Finite Group Databasebeta

← #27142232

Subgroups — #27142232

Order 12416.

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

normal subgroups only

This group has 310 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#20261213C211yesmaximal, characteristic, complemented
2#20260752C211yesmaximal, characteristic
3#20260744C211yesmaximal, characteristic, complemented
4#20260709C211yesmaximal, characteristic
5#20260506C211yesmaximal, characteristic, complemented
6#20260492C211yesmaximal, characteristic, complemented
7C2×#137×C97C211yesmaximal, characteristic
8C2×#13095495C2211yescharacteristic
9Q8×Dic97—22no—
10Q8×Dic97—22no—
11#15690769—22no—
12#15690769C2211yescharacteristic
13#15690769C2211yescharacteristic
14#15690769C2211yescharacteristic
15#15690737—22no—
16#15690734—22no—
17#15690725—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—21no—
26#13095495—21no—
27C2×#11715895—21no—
28C4⋊Dic97—24no—
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C22⋊Dic97—21no—
33C22⋊Dic97—21no—
34C22⋊Dic97D412yes—
35C4×Dic97—24no—
36C4×Dic97—21no—
37C4×Dic97—21no—
38C4×Dic97D412yes—
39#41×C97—21no—
40#41×C97—21no—
41#41×C97—21no—
42#41×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97C2×C411yescharacteristic
47C2×Q8×C97C2×C412yes—
48C2×Q8×C97D412yes—
49C2×Q8×C97D412yes—
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×C97C2×D411yescharacteristic
62Q8×C97—22no—
63Q8×C97—22no—
64Q8×C97—22no—
65Q8×C97—21no—
66Q8×C97—21no—
67Q8×C97C22⋊C414yes—
68D4×C97—21no—
69D4×C97—21no—
70D4×C97—41no—
71D4×C97—41no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—21no—
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×C97#4111yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—41no—
87C8×C97—22no—
88#11715895—41no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—81no—
92C22×C97C4×D411yescharacteristic
93C22×C97#12011yescharacteristic
94C22×C97#12111yescharacteristic
95C4×C97—24no—
96C4×C97—41no—
97C4×C97C2×C22⋊C411yescharacteristic
98C4×C97C4×D411yescharacteristic
99C4×C97#12111yescharacteristic
100C4×C97#12311yescharacteristic
101Dic97—42no—
102Dic97—42no—
103Dic97—81no—
104C2×C97—21no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97—81no—
108C2×C97#38611yescharacteristic
109#1905—971nomaximal, Sylow
110C97#190511yescharacteristic, complemented, Sylow
111C2×#137D9711yescharacteristic
112#519—971no—
113#421—971no—
114#419—971no—
115#410—971no—
116#344—971no—
117H(3,ℤ/4)—971no—
118C2×#41C2×D9711yescharacteristic
119C22×Q8C2×D9711yescharacteristic
120#137—22no—
121#137—22no—
122C2×Q16Dic9712yescomplemented
123C2×SD16Dic9712yescomplemented
124C2×M16—971no—
125C2×M16C2×D9711yescharacteristic
126C4×Q8—1942no—
127C4×Q8—1942no—
128#117—971no—
129#117—971no—
130#117—971no—
131#117—1942no—
132#104—1942no—
133#103—1942no—
134#99—1942no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C2×Dic9711yescharacteristic
143C2×Q8C2×Dic9712yes—
144C2×Q8C22⋊D9712yes—
145C2×Q8C22⋊D9712yes—
146C2×D4—21no—
147C2×D4C2×Dic9711yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4C22×D9711yescharacteristic
151Q16—24no—
152SD16—24no—
153M16—21no—
154M16—21no—
155M16—1941no—
156M16—1941no—
157C2×C8—1941no—
158C2×C8C2×Dic9712yes—
159C4⋊C4—1944no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C4⋊C4—1941no—
163C22⋊C4—972no—
164C22⋊C4—1941no—
165C22⋊C4—1941no—
166C42—972no—
167C42—1944no—
168C42—1941no—
169C42—1941no—
170C23—21no—
171C23D4×D9711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—22no—
175Q8—21no—
176Q8—21no—
177Q8C22⋊Dic9714yes—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—3881no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4C22×Dic9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4C2×C22⋊D9711yescharacteristic
194C2×C4C2×C22⋊D9711yescharacteristic
195C2×C4#1309551011yescharacteristic
196C2×C4#1309551011yescharacteristic
197C8—22no—
198C8—3881no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22D4×Dic9711yescharacteristic
203C22#1569077811yescharacteristic
204C22#1569078211yescharacteristic
205C4—24no—
206C4—41no—
207C4—7761no—
208C4—3882no—
209C4—3882no—
210C4C2×C22⋊Dic9711yescharacteristic
211C4D4×Dic9711yescharacteristic
212C4#1569078211yescharacteristic
213C4#1569079111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#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#1905
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
1. #20261213C2semidirect
3. #20260744C2semidirect
5. #20260506C2semidirect
6. #20260492C2semidirect
22. C2×Q16×C97C4semidirect
23. C2×SD16×C97C4semidirect
110. C97#1905semidirect
122. C2×Q16Dic97semidirect
123. C2×SD16Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612132
#202607522
#202607442
#202607092
#202605062
#202604922
C2×#137×C972
#190597