Finite Group Databasebeta

← #27143430

Subgroups — #27143430

Order 12416.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261306C211yesmaximal, characteristic
2#20260719C211yesmaximal, characteristic
3#20260533C214yesmaximal, complemented
4#413×C97C211yesmaximal, characteristic
5C22×Dic194C411yescharacteristic
6C2×#13095495—21no—
7#15690808C2212yes—
8#15690797C412yes—
9#15690791—21no—
10#15690791—21no—
11#15690789—21no—
12#15690789—21no—
13#15690769C411yescharacteristic
14#15690729C2212yes—
15#15690727—21no—
16C2×#41×C97C2211yescharacteristic
17C2×M16×C97—21no—
18#101×C97C2212yes—
19#100×C97—21no—
20C2×Dic194—21no—
21C2×Dic194—41no—
22C2×Dic194—41no—
23C2×Dic194C2×C412yes—
24C22×Dic97—21no—
25#13095495—22no—
26#13095495—41no—
27C2×#11715895—41no—
28C4⋊Dic97—21no—
29#13095490—22no—
30#13095490—22no—
31C22⋊Dic97—21no—
32C22⋊Dic97—22no—
33C22⋊Dic97—22no—
34C4×Dic97C2×C412yes—
35#41×C97—41no—
36#41×C97—41no—
37C2×Q8×C97C2311yescharacteristic
38C2×D4×C97C2×C411yescharacteristic
39C2×D4×C97D411yescharacteristic
40C2×D4×C97D411yescharacteristic
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C22×C4×C97D411yescharacteristic
44M16×C97—22no—
45M16×C97—41no—
46C2×C8×C97—41no—
47C23×C97C22⋊C411yescharacteristic
48C23×C97C22⋊C411yescharacteristic
49C23×C97C22⋊C411yescharacteristic
50Q8×C97—41no—
51D4×C97—41no—
52D4×C97—41no—
53D4×C97—41no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97C22⋊C411yescharacteristic
61C2×C4×C97C22×C411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—22no—
65C2×Dic97—22no—
66C2×Dic97—42no—
67Dic194—41no—
68Dic194—41no—
69Dic194—42no—
70C8×C97—42no—
71#11715895—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97C2×C22⋊C411yescharacteristic
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—21no—
82C4×C97#9911yescharacteristic
83C4×C97#9911yescharacteristic
84Dic97—42no—
85Dic97—42no—
86C2×C97—21no—
87C2×C97—41no—
88C2×C97—41no—
89C2×C97—41no—
90C2×C97C2×#9911yescharacteristic
91#2135—971nomaximal, Sylow
92C97#213511yescharacteristic, complemented, Sylow
93#536—971no—
94#413—971no—
95#413D9711yescharacteristic
96#356—974no—
97C2×#41C2×D9711yescharacteristic
98C22×Q8—971no—
99C2×M16—21no—
100C2×M16—1941no—
101C4⋊Q8—972no—
102#124—972no—
103#123—1941no—
104#123—1941no—
105C22⋊Q8—1941no—
106C22⋊Q8—1941no—
107#117—971no—
108#101—972no—
109#101C2×D9712yes—
110#100—21no—
111#100—1941no—
112#41—41no—
113#41—41no—
114C2×Q8—972no—
115C2×Q8—3881no—
116C2×Q8—3881no—
117C2×Q8—1941no—
118C2×Q8C22×D9711yescharacteristic
119C2×D4C4×D9711yescharacteristic
120C2×D4D38811yescharacteristic
121C2×D4C22⋊D9711yescharacteristic
122C22×C4—1941no—
123C22×C4C4×D9711yescharacteristic
124C22×C4D38811yescharacteristic
125C22×C4C22⋊D9711yescharacteristic
126M16—22no—
127M16—41no—
128M16—3881no—
129M16—1942no—
130C2×C8—41no—
131C2×C8—3881no—
132C4⋊C4—1941no—
133C4⋊C4—1942no—
134C4⋊C4—1942no—
135C22⋊C4—1941no—
136C22⋊C4—1942no—
137C22⋊C4—1942no—
138C42—972no—
139C23#1309548811yescharacteristic
140C23#1309548811yescharacteristic
141C23#1309548811yescharacteristic
142Q8—41no—
143Q8—3882no—
144Q8—3881no—
145Q8—3881no—
146D4—41no—
147D4—41no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—3882no—
156C2×C4—1942no—
157C2×C4—1942no—
158C2×C4#1309548811yescharacteristic
159C2×C4C2×C4×D9711yescharacteristic
160C2×C4C2×C22⋊D9711yescharacteristic
161C2×C4C2×D38811yescharacteristic
162C8—42no—
163C8—3882no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×#1309548811yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—3882no—
175C4—3882no—
176C4#1569072411yescharacteristic
177C4#1569072411yescharacteristic
178C2—21no—
179C2—41no—
180C2—41no—
181C2—41no—
182C2C2×#1569072411yescharacteristic
subgroupG/H
centerC2C2×#15690724
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting#413×C97C2
socleC2×C97C2×#99
prime radicalC2×#41×C97C22
cosocleC2×Q8×C97C23
hypercenter#413D97
hypocenterC97#2135
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #20260533C2semidirect
92. C97#2135semidirect

Maximal subgroups

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

subgroupindex
#202613062
#202607192
#202605332
#413×C972
#213597