Finite Group Databasebeta

← #27157703

Subgroups — #27157703

Order 12416.

270 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 70 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×#15690897C211yesmaximal, characteristic
2C2×#15690881C211yesmaximal, characteristic, complemented
3#20261742C211yesmaximal, characteristic, complemented
4C2×#15690874C211yesmaximal, characteristic
5#20261605C211yesmaximal, characteristic, complemented
6#20261600C211yesmaximal, characteristic, complemented
7#20260728C211yesmaximal, characteristic
8#15690897—22no—
9#15690897—22no—
10C2×#13095521C2211yescharacteristic
11C2×#13095521C2211yescharacteristic
12C2×#13095520—21no—
13C2×#13095519C2211yescharacteristic
14C22×#11715897—21no—
15C2×C4×#87604C2211yescharacteristic
16#15690881—22no—
17#15690877—21no—
18C4⋊#11715897—21no—
19#15690874—22no—
20C2×#13095510C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22#15690768C2211yescharacteristic
23#15690735C411yescharacteristic, complemented
24#15690733C411yescharacteristic, complemented
25#15690731C411yescharacteristic, complemented
26#102×C97C411yescharacteristic, complemented
27C22×#87604—21no—
28#13095521—21no—
29#13095521—21no—
30#13095521—21no—
31#13095521D412yes—
32#13095520—24no—
33#13095519—21no—
34#13095519—21no—
35C2×#11715897—24no—
36C2×#11715897—21no—
37C2×#11715897—21no—
38C2×#11715897D411yescharacteristic
39C2×#11715897D411yescharacteristic
40C4×#87604—21no—
41C4×#87604—21no—
42#13095510—21no—
43#13095510—21no—
44#13095510D412yes—
45#13095510Q812yes—
46C2×Dic194C2×C411yescharacteristic
47C2×C22⋊D97—21no—
48C22×Dic97—21no—
49C2×C4×D97C2311yescharacteristic
50C2×#11715895C2×C411yescharacteristic
51C8×D97—21no—
52C8×D97—21no—
53C4⋊Dic97C2×C411yescharacteristic
54#13095488—21no—
55C4×Dic97—21no—
56C2×D4×C97C2×C411yescharacteristic
57C2×C8×C97C2×C411yescharacteristic
58C4⋊C4×C97C2×C411yescharacteristic
59C23×C97—21no—
60C22×D97#4111yescharacteristic
61C2×#87604—22no—
62C2×#87604—41no—
63C2×#87604—41no—
64D4×C97—21no—
65D4×C97C4⋊C412yes—
66C2×C4×C97—21no—
67C2×C4×C97C22×C411yescharacteristic
68C22⋊D97—22no—
69C22⋊D97—22no—
70C2×Dic97—21no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—21no—
74C2×Dic97C2×D411yescharacteristic
75C4×D97C2×D411yescharacteristic
76C4×D97C2×D411yescharacteristic
77C4×D97C2×Q811yescharacteristic
78C4×D97#4111yescharacteristic
79Dic194—21no—
80Dic194C4⋊C412yes—
81#11715897—21no—
82#11715897—22no—
83#11715897—21no—
84#11715897—41no—
85C8×C97—21no—
86#11715895—21no—
87C22×C97—22no—
88C22×C97—41no—
89C22×C97—41no—
90C22×C97C4×D411yescharacteristic
91C2×D97—21no—
92C2×D97—21no—
93C2×D97C22⋊Q811yescharacteristic
94C2×D97#12311yescharacteristic
95#87604—42no—
96C4×C97—41no—
97C4×C97C2×C4⋊C411yescharacteristic
98C4×C97C4×D411yescharacteristic
99Dic97—22no—
100Dic97—41no—
101Dic97#12011yescharacteristic
102Dic97C22⋊Q811yescharacteristic
103C2×C97—42no—
104C2×C97#38511yescharacteristic
105C2×C97#44311yescharacteristic
106C2×C97#44411yescharacteristic
107D97—21no—
108D97—21no—
109#1866—971nomaximal, Sylow
110C97#186611yescharacteristic, complemented, Sylow
111C2×#131—971no—
112C2×#108—971no—
113#424—971no—
114C2×#104—971no—
115#416—971no—
116#339—971no—
117#336—971no—
118C2×#41—971no—
119#131—1942no—
120#131—1942no—
121C2×M16—971no—
122C2×M16—971no—
123C2×M16—1941no—
124C22×C8—971no—
125C22×C8—971no—
126C22×C8—1941no—
127#117—971no—
128C2×C42—971no—
129#108—1942no—
130C4⋊C8—1941no—
131C4⋊C8—1941no—
132#104—1942no—
133#103—971no—
134#103—971no—
135#102—971no—
136#102#8760411yescharacteristic, complemented
137#41—972no—
138#41—972no—
139#41—1941no—
140#41—1941no—
141C2×Q8—971no—
142C2×D4—1941no—
143C2×D4C2×#8760411yescharacteristic
144C22×C4—971no—
145C22×C4—1941no—
146C22×C4—1941no—
147M16—972no—
148M16—1941no—
149M16—1941no—
150M16—1941no—
151M16—1944no—
152C2×C8—971no—
153C2×C8—971no—
154C2×C8—971no—
155C2×C8—1941no—
156C2×C8—1941no—
157C2×C8—1941no—
158C2×C8—1941no—
159C2×C8—1941no—
160C2×C8—1941no—
161C2×C8—1944no—
162C2×C8C2×#8760411yescharacteristic
163C4⋊C4—971no—
164C4⋊C4C2×#8760411yescharacteristic
165C22⋊C4—1941no—
166C42—1941no—
167C42—1941no—
168C42—1941no—
169C23—21no—
170C23—971no—
171Q8—972no—
172Q8—1941no—
173D4—21no—
174D4—1942no—
175D4—1942no—
176D4C4⋊#8760412yes—
177C2×C4—21no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—971no—
182C2×C4—971no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—1942no—
186C2×C4—1942no—
187C2×C4—1942no—
188C2×C4—1941no—
189C2×C4—1941no—
190C2×C4C22×#8760411yescharacteristic
191C8—21no—
192C8—3881no—
193C8—1941no—
194C8—1941no—
195C8—1942no—
196C8—1941no—
197C22—22no—
198C22—41no—
199C22—41no—
200C22—971no—
201C22—971no—
202C22—1941no—
203C22—1941no—
204C22D4×#8760411yescharacteristic
205C4—41no—
206C4—971no—
207C4—971no—
208C4—3882no—
209C4—3881no—
210C4—1942no—
211C4C2×C4⋊#8760411yescharacteristic
212C4D4×#8760411yescharacteristic
213C2—42no—
214C2—1941no—
215C2—1941no—
216C2#2026168211yescharacteristic
217C2#2026177711yescharacteristic
218C2#2026178011yescharacteristic
subgroupG/H
centerC22D4×#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#102×C97C4
socleC22×C97C4×D4
prime radicalC2×#13095510C22
cosocleC2×C4×D97C23
hypercenter#102#87604
hypocenterC97#1866
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#15690881C2semidirect
3. #20261742C2semidirect
5. #20261605C2semidirect
6. #20261600C2semidirect
23. #15690735C4semidirect
24. #15690733C4semidirect
25. #15690731C4semidirect
26. #102×C97C4semidirect
110. C97#1866semidirect
136. #102#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908972
C2×#156908812
#202617422
C2×#156908742
#202616052
#202616002
#202607282
#186697