Finite Group Databasebeta

← #27157749

Subgroups — #27157749

Order 12416.

380 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×#13095519C211yesmaximal, characteristic
2C2×#15690873C212yesmaximal, complemented
3#20261677C211yesmaximal, characteristic
4#20261608C212yesmaximal, complemented
5C22⋊Q8×D97C211yesmaximal, characteristic
6C2×#13095519—24no—
7C2×#13095519—21no—
8C2×#13095519C2212yes—
9C22×#11715897—21no—
10C2×C4⋊#87604C2212yes—
11C2×C22⋊#87604—21no—
12#15690873—24no—
13(C2×#87604)⋊C4—21no—
14C2×Q8×D97C2211yescharacteristic
15C22×C4×D97C2211yescharacteristic
16#15690790C411yescharacteristic, complemented
17#15690789C411yescharacteristic, complemented
18#15690788—21no—
19#15690787—21no—
20#15690786C411yescharacteristic, complemented
21C4⋊C4×D97—21no—
22C4⋊C4×D97C2211yescharacteristic
23C22⋊C4×D97—21no—
24C22⋊Q8×C97C411yescharacteristic, complemented
25C22×#87604—22no—
26#13095519—24no—
27#13095519—22no—
28#13095519D414yes—
29C2×#11715897—24no—
30C2×#11715897—22no—
31C4⋊#87604—22no—
32C4⋊#87604—22no—
33C22⋊#87604—42no—
34C23×D97D411yescharacteristic
35C2×Dic194—21no—
36C2×Dic194C2×C411yescharacteristic
37Q8×D97—22no—
38Q8×D97—22no—
39C22×Dic97D411yescharacteristic
40C2×C4×D97—22no—
41C2×C4×D97—22no—
42C2×C4×D97—21no—
43C2×C4×D97—21no—
44C2×C4×D97C2×C411yescharacteristic
45C2×C4×D97D411yescharacteristic
46C2×C4×D97D411yescharacteristic
47C2×C4×D97C2311yescharacteristic
48C4⋊Dic97—21no—
49C4⋊Dic97C2×C411yescharacteristic
50#13095490—21no—
51#13095490—21no—
52#13095490—21no—
53C22⋊Dic97—21no—
54#13095488—21no—
55#13095488—21no—
56C2×Q8×C97C2×C411yescharacteristic
57C22×C4×C97C2×C411yescharacteristic
58C4⋊C4×C97—21no—
59C4⋊C4×C97C2×C411yescharacteristic
60C22⋊C4×C97—21no—
61C23×C97C22⋊C411yescharacteristic
62C22×D97—22no—
63C22×D97—22no—
64C22×D97C22⋊C411yescharacteristic
65C22×D97SD1612yes—
66C22×D97Q1612yes—
67C22×D97C2×D411yescharacteristic
68C2×#87604—42no—
69C2×#87604—42no—
70C2×#87604—42no—
71Q8×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97C22⋊C411yescharacteristic
76C2×C4×C97C22×C411yescharacteristic
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97—21no—
81C2×Dic97C22⋊C411yescharacteristic
82C2×Dic97C2×D411yescharacteristic
83C4×D97—22no—
84C4×D97—22no—
85C4×D97—21no—
86C4×D97—21no—
87C4×D97—42no—
88C4×D97—42no—
89C4×D97C2×D412yes—
90C4×D97#4112yes—
91Dic194—22no—
92Dic194—41no—
93Dic194—41no—
94#11715897—24no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97#10312yescomplemented
98C22×C97C2×C22⋊C411yescharacteristic
99C2×D97—21no—
100C2×D97—21no—
101C2×D97—24no—
102C2×D97—24no—
103C2×D97#10312yescomplemented
104C2×D97#12112yes—
105C2×D97C2×SD1611yescharacteristic
106C2×D97C2×Q1611yescharacteristic
107C2×D97#13512yes—
108#87604—82no—
109C4×C97—21no—
110C4×C97—41no—
111C4×C97—41no—
112C4×C97C4×D412yes—
113Dic97—21no—
114Dic97—41no—
115Dic97—41no—
116Dic97#12011yescharacteristic
117Dic97#12311yescharacteristic
118C2×C97—22no—
119C2×C97#38611yescharacteristic
