Finite Group Databasebeta

← #27141729

Subgroups — #27141729

Order 12416.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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×C4×Dic194C211yesmaximal, characteristic
2#20260759C211yesmaximal, characteristic
3C2×#15690728C212yesmaximal
4#20260503C212yesmaximal
5#423×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C2×#13095495C2212yes—
8C4×Dic194—21no—
9C4×Dic194—21no—
10C4×Dic194—21no—
11C4×Dic194—21no—
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490C412yes—
14C2×C4×Dic97C412yes—
15C4⋊#11715895—21no—
16C4⋊#11715895—21no—
17#15690728—24no—
18C2×M16×C97C2212yes—
19C2×C42×C97C2211yescharacteristic
20C4⋊C8×C97—21no—
21C4⋊C8×C97—21no—
22C2×Dic194—21no—
23C2×Dic194—21no—
24C2×Dic194—21no—
25C2×Dic194—21no—
26C2×Dic194D412yes—
27C2×Dic194Q812yes—
28C22×Dic97C2×C412yes—
29C22×Dic97C2×C412yes—
30#13095495—22no—
31#13095495—22no—
32C2×#11715895—22no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35#13095490—22no—
36#13095490—22no—
37C4×Dic97—22no—
38C4×Dic97—22no—
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2×C411yescharacteristic
41C22×C4×C97C2311yescharacteristic
42M16×C97—22no—
43M16×C97—22no—
44C2×C8×C97—22no—
45C42×C97D411yescharacteristic
46C42×C97D411yescharacteristic
47C42×C97D411yescharacteristic
48C42×C97D411yescharacteristic
49C23×C97C22×C411yescharacteristic
50C2×C4×C97—22no—
51C2×C4×C97—41no—
52C2×C4×C97C22⋊C411yescharacteristic
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×Q811yescharacteristic
60C2×C4×C97#4111yescharacteristic
61C2×C4×C97#4111yescharacteristic
62C2×Dic97—22no—
63C2×Dic97—22no—
64C2×Dic97—22no—
65C2×Dic97—42no—
66C2×Dic97C4⋊C414yes—
67Dic194—22no—
68Dic194—22no—
69Dic194—22no—
70Dic194—22no—
71C8×C97—42no—
72#11715895—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97C2×C22⋊C411yescharacteristic
76C22×C97C2×C4⋊C411yescharacteristic
77C22×C97#11711yescharacteristic
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97#10112yescomplemented
82C4×C97#12111yescharacteristic
83C4×C97#12111yescharacteristic
84C4×C97C22⋊Q811yescharacteristic
85C4×C97C22⋊Q811yescharacteristic
86Dic97—24no—
87Dic97—42no—
88C2×C97—21no—
89C2×C97—21no—
90C2×C97#38011yescharacteristic
91C2×C97C2×#10111yescharacteristic
92C2×C97#41311yescharacteristic
93#1801—971nomaximal, Sylow
94C97#180111yescharacteristic, complemented, Sylow
95C2×C4×Q8—971no—
96#423—971no—
97#423D9711yescharacteristic
98C2×#101—972no—
99#343—972no—
100C22×Q8—971no—
101C2×M16—972no—
102C2×M16C2×D9712yes—
103C4×Q8—1941no—
104C4×Q8—1941no—
105C4×Q8—1941no—
106C4×Q8—1941no—
107C2×C4⋊C4—972no—
108C2×C4⋊C4—971no—
109C2×C42—972no—
110C2×C42C2×D9711yescharacteristic
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—1941no—
114C4⋊C8—1941no—
115#101—1944no—
116C2×Q8—972no—
117C2×Q8—972no—
118C2×Q8—1941no—
119C2×Q8—1941no—
120C2×Q8—1941no—
121C2×Q8—1941no—
122C22×C4—972no—
123C22×C4—972no—
124C22×C4C4×D9711yescharacteristic
125C22×C4C4×D9711yescharacteristic
126C22×C4C22×D9711yescharacteristic
127M16—22no—
128M16—22no—
129M16—1942no—
130M16—1942no—
131C2×C8—22no—
132C2×C8—1942no—
133C4⋊C4—1942no—
134C4⋊C4—1942no—
135C4⋊C4—1941no—
136C4⋊C4—1941no—
137C42—1942no—
138C42—1942no—
139C42D38811yescharacteristic
140C42D38811yescharacteristic
141C42C22⋊D9711yescharacteristic
142C42C22⋊D9711yescharacteristic
143C23C2×C4×D9711yescharacteristic
144Q8—1942no—
145Q8—1942no—
146Q8—1942no—
147Q8—1942no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—974no—
151C2×C4—3882no—
152C2×C4—1942no—
153C2×C4—1942no—
154C2×C4—1942no—
155C2×C4#1309548811yescharacteristic
156C2×C4#1309548811yescharacteristic
157C2×C4#1309548811yescharacteristic
158C2×C4#1309548811yescharacteristic
159C2×C4D4×D9711yescharacteristic
160C2×C4C2×C22⋊D9711yescharacteristic
161C2×C4C2×D38811yescharacteristic
162C2×C4Q8×D9711yescharacteristic
163C2×C4#1309551011yescharacteristic
164C2×C4#1309551211yescharacteristic
165C8—42no—
166C8—3882no—
167C22—21no—
168C22—21no—
169C22C2×#1309548811yescharacteristic
170C22C4⋊C4×D9711yescharacteristic
171C22#1569076811yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—3882no—
176C4—1944no—
177C4#1569072812yescomplemented
178C4#1569078111yescharacteristic
179C4#1569078511yescharacteristic
180C4#1569078611yescharacteristic
181C4#1569079011yescharacteristic
182C2—21no—
183C2—21no—
184C2#2026059211yescharacteristic
185C2C2×#1569072811yescharacteristic
186C2#2026071711yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutator, prime radicalC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#423×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#423D97
hypocenterC97#1801
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
81. C4×C97#101semidirect
94. C97#1801semidirect
177. C4#15690728semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic1942
#202607592
C2×#156907282
#202605032
#423×C972
#180197