21no—
44#13095488—21no—
45#13095488—21no—
46#13095488—21no—
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97D411yescharacteristic
49C4×Dic97D411yescharacteristic
50C2×Q8×C97C2×C411yescharacteristic
51C2×D4×C97C2×C411yescharacteristic
52C2×C8×C97—21no—
53C2×C8×C97—21no—
54C22⋊C4×C97—21no—
55C22⋊C4×C97—21no—
56C42×C97C2311yescharacteristic
57C23×C97—21no—
58C22×D97—21no—
59C22×D97—42no—
60C22×D97—42no—
61C22×D97—42no—
62C22×D97C22⋊C411yescharacteristic
63Q8×C97—41no—
64D4×C97—41no—
65C2×C4×C97—21no—
66C2×C4×C97C22×C411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C22⋊D97—42no—
70C2×Dic97—21no—
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C22⋊C411yescharacteristic
73C2×Dic97C22⋊C411yescharacteristic
74D388—41no—
75C4×D97—22no—
76C4×D97—22no—
77C4×D97—41no—
78C4×D97—42no—
79Dic194—42no—
80Dic194—41no—
81C8×C97—22no—
82C8×C97—41no—
83#11715895—22no—
84#11715895—41no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97C2×C22⋊C411yescharacteristic
89C2×D97—21no—
90C2×D97—21no—
91C2×D97—41no—
92C2×D97—41no—
93C2×D97—41no—
94C2×D97—84no—
95C2×D97#10412yes—
96C4×C97—21no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—41no—
100Dic97—21no—
101Dic97—21no—
102Dic97—41no—
103Dic97#10012yes—
104C2×C97—81no—
105C2×C97C2×#10011yescharacteristic
106C2×C97C2×#10411yescharacteristic
107C2×C97#42111yescharacteristic
108D97—22no—
109D97—81no—
110#1527—971nomaximal, Sylow
111C97#152711yescharacteristic, complemented, Sylow
112C2×#124—971no—
113#432—971no—
114C4×M16—971no—
115#329—971no—
116#329—971no—
117#329—971no—
118#329D9711yescharacteristic, complemented
119C22×Q8—971no—
120C22×D4—971no—
121C2×M16—1941no—
122#124—971no—
123#124—971no—
124#124—971no—
125#124—1941no—
126#124—1941no—
127#124C2×D9711yescharacteristic
128C2×C22⋊C4—1941no—
129C2×C22⋊C4—1941no—
130C2×C42—971no—
131C4⋊C8—1941no—
132C22⋊C8—1941no—
133#97—971no—
134#97—971no—
135#97C2×D9711yescharacteristic, complemented
136#97C2×D9711yescharacteristic, complemented
137C4×C8—1941no—
138C24—1941no—
139C2×Q8—971no—
140C2×Q8—3882no—
141C2×Q8—1941no—
142C2×Q8C4×D9711yescharacteristic
143C2×D4—971no—
144C2×D4—3882no—
145C2×D4—1941no—
146C2×D4C4×D9711yescharacteristic
147C22×C4—971no—
148C22×C4—971no—
149C22×C4—971no—
150C22×C4—1941no—
151M16—1944no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—1941no—
155C2×C8—1941no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—1941no—
159C22⋊C4—1941no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C22⋊C4—1941no—
164C42—971no—
165C42—971no—
166C42—971no—
167C42C22×D9711yescharacteristic
168C23—21no—
169C23—971no—
170C23—3882no—
171C23—3882no—
172C23—3882no—
173C23—1941no—
174Q8—41no—
175Q8—3881no—
176Q8—3882no—
177D4—41no—
178D4—3881no—
179D4—3882no—
180C2×C4—21no—
181C2×C4—971no—
182C2×C4—971no—
183C2×C4—971no—
184C2×C4—3881no—
185C2×C4—3882no—
186C2×C4—1942no—
187C2×C4—1942no—
188C2×C4—1941no—
189C2×C4C2×C4×D9711yescharacteristic
190C2×C4D4×D9711yescharacteristic
191C2×C4D4×D9711yescharacteristic
192C8—22no—
193C8—41no—
194C8—3881no—
195C8—1942no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—7764no—
200C22—972no—
201C22—3881no—
202C22—3881no—
203C22—3881no—
204C22—1941no—
205C22—1941no—
206C22C22⋊C4×D9711yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—41no—
211C4—972no—
212C4—3881no—
213C4—1941no—
214C4—1941no—
215C2—81no—
216C2—7761no—
217C2—1942no—
218C2#100×D9711yescharacteristic
219C2#104×D9711yescharacteristic
220C2#2026075011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#329×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC42×C97C23
hypercenter#329D97
hypocenterC97#1527
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20260790C2semidirect
3. #20260687C2semidirect
4. #20260475C2semidirect
5. #20260474C2semidirect
6. #20260474C2semidirect
7. #329×C97C2semidirect
21. #15690721C22semidirect
22. #15690721C22semidirect
25. #97×C97C22semidirect
26. #97×C97C22semidirect
111. C97#1527semidirect
118. #329D97semidirect
135. #97C2×D97semidirect
136. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#124×D972
#202607902
#202606872
#202604752
#202604742
#329×C972
#152797