Finite Group Databasebeta

← #27142782

Subgroups — #27142782

Order 12416.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#15690841C212yesmaximal, complemented
2#20261316C211yesmaximal, characteristic
3C2×#15690738C212yesmaximal, complemented
4#20260504C211yesmaximal, characteristic, complemented
5#391×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2212yes—
7#15690841—24no—
8#15690841—24no—
9C2×#13095501—22no—
10C2×#13095498—22no—
11C2×#13095495C2212yes—
12#15690790—22no—
13#15690785—22no—
14#15690784—22no—
15#15690738—24no—
16C22×Q8×C97C2211yescharacteristic
17C2×C4⋊C4×C97C2211yescharacteristic
18C2×C42×C97C2211yescharacteristic
19#95×C97—22no—
20#13095511—22no—
21#13095511—22no—
22#13095511—22no—
23#13095511—22no—
24C2×Dic194D412yes—
25C2×D388—22no—
26C2×D388D412yes—
27C2×C22⋊D97—22no—
28C2×C4×D97—22no—
29#13095501—28no—
30#13095498—28no—
31#13095495—22no—
32#13095495D414yescomplemented
33C2×#11715895—22no—
34C4⋊Dic97—22no—
35#13095488—24no—
36C2×Q8×C97—21no—
37C2×Q8×C97—22no—
38C2×Q8×C97—22no—
39C2×Q8×C97D412yescomplemented
40C22×C4×C97—21no—
41C22×C4×C97—21no—
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—22no—
45C42×C97—22no—
46C23×C97C2×D411yescharacteristic
47C22×D97—22no—
48Q8×C97—24no—
49Q8×C97—42no—
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97—42no—
58C2×C4×C97C2×D412yes—
59C2×C4×C97C2×D412yes—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97#4111yescharacteristic
62C22⋊D97—42no—
63C22⋊D97—42no—
64C2×Dic97—22no—
65D388—22no—
66D388—22no—
67D388—82no—
68C4×D97—42no—
69C4×D97—42no—
70Dic194—22no—
71Dic194—22no—
72#11715895—24no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97#12011yescharacteristic
76C22×C97#12111yescharacteristic
77C22×C97#12711yescharacteristic
78C2×D97—42no—
79C2×D97—42no—
80C2×D97—42no—
81C4×C97—42no—
82C4×C97—42no—
83C4×C97—42no—
84C4×C97#12012yes—
85C4×C97#12112yes—
86Dic97—42no—
87C2×C97—21no—
88C2×C97—21no—
89C2×C97#39211yescharacteristic
90C2×C97#45511yescharacteristic
91C2×C97#45611yescharacteristic
92D97—82no—
93#2007—971nomaximal, Sylow
94C97#200711yescharacteristic, complemented, Sylow
95C2×#137—972no—
96#537—971no—
97C2×#104—972no—
98#391D9711yescharacteristic, complemented
99#343—971no—
100C2×#41—972no—
101C22×Q8C2×D9711yescharacteristic
102#137—1944no—
103#137—1944no—
104C2×Q16—1942no—
105C2×SD16—1942no—
106C2×M16—972no—
107C22⋊Q8—1942no—
108#121—1942no—
109#121—1942no—
110C2×C4⋊C4C2×D9711yescharacteristic
111C2×C42C2×D9711yescharacteristic
112#104—1944no—
113#95—22no—
114#41—1942no—
115#41—1942no—
116#41—1942no—
117#41—1942no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—972no—
122C2×Q8D38812yescomplemented
123C2×D4—972no—
124C2×D4—1942no—
125C2×D4—1942no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—1942no—
129C22×C4C22⋊D9712yes—
130C22×C4C22×D9711yescharacteristic
131Q16—1948no—
132SD16—1948no—
133M16—974no—
134M16—1942no—
135C2×C8—1942no—
136C4⋊C4—22no—
137C4⋊C4—1942no—
138C22⋊C4—1944no—
139C42—22no—
140C23—1942no—
141C23C2×C22⋊D9711yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—1942no—
145Q8—1942no—
146D4—7762no—
147D4—3882no—
148D4—3882no—
149D4—1942no—
150D4—1942no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—3882no—
160C2×C4—3882no—
161C2×C4—1942no—
162C2×C4D4×D9712yes—
163C2×C4D4×D9712yes—
164C2×C4C2×D38811yescharacteristic
165C2×C4#1309551111yescharacteristic
166C8—1944no—
167C22—21no—
168C22—21no—
169C22—3882no—
170C22—3882no—
171C22—3882no—
172C22#1569077811yescharacteristic
173C22#1569078411yescharacteristic
174C22#1569080611yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—3882no—
179C4#1569078012yes—
180C4#1569078312yes—
181C2—21no—
182C2—21no—
183C2—7762no—
184C2#2026064411yescharacteristic
185C2#2026091211yescharacteristic
186C2#2026091711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#391×C97C2
socleC22×C97#120
prime radical#20261316C2
hypercenter#391D97
hypocenterC97#2007
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
3. C2×#15690738C2semidirect
4. #20260504C2semidirect
5. #391×C97C2semidirect
32. #13095495D4semidirect
39. C2×Q8×C97D4semidirect
94. C97#2007semidirect
98. #391D97semidirect
122. C2×Q8D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
#202613162
C2×#156907382
#202605042
#391×C972
#200797