120C2×C97C2×#10311yescharacteristic
121C2×C97#41611yescharacteristic
122D97—22no—
123D97#46512yes—
124D97#46712yes—
125#1892—971nomaximal, Sylow
126C97#189211yescharacteristic, complemented, Sylow
127C23×C8—971no—
128C2×C22⋊Q8—971no—
129C2×#103—972no—
130#380—971no—
131#340—972no—
132C22×Q8—971no—
133C23×C4—971no—
134C22×C8—972no—
135C22×C8—1944no—
136C22×C8—1941no—
137C22×C8—1941no—
138C22⋊Q8—971no—
139C22⋊Q8—971no—
140C22⋊Q8—971no—
141C22⋊Q8—1941no—
142C22⋊Q8—1941no—
143C22⋊Q8#8760411yescharacteristic, complemented
144C2×C4⋊C4—971no—
145C2×C4⋊C4—972no—
146C2×C4⋊C4—1941no—
147C2×C22⋊C4—1941no—
148C2×C22⋊C4—1941no—
149#103—1944no—
150#95—1941no—
151C24—971no—
152C2×Q8—971no—
153C2×Q8—1941no—
154C2×Q8—1942no—
155C2×Q8—1942no—
156C2×Q8C2×#8760411yescharacteristic
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—1942no—
163C22×C4—1942no—
164C22×C4—1941no—
165C22×C4—1941no—
166C22×C4—1942no—
167C22×C4C2×#8760411yescharacteristic
168C2×C8—974no—
169C2×C8—1942no—
170C2×C8—1942no—
171C2×C8—1944no—
172C2×C8—1944no—
173C4⋊C4—21no—
174C4⋊C4—971no—
175C4⋊C4—1941no—
176C4⋊C4—1941no—
177C4⋊C4—1941no—
178C4⋊C4—1942no—
179C4⋊C4—1942no—
180C4⋊C4—1941no—
181C4⋊C4C2×#8760411yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—3882no—
184C22⋊C4—1941no—
185C22⋊C4—1941no—
186C22⋊C4—1941no—
187C23—971no—
188C23—972no—
189C23—972no—
190C23—971no—
191C23—1942no—
192C23—1942no—
193C23C22⋊#8760411yescharacteristic
194Q8—22no—
195Q8—3881no—
196Q8—3881no—
197Q8—1942no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—971no—
202C2×C4—972no—
203C2×C4—972no—
204C2×C4—971no—
205C2×C4—3882no—
206C2×C4—3882no—
207C2×C4—3882no—
208C2×C4—3882no—
209C2×C4—3882no—
210C2×C4—1941no—
211C2×C4—1941no—
212C2×C4—1942no—
213C2×C4—1941no—
214C2×C4—1941no—
215C2×C4—1941no—
216C2×C4—1941no—
217C2×C4—1942no—
218C2×C4C22⋊#8760411yescharacteristic
219C2×C4C22×#8760411yescharacteristic
220C8—1944no—
221C22—21no—
222C22—21no—
223C22—972no—
224C22—971no—
225C22—971no—
226C22—972no—
227C22—972no—
228C22—1944no—
229C22—1944no—
230C22—1941no—
231C22—1941no—
232C22#1569087312yescomplemented
233C22C2×C22⋊#8760411yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—7762no—
238C4—971no—
239C4—971no—
240C4—3881no—
241C4—3881no—
242C4—1941no—
243C4D4×#8760412yes—
244C2—22no—
245C2—972no—
246C2—972no—
247C2—1942no—
248C2#2026168311yescharacteristic
249C2C2×#1569087311yescharacteristic
250C2#2026172511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC22⋊Q8#87604
hypocenterC97#1892
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15690873C2semidirect
4. #20261608C2semidirect
16. #15690790C4semidirect
17. #15690789C4semidirect
20. #15690786C4semidirect
24. C22⋊Q8×C97C4semidirect
97. C22×C97#103semidirect
103. C2×D97#103semidirect
126. C97#1892semidirect
143. C22⋊Q8#87604semidirect
232. C22#15690873semidirect

Maximal subgroups

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

subgroupindex
C22×#130955192
C2×#156908732
#202616772
#202616082
C22⋊Q8×D972
#189297