Finite Group Databasebeta

← #27157721

Subgroups — #27157721

Order 12416.

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

normal subgroups only

This group has 404 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×C4⋊#87604C211yesmaximal, characteristic
2C2×#15690873C212yesmaximal, complemented
3#20261703C211yesmaximal, characteristic
4#20261608C212yesmaximal, complemented
5C22⋊Q8×D97C211yesmaximal, characteristic
6C23×#87604—21no—
7C2×#13095519C2212yes—
8C2×C4⋊#87604—24no—
9C2×C4⋊#87604—21no—
10C2×C4⋊#87604C2212yes—
11#15690873—24no—
12#15690867—21no—
13C22⋊#11715897—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—24no—
26C22×#87604—22no—
27C22×#87604—42no—
28C22×#87604—42no—
29#13095519—22no—
30#13095519—22no—
31C2×#11715897—22no—
32C4⋊#87604—24no—
33C4⋊#87604—22no—
34C4⋊#87604D414yes—
35C23×D97D411yescharacteristic
36C2×Dic194—21no—
37C2×Dic194C2×C411yescharacteristic, complemented
38Q8×D97—22no—
39Q8×D97—22no—
40C22×Dic97D411yescharacteristic
41C2×C4×D97—22no—
42C2×C4×D97—22no—
43C2×C4×D97—21no—
44C2×C4×D97—21no—
45C2×C4×D97C2×C411yescharacteristic
46C2×C4×D97D411yescharacteristic
47C2×C4×D97D411yescharacteristic
48C2×C4×D97C2311yescharacteristic
49C4⋊Dic97—21no—
50C4⋊Dic97C2×C411yescharacteristic, complemented
51#13095490—21no—
52#13095490—21no—
53#13095490—21no—
54C22⋊Dic97—21no—
55#13095488—21no—
56#13095488—21no—
57C2×Q8×C97C2×C411yescharacteristic, complemented
58C22×C4×C97C2×C411yescharacteristic
59C4⋊C4×C97—21no—
60C4⋊C4×C97C2×C411yescharacteristic, complemented
61C22⋊C4×C97—21no—
62C23×C97C22⋊C411yescharacteristic
63C22×D97—21no—
64C22×D97—21no—
65C22×D97—21no—
66C22×D97—21no—
67C22×D97C22⋊C411yescharacteristic
68C22×D97SD1612yes—
69C22×D97Q1612yes—
70C22×D97C2×D411yescharacteristic
71C2×#87604—24no—
72C2×#87604—48no—
73C2×#87604—42no—
74C2×#87604—42no—
75Q8×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—21no—
78C2×C4×C97—21no—
79C2×C4×C97C22⋊C411yescharacteristic
80C2×C4×C97C22×C411yescharacteristic
81C2×Dic97—22no—
82C2×Dic97—21no—
83C2×Dic97—21no—
84C2×Dic97C22⋊C411yescharacteristic
85C2×Dic97C2×D411yescharacteristic
86C4×D97—21no—
87C4×D97—21no—
88C4×D97—22no—
89C4×D97—22no—
90C4×D97—41no—
91C4×D97—41no—
92C4×D97—41no—
93C4×D97—41no—
94C4×D97C2×D412yes—
95C4×D97#4112yes—
96Dic194—22no—
97Dic194—42no—
98#11715897—42no—
99C22×C97—21no—
100C22×C97—21no—
101C22×C97#10312yescomplemented
102C22×C97C2×C22⋊C411yescharacteristic
103C2×D97—21no—
104C2×D97—21no—
105C2×D97—22no—
106C2×D97—22no—
107C2×D97—22no—
108C2×D97—22no—
109C2×D97#10312yescomplemented
110C2×D97#12011yescharacteristic
111C2×D97#12311yescharacteristic
112C2×D97C2×SD1611yescharacteristic
113C2×D97C2×Q1611yescharacteristic
114C2×D97AGL(1,ℤ/8)11yescharacteristic
115C2×D97#13711yescharacteristic
116#87604—44no—
117C4×C97—21no—
118C4×C97—41no—
119C4×C97—41no—
120C4×C97C4×D412yes—
121Dic97—21no—
122Dic97—41no—
123Dic97—41no—
124Dic97#12112yes—
125C2×C97—22no—
126C2×C97#38611yescharacteristic
127C2×C97C2×#10311yescharacteristic
128C2×C97#41711yescharacteristic
129D97—22no—
130D97#45011yescharacteristic
131D97#45111yescharacteristic
