Finite Group Databasebeta

← #27142211

Subgroups — #27142211

Order 12416.

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

normal subgroups only

This group has 304 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
2#20260752C211yesmaximal, characteristic
3#20260744C211yesmaximal, characteristic, complemented
4C2×#15690729C211yesmaximal, characteristic, complemented
5#20260506C211yesmaximal, characteristic, complemented
6#20260493C211yesmaximal, characteristic, complemented
7C2×#137×C97C211yesmaximal, characteristic
8#15690817—22no—
9#15690817—22no—
10C2×#13095495—22no—
11C2×#13095495C2211yescharacteristic
12C2×#13095495C2211yescharacteristic
13C2×#13095495C2211yescharacteristic
14#15690769C2211yescharacteristic
15#15690737—22no—
16#15690734—22no—
17#15690729—22no—
18C2×#41×C97C2211yescharacteristic
19C22×Q8×C97C2211yescharacteristic
20#137×C97—22no—
21#137×C97—22no—
22C2×Q16×C97C412yescomplemented
23C2×SD16×C97C412yescomplemented
24C2×M16×C97C2211yescharacteristic
25#13095495—24no—
26#13095495—21no—
27#13095495—21no—
28#13095495—21no—
29#13095495—21no—
30#13095495—21no—
31#13095495D412yes—
32C2×#11715895—24no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895D412yes—
36C4⋊Dic97—21no—
37C22⋊Dic97—21no—
38C4×Dic97—21no—
39#41×C97—21no—
40#41×C97—21no—
41#41×C97D412yes—
42#41×C97D412yes—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97C2×C412yes—
49C2×Q8×C97C2×C411yescharacteristic
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C22×C4×C97—21no—
54C22×C4×C97C2311yescharacteristic
55Q16×C97—24no—
56SD16×C97—24no—
57M16×C97—21no—
58M16×C97—21no—
59C2×C8×C97C2×C412yes—
60C23×C97—21no—
61C23×C97#4111yescharacteristic
62Q8×C97—21no—
63Q8×C97—22no—
64Q8×C97—22no—
65Q8×C97—41no—
66Q8×C97—41no—
67Q8×C97C22⋊C412yes—
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—21no—
71D4×C97C22⋊C412yes—
72C2×C4×C97—21no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97C22×C411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C2×Dic97—41no—
85C8×C97—22no—
86#11715895—22no—
87#11715895—22no—
88#11715895—41no—
89C22×C97—22no—
90C22×C97—41no—
91C22×C97—81no—
92C22×C97C4×D411yescharacteristic
93C22×C97#12111yescharacteristic
94C22×C97#12311yescharacteristic
95C4×C97—22no—
96C4×C97—42no—
97C4×C97C2×C22⋊C411yescharacteristic
98C4×C97C4×D411yescharacteristic
99C4×C97#12011yescharacteristic
100C4×C97#12111yescharacteristic
101Dic97—81no—
102C2×C97—21no—
103C2×C97—21no—
104C2×C97—21no—
105C2×C97—42no—
106C2×C97#38611yescharacteristic
107#1902—971nomaximal, Sylow
108C97#190211yescharacteristic, complemented, Sylow
109C2×#137D9711yescharacteristic
110#568—971no—
111#421—971no—
112#419—971no—
113C2×#101—971no—
114#344—971no—
115#338—971no—
116C2×#41C2×D9711yescharacteristic
117C22×Q8C2×D9711yescharacteristic
118#137—22no—
119#137—22no—
120C2×Q16Dic9712yescomplemented
121C2×SD16Dic9712yescomplemented
122#131—1942no—
123#131—1942no—
124C2×M16—971no—
125C2×M16—971no—
126C2×M16—971no—
127C2×M16—1942no—
128C2×M16C2×D9711yescharacteristic
129#117—971no—
130#104—1942no—
131#103—1942no—
132#101—1942no—
133#41—21no—
134#41—21no—
135#41C22⋊D9712yes—
136#41C22⋊D9712yes—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C2×Dic9712yes—
143C2×Q8C2×Dic9711yescharacteristic
144C2×D4—21no—
145C2×D4C2×Dic9711yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22×D9711yescharacteristic
149Q16—24no—
150SD16—24no—
151M16—21no—
152M16—21no—
153M16—972no—
154M16—1941no—
155M16—1941no—
156M16—1944no—
157M16—1941no—
158M16—1941no—
159M16—1941no—
160C2×C8—972no—
161C2×C8—1941no—
162C2×C8—1944no—
163C2×C8—1941no—
164C2×C8C2×Dic9712yes—
165C4⋊C4—1941no—
166C22⋊C4—1941no—
167C42—1941no—
168C23—21no—
169C23#1309551011yescharacteristic
170Q8—21no—
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175Q8C22⋊Dic9712yes—
176D4—22no—
177D4—22no—
178D4—21no—
179D4C22⋊Dic9712yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3881no—
187C2×C4C22×Dic9711yescharacteristic
188C2×C4D4×D9711yescharacteristic
189C2×C4D4×D9711yescharacteristic
190C2×C4C2×C22⋊D9711yescharacteristic
191C2×C4C2×C22⋊D9711yescharacteristic
192C2×C4#1309551011yescharacteristic
193C8—22no—
194C8—3881no—
195C8—1942no—
196C8—1942no—
197C22—22no—
198C22—41no—
199C22—81no—
200C22D4×Dic9711yescharacteristic
201C22#1569078211yescharacteristic
202C22#1569079111yescharacteristic
203C4—22no—
204C4—42no—
205C4—7761no—
206C4C2×C22⋊Dic9711yescharacteristic
207C4D4×Dic9711yescharacteristic
208C4#1569077811yescharacteristic
209C4#1569078211yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—42no—
214C2#2026061511yescharacteristic
subgroupG/H
centerC2#20260615
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#137×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenterC2×#137D97
hypocenterC97#1902
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
3. #20260744C2semidirect
4. C2×#15690729C2semidirect
5. #20260506C2semidirect
6. #20260493C2semidirect
22. C2×Q16×C97C4semidirect
23. C2×SD16×C97C4semidirect
108. C97#1902semidirect
120. C2×Q16Dic97semidirect
121. C2×SD16Dic97semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202607522
#202607442
C2×#156907292
#202605062
#202604932
C2×#137×C972
#190297