Finite Group Databasebeta

← #27142119

Subgroups — #27142119

Order 12416.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261281C211yesmaximal, characteristic
2#20260751C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic, complemented
4#20260496C212yesmaximal, complemented
5#568×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2211yescharacteristic
7#15690808C411yescharacteristic, complemented
8#15690806C411yescharacteristic, complemented
9#15690799C411yescharacteristic, complemented
10#15690797C411yescharacteristic, complemented
11#15690782—22no—
12C4×C22⋊D97—22no—
13#15690769C2212yes—
14C2×C4×Dic97C2211yescharacteristic
15#15690736—24no—
16#15690725—22no—
17C2×#41×C97C2211yescharacteristic
18#131×C97—24no—
19C2×M16×C97—22no—
20C2×M16×C97C2212yes—
21#13095511—22no—
22#13095511—22no—
23C2×Dic194C2×C411yescharacteristic
24C2×D388C2×C411yescharacteristic
25C2×C22⋊D97—22no—
26C2×C22⋊D97—21no—
27C22×Dic97—21no—
28C2×C4×D97—21no—
29C4⋊Dic97—22no—
30#13095490—22no—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33#13095488—22no—
34C4×Dic97—21no—
35C4×Dic97—22no—
36C4×Dic97C2×C411yescharacteristic
37C4×Dic97C2×C411yescharacteristic
38#41×C97—22no—
39#41×C97—22no—
40C2×Q8×C97C2×C411yescharacteristic
41C2×D4×C97C2×C411yescharacteristic
42C2×D4×C97D412yes—
43C22×C4×C97D412yes—
44C22×C4×C97C2311yescharacteristic
45M16×C97—24no—
46M16×C97—22no—
47M16×C97D414yescomplemented
48C2×C8×C97—24no—
49C2×C8×C97—22no—
50C23×C97C22⋊C412yes—
51C23×C97C2×D411yescharacteristic
52C22×D97—21no—
53Q8×C97—22no—
54D4×C97—24no—
55D4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97C22⋊C412yes—
59C2×C4×C97C22×C411yescharacteristic
60C2×C4×C97C2×D412yes—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97#4112yes—
63C22⋊D97—44no—
64C22⋊D97—42no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97—21no—
68C2×Dic97—42no—
69C2×Dic97—42no—
70D388—22no—
71C4×D97—41no—
72C4×D97—41no—
73Dic194—22no—
74C8×C97—24no—
75C22×C97—22no—
76C22×C97—41no—
77C22×C97—42no—
78C22×C97C2×C22⋊C411yescharacteristic
79C22×C97#12112yes—
80C2×D97—42no—
81C2×D97—41no—
82C4×C97—22no—
83C4×C97C4×D412yes—
84C4×C97#12011yescharacteristic
85C4×C97#12311yescharacteristic
86Dic97—22no—
87Dic97—41no—
88Dic97—41no—
89Dic97—82no—
90C2×C97—21no—
91C2×C97—22no—
92C2×C97—42no—
93C2×C97#38611yescharacteristic
94D97—81no—
95#1884—971nomaximal, Sylow
96C97#188411yescharacteristic, complemented, Sylow
97#568D9711yescharacteristic, complemented
98#532—971no—
99#421—972no—
100#410—971no—
101#339—972no—
102C2×#41—971no—
103C2×#41C2×D9711yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D9712yes—
107C4⋊Q8—971no—
108#127—971no—
109#124—971no—
110#124—971no—
111#121—1942no—
112C4×D4—1942no—
113#117—972no—
114C2×C42—971no—
115#104—1944no—
116#99—1942no—
117#41—22no—
118#41—22no—
119#41—1942no—
120#41—1942no—
121C2×Q8—971no—
122C2×Q8C4×D9711yescharacteristic
123C2×D4—971no—
124C2×D4—1942no—
125C2×D4—1941no—
126C2×D4C4×D9711yescharacteristic
127C2×D4D38812yes—
128C22×C4—1941no—
129C22×C4—1941no—
130C22×C4C22⋊D9712yes—
131C22×C4C22×D9711yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D9714yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—1942no—
138C4⋊C4—1942no—
139C22⋊C4—1942no—
140C22⋊C4—1942no—
141C22⋊C4—1942no—
142C42—971no—
143C42—971no—
144C42—1942no—
145C42—1941no—
146C23—1941no—
147C23#1309548812yes—
148C23C2×D38811yescharacteristic
149Q8—22no—
150Q8—1942no—
151D4—24no—
152D4—22no—
153D4—3882no—
154D4—3884no—
155D4—1942no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—3882no—
159C2×C4—3882no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—1941no—
163C2×C4—1941no—
164C2×C4—1941no—
165C2×C4#1309548812yes—
166C2×C4C2×C4×D9711yescharacteristic
167C2×C4C2×C22⋊D9712yes—
168C2×C4C2×C22⋊D9711yescharacteristic
169C2×C4#1309551112yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—3882no—
175C22—3881no—
176C22C2×#1309548811yescharacteristic
177C22#1569078412yes—
178C4—22no—
179C4—7762no—
180C4—3881no—
181C4—3881no—
182C4—1942no—
183C4C4×C22⋊D9712yes—
184C4#1569077911yescharacteristic
185C4#1569079311yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—7761no—
190C2#2026061411yescharacteristic
subgroupG/H
centerC4#15690779
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#568×C97C2
socleC2×C97#386
prime radical#20261281C2
cosocleC22×C4×C97C23
hypercenter#568D97
hypocenterC97#1884
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20260751C2semidirect
3. #20260709C2semidirect
4. #20260496C2semidirect
5. #568×C97C2semidirect
7. #15690808C4semidirect
8. #15690806C4semidirect
9. #15690799C4semidirect
10. #15690797C4semidirect
47. M16×C97D4semidirect
96. C97#1884semidirect
97. #568D97semidirect
134. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612812
#202607512
#202607092
#202604962
#568×C972
#188497