132D97#48311yescharacteristic
133D97#48411yescharacteristic
134#1872—971nomaximal, Sylow
135C97#187211yescharacteristic, complemented, Sylow
136C2×C22⋊Q8—971no—
137C22×C4⋊C4—971no—
138C2×#103—972no—
139C2×C22⋊C8—971no—
140#340—972no—
141C22×Q8—971no—
142C23×C4—971no—
143C23×C4—1941no—
144C22×C8—972no—
145C22⋊Q8—971no—
146C22⋊Q8—971no—
147C22⋊Q8—971no—
148C22⋊Q8—1941no—
149C22⋊Q8—1941no—
150C22⋊Q8#8760411yescharacteristic, complemented
151C2×C4⋊C4—972no—
152C2×C4⋊C4—971no—
153C2×C4⋊C4—1941no—
154C2×C4⋊C4—1944no—
155C2×C4⋊C4—1941no—
156C2×C22⋊C4—1941no—
157#103—1944no—
158C22⋊C8—1941no—
159C22⋊C8—1941no—
160C24—971no—
161C2×Q8—971no—
162C2×Q8—1941no—
163C2×Q8—1942no—
164C2×Q8—1942no—
165C2×Q8C2×#8760411yescharacteristic, complemented
166C22×C4—971no—
167C22×C4—971no—
168C22×C4—971no—
169C22×C4—971no—
170C22×C4—971no—
171C22×C4—3882no—
172C22×C4—3882no—
173C22×C4—1942no—
174C22×C4—1942no—
175C22×C4—1941no—
176C22×C4—1941no—
177C22×C4—1942no—
178C22×C4—1944no—
179C22×C4C2×#8760411yescharacteristic
180C2×C8—1942no—
181C2×C8—1942no—
182C2×C8—1942no—
183C4⋊C4—21no—
184C4⋊C4—974no—
185C4⋊C4—971no—
186C4⋊C4—1941no—
187C4⋊C4—1941no—
188C4⋊C4—1941no—
189C4⋊C4—1942no—
190C4⋊C4—1944no—
191C4⋊C4—1941no—
192C4⋊C4C2×#8760411yescharacteristic, complemented
193C22⋊C4—21no—
194C22⋊C4—1941no—
195C22⋊C4—1941no—
196C22⋊C4—1941no—
197C23—971no—
198C23—972no—
199C23—972no—
200C23—971no—
201C23—1941no—
202C23—1941no—
203C23—1941no—
204C23—1941no—
205C23C22⋊#8760411yescharacteristic
206Q8—22no—
207Q8—3882no—
208Q8—1942no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—971no—
213C2×C4—972no—
214C2×C4—972no—
215C2×C4—971no—
216C2×C4—3881no—
217C2×C4—3881no—
218C2×C4—3881no—
219C2×C4—3881no—
220C2×C4—3882no—
221C2×C4—3882no—
222C2×C4—3888no—
223C2×C4—1942no—
224C2×C4—1941no—
225C2×C4—1941no—
226C2×C4—1942no—
227C2×C4—1944no—
228C2×C4—1941no—
229C2×C4—1941no—
230C2×C4—1942no—
231C2×C4C22⋊#8760411yescharacteristic
232C2×C4C22×#8760411yescharacteristic
233C8—3882no—
234C22—21no—
235C22—21no—
236C22—971no—
237C22—971no—
238C22—971no—
239C22—971no—
240C22—971no—
241C22—971no—
242C22—972no—
243C22—1942no—
244C22—1942no—
245C22—1942no—
246C22—1942no—
247C22—1941no—
248C22—1941no—
249C22#1569087312yescomplemented
250C22C2×C22⋊#8760411yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—972no—
255C4—3884no—
256C4—3881no—
257C4—3881no—
258C4—1941no—
259C4D4×#8760412yes—
260C2—22no—
261C2—971no—
262C2—971no—
263C2—971no—
264C2—971no—
265C2—1942no—
266C2#2026168311yescharacteristic
267C2C2×#1569087311yescharacteristic
268C2#2026172611yescharacteristic
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#1872
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
37. C2×Dic194C2×C4semidirect
50. C4⋊Dic97C2×C4semidirect
57. C2×Q8×C97C2×C4semidirect
60. C4⋊C4×C97C2×C4semidirect
101. C22×C97#103semidirect
109. C2×D97#103semidirect
135. C97#1872semidirect
150. C22⋊Q8#87604semidirect
165. C2×Q8C2×#87604semidirect
192. C4⋊C4C2×#87604semidirect
249. C22#15690873semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#876042
C2×#156908732
#202617032
#202616082
C22⋊Q8×D972
#187297