Finite Group Databasebeta

← #27142045

Subgroups — #27142045

Order 12416.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15690786C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic
3C2×#15690734C211yesmaximal, characteristic, complemented
4C2×#15690722C211yesmaximal, characteristic
5#20260498C211yesmaximal, characteristic, complemented
6#20260497C211yesmaximal, characteristic, complemented
7C2×#103×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10#15690786—22no—
11#15690786C414yescomplemented
12#15690768—22no—
13#15690768—22no—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490—21no—
17C2×#13095488—21no—
18C2×#13095488—21no—
19C2×C4×Dic97—21no—
20#15690734—22no—
21#15690722—22no—
22C22×Q8×C97C2211yescharacteristic
23C22×C8×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25#103×C97—22no—
26C23×D97D411yescharacteristic
27C22×Dic97—21no—
28C22×Dic97—21no—
29C22×Dic97D411yescharacteristic
30C2×C4×D97—21no—
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97C2×C412yes—
35C2×C4×D97D412yes—
36C2×#11715895—21no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97—21no—
41C4⋊Dic97C2×C412yes—
42C4⋊Dic97D412yes—
43#13095490—24no—
44#13095488—24no—
45#13095488—24no—
46C4×Dic97—24no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97—21no—
50C2×Q8×C97—21no—
51C2×Q8×C97—21no—
52C2×Q8×C97C2×C412yes—
53C22×C4×C97—21no—
54C22×C4×C97—21no—
55C22×C4×C97C2311yescharacteristic
56C2×C8×C97—21no—
57C2×C8×C97—21no—
58C2×C8×C97—21no—
59C4⋊C4×C97—21no—
60C4⋊C4×C97D412yes—
61C23×C97C2×D411yescharacteristic
62C22×D97—21no—
63C22×D97—21no—
64C22×D97—21no—
65C22×D97—21no—
66C22×D97—21no—
67C22×D97—21no—
68C22×D97C22⋊C412yes—
69Q8×C97—22no—
70Q8×C97—22no—
71Q8×C97—41no—
72Q8×C97—41no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97—41no—
79C2×C4×C97C22×C411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C2×C4×C97#4111yescharacteristic
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97—41no—
92C2×Dic97—41no—
93C2×Dic97C22⋊C412yes—
94C4×D97—22no—
95C4×D97—22no—
96C4×D97—22no—
97C4×D97—22no—
98C8×C97—41no—
99#11715895—41no—
100C22×C97C2×C22⋊C411yescharacteristic
101C22×C97#12011yescharacteristic
102C22×C97#12311yescharacteristic
103C22×C97#13511yescharacteristic
104C22×C97#13511yescharacteristic
105C22×C97#13711yescharacteristic
106C22×C97#13711yescharacteristic
107C2×D97—22no—
108C2×D97—22no—
109C2×D97—22no—
110C2×D97—41no—
111C2×D97—41no—
112C2×D97—41no—
113C2×D97—41no—
114C4×C97—42no—
115C4×C97—42no—
116C4×C97C4×D411yescharacteristic
117C4×C97C4×D411yescharacteristic
118C4×C97#12111yescharacteristic
119C4×C97#12111yescharacteristic
120Dic97—22no—
121Dic97—42no—
122Dic97—42no—
123C2×C97#38611yescharacteristic
124C2×C97#41611yescharacteristic
125C2×C97#41911yescharacteristic
126C2×C97#45211yescharacteristic
127C2×C97#45211yescharacteristic
128C2×C97#48511yescharacteristic
129C2×C97#48511yescharacteristic
130D97—42no—
131#1873—971nomaximal, Sylow
132C97#187311yescharacteristic, complemented, Sylow
133C2×C22⋊Q8—971no—
134C2×#117—971no—
135C2×#103—971no—
136C2×#103D9711yescharacteristic, complemented
137C2×C22⋊C8—971no—
138#340—971no—
139#340—971no—
140C22×Q8C2×D9711yescharacteristic
141C23×C4—971no—
142C22×C8—971no—
143C22×C8C2×D9711yescharacteristic
144C22⋊Q8—974no—
145C22⋊Q8—1942no—
146#117—1942no—
147#117—1942no—
148C2×C4⋊C4—971no—
149C2×C4⋊C4—971no—
150C2×C4⋊C4—1941no—
151C2×C4⋊C4C2×D9711yescharacteristic
152C2×C22⋊C4—1941no—
153C2×C22⋊C4—1941no—
154C2×C42—1941no—
155#103—22no—
156#103—1942no—
157C22⋊C8—1942no—
158C24—971no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8C4×D9712yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—971no—
168C22×C4—972no—
169C22×C4—972no—
170C22×C4—1941no—
171C22×C4—1941no—
172C22×C4—1941no—
173C22×C4—1941no—
174C22×C4—1941no—
175C22×C4—1941no—
176C22×C4C22×D9711yescharacteristic
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—1941no—
181C2×C8—1941no—
182C2×C8—1941no—
183C4⋊C4—21no—
184C4⋊C4—972no—
185C4⋊C4—972no—
186C4⋊C4—1944no—
187C4⋊C4—1941no—
188C4⋊C4—1941no—
189C4⋊C4C22⋊D9712yes—
190C22⋊C4—1944no—
191C22⋊C4—1944no—
192C42—1944no—
193C23—972no—
194C23—1941no—
195C23—1941no—
196C23—1941no—
197C23—1941no—
198C23—1941no—
199C23—1941no—
200C23D4×D9711yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—41no—
204Q8—41no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—972no—
212C2×C4—3881no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—3881no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1942no—
223C2×C4—1942no—
224C2×C4C2×C4×D9711yescharacteristic
225C2×C4D4×D9711yescharacteristic
226C2×C4D4×D9711yescharacteristic
227C2×C4C2×C22⋊D9711yescharacteristic
228C2×C4#1309551011yescharacteristic
229C2×C4#1309551111yescharacteristic
230C8—41no—
231C8—3881no—
232C22—3881no—
233C22—3881no—
234C22—3881no—
235C22—3881no—
236C22—1942no—
237C22—1942no—
238C22—1942no—
239C22C22⋊C4×D9711yescharacteristic
240C22#1569077811yescharacteristic
241C22#1569079211yescharacteristic
242C22#1569083111yescharacteristic
243C22#1569083311yescharacteristic
244C22#1569083911yescharacteristic
245C22#1569084211yescharacteristic
246C4—42no—
247C4—42no—
248C4—3882no—
249C4—3882no—
250C4—1942no—
251C4#1569077011yescharacteristic
252C4C4×C22⋊D9711yescharacteristic
253C4#1569078111yescharacteristic
254C4#1569078311yescharacteristic
255C2—3882no—
256C2#2026061111yescharacteristic
257C2#2026073011yescharacteristic
258C2#2026074211yescharacteristic
259C2#2026089411yescharacteristic
260C2#2026089611yescharacteristic
261C2#2026107411yescharacteristic
262C2#2026107511yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socleC23×C97C2×D4
prime radicalC23×D97D4
cosocleC22×C4×C97C23
hypercenterC2×#103D97
hypocenterC97#1873
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#15690734C2semidirect
5. #20260498C2semidirect
6. #20260497C2semidirect
7. C2×#103×C97C2semidirect
11. #15690786C4semidirect
132. C97#1873semidirect
136. C2×#103D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907862
C2×#156907682
C2×#156907342
C2×#156907222
#202604982
#202604972
C2×#103×C972
#187397