Finite Group Databasebeta

← #27142120

Subgroups — #27142120

Order 12416.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#20261502C211yesmaximal, characteristic, complemented
2#20260752C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic, complemented
4#20260495C212yesmaximal, complemented
5#532×C97C211yesmaximal, characteristic
6#15690817—24no—
7C2×#13095495—22no—
8C2×#13095495C2212yes—
9#15690769C2212yes—
10#15690737—24no—
11#15690725—22no—
12C2×#41×C97C2211yescharacteristic
13C2×#41×C97C2211yescharacteristic
14C4⋊Q8×C97C411yescharacteristic, complemented
15#127×C97C411yescharacteristic, complemented
16#124×C97C411yescharacteristic, complemented
17#124×C97C411yescharacteristic, complemented
18#121×C97—22no—
19C4×D4×C97—22no—
20C2×C42×C97C2211yescharacteristic
21#13095495—24no—
22#13095495—22no—
23#13095495D414yescomplemented
24C2×#11715895—24no—
25C2×#11715895—22no—
26C4⋊Dic97—22no—
27C22⋊Dic97—22no—
28C4×Dic97—22no—
29#41×C97—22no—
30#41×C97—22no—
31#41×C97—22no—
32#41×C97—22no—
33C2×Q8×C97C2×C411yescharacteristic
34C2×Q8×C97C2×C411yescharacteristic
35C2×D4×C97—22no—
36C2×D4×C97—21no—
37C2×D4×C97C2×C411yescharacteristic
38C2×D4×C97C2×C411yescharacteristic
39C2×D4×C97D412yes—
40C22×C4×C97—21no—
41C22×C4×C97—21no—
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—22no—
45C22⋊C4×C97—22no—
46C22⋊C4×C97—22no—
47C42×C97—21no—
48C42×C97C2×C411yescharacteristic
49C42×C97C2×C411yescharacteristic
50C23×C97—21no—
51C23×C97C22⋊C412yes—
52C23×C97C2×D411yescharacteristic
53Q8×C97—22no—
54Q8×C97—22no—
55D4×C97—24no—
56D4×C97—22no—
57D4×C97—22no—
58D4×C97—44no—
59D4×C97—42no—
60C2×C4×C97—21no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97—42no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97C22⋊C412yes—
69C2×C4×C97C22×C411yescharacteristic
70C2×C4×C97C2×D412yes—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97#4112yes—
73C2×Dic97—42no—
74#11715895—24no—
75C22×C97—22no—
76C22×C97—41no—
77C22×C97—42no—
78C22×C97—42no—
79C22×C97—41no—
80C22×C97C2×C22⋊C411yescharacteristic
81C22×C97#12112yes—
82C4×C97—22no—
83C4×C97—22no—
84C4×C97—41no—
85C4×C97—41no—
86C4×C97C4×D412yes—
87C4×C97#12011yescharacteristic
88C4×C97#12311yescharacteristic
89Dic97—82no—
90C2×C97—22no—
91C2×C97—21no—
92C2×C97—42no—
93C2×C97—81no—
94C2×C97#38611yescharacteristic
95#1884—971nomaximal, Sylow
96C97#188411yescharacteristic, complemented, Sylow
97#568—971no—
98#532D9711yescharacteristic
99#421—972no—
100#410—971no—
101#339—972no—
102C2×#41C2×D9711yescharacteristic
103C2×#41C2×D9711yescharacteristic
104#131—1944no—
105C2×M16—972no—
106C2×M16—1942no—
107C4⋊Q8Dic9711yescharacteristic, complemented
108#127Dic9711yescharacteristic, complemented
109#124Dic9711yescharacteristic, complemented
110#124Dic9711yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117—972no—
114C2×C42C2×D9711yescharacteristic
115#104—1944no—
116#99—1942no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×Dic9711yescharacteristic
122C2×Q8C2×Dic9711yescharacteristic
123C2×D4—22no—
124C2×D4—21no—
125C2×D4C2×Dic9711yescharacteristic
126C2×D4C2×Dic9711yescharacteristic
127C2×D4C22⋊D9712yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊D9712yes—
131C22×C4C22×D9711yescharacteristic
132M16—974no—
133M16—1944no—
134M16—1942no—
135C2×C8—1944no—
136C2×C8—1942no—
137C4⋊C4—22no—
138C4⋊C4—1942no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—1942no—
142C42—21no—
143C42—1942no—
144C42C2×Dic9711yescharacteristic
145C42C2×Dic9711yescharacteristic
146C23—21no—
147C23C22⋊Dic9712yes—
148C23C2×C22⋊D9711yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—3882no—
165C2×C4C22⋊Dic9712yes—
166C2×C4C22×Dic9711yescharacteristic
167C2×C4D4×D9712yes—
168C2×C4C2×C22⋊D9711yescharacteristic
169C2×C4#1309551012yes—
170C8—1944no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic9711yescharacteristic
177C22#1569078212yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—7762no—
183C4D4×Dic9712yes—
184C4#1569077811yescharacteristic
185C4#1569079111yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#2026061511yescharacteristic
subgroupG/H
centerC4#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#532×C97C2
socleC2×C97#386
cosocleC22×C4×C97C23
hypercenter#532D97
hypocenterC97#1884
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20261502C2semidirect
2. #20260752C2semidirect
3. #20260709C2semidirect
4. #20260495C2semidirect
14. C4⋊Q8×C97C4semidirect
15. #127×C97C4semidirect
16. #124×C97C4semidirect
17. #124×C97C4semidirect
23. #13095495D4semidirect
96. C97#1884semidirect
107. C4⋊Q8Dic97semidirect
108. #127Dic97semidirect
109. #124Dic97semidirect
110. #124Dic97semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202607522
#202607092
#202604952
#532×C972
#188497