Finite Group Databasebeta

← #27142076

Subgroups — #27142076

Order 12416.

338 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 103 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×Dic388C211yesmaximal, characteristic
2C2×C4×Dic194C211yesmaximal, characteristic
3C2×#15690735C211yesmaximal, characteristic
4C2×#15690735C211yesmaximal, characteristic
5#20260641C211yesmaximal, characteristic
6#20260498C211yesmaximal, characteristic
7#336×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C22×Dic194C2211yescharacteristic
10C2×Dic388—22no—
11C2×Dic388—22no—
12C2×Dic388C414yescomplemented
13C4×Dic194—22no—
14C4×Dic194—22no—
15C2×C4⋊Dic97C2211yescharacteristic
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490—21no—
18C2×C4×Dic97—21no—
19#15690735—22no—
20#15690735—22no—
21#15690717—21no—
22#15690717—21no—
23C22×C8×C97C2211yescharacteristic
24C22×C8×C97C2211yescharacteristic
25C2×C42×C97C2211yescharacteristic
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194—21no—
32C2×Dic194—21no—
33C2×Dic194—21no—
34C2×Dic194C2×C412yes—
35C2×Dic194C2×C412yes—
36C2×Dic194D412yes—
37C2×Dic194D412yes—
38C22×Dic97—21no—
39C22×Dic97—21no—
40C22×Dic97—21no—
41C22×Dic97—21no—
42Dic388—28no—
43C4⋊Dic97—21no—
44C4⋊Dic97—21no—
45C4⋊Dic97—21no—
46C4⋊Dic97D412yes—
47#13095490—24no—
48C4×Dic97—24no—
49C22×C4×C97D411yescharacteristic
50C22×C4×C97D411yescharacteristic
51C22×C4×C97C2311yescharacteristic
52C2×C8×C97—21no—
53C2×C8×C97—21no—
54C2×C8×C97—21no—
55C2×C8×C97—21no—
56C2×C8×C97—21no—
57C2×C8×C97C2×C412yes—
58C42×C97—21no—
59C42×C97—21no—
60C23×C97C2×D411yescharacteristic
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97Q1612yescomplemented
67C2×C4×C97C22×C411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97#4111yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97—41no—
84C2×Dic97—41no—
85Dic194—22no—
86Dic194—22no—
87Dic194—22no—
88Dic194—22no—
89Dic194—22no—
90Dic194—41no—
91Dic194—41no—
92Dic194C22⋊C414yes—
93C8×C97—22no—
94C8×C97—41no—
95C22×C97C4×D411yescharacteristic
96C22×C97#12011yescharacteristic
97C22×C97#12111yescharacteristic
98C22×C97C2×Q1611yescharacteristic
99C22×C97#13511yescharacteristic
100C22×C97#13711yescharacteristic
101C22×C97#13711yescharacteristic
102C4×C97—22no—
103C4×C97—41no—
104C4×C97C2×C22⋊C411yescharacteristic
105C4×C97C4×D411yescharacteristic
106C4×C97#12111yescharacteristic
107C4×C97#12311yescharacteristic
108Dic97—24no—
109Dic97—42no—
110Dic97—42no—
111Dic97—81no—
112C2×C97#38611yescharacteristic
113C2×C97C4×Q1611yescharacteristic
114C2×C97#44111yescharacteristic
115C2×C97#45111yescharacteristic
116C2×C97#45211yescharacteristic
117C2×C97#46211yescharacteristic
118C2×C97#46411yescharacteristic
119#1878—971nomaximal, Sylow
120C97#187811yescharacteristic, complemented, Sylow
121C22×Q16—971no—
122C2×C4×Q8—971no—
123C2×#103—971no—
124C2×#103—971no—
125#391—971no—
126#340—971no—
127#336D9711yescharacteristic
128C22×Q8—971no—
129C22×Q8—971no—
130C2×Q16—974no—
131C2×Q16—1942no—
132C2×Q16—1942no—
133C22×C8C2×D9711yescharacteristic
134C22×C8C2×D9711yescharacteristic
135C4×Q8—1942no—
136C4×Q8—1942no—
137C2×C4⋊C4—971no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—1941no—
140C2×C42—1941no—
141C2×C42C2×D9711yescharacteristic
142#103—1942no—
143#103—1942no—
144#95—1941no—
145#95—1941no—
146C2×Q8—972no—
147C2×Q8—972no—
148C2×Q8—972no—
149C2×Q8—972no—
150C2×Q8—1941no—
151C2×Q8—1941no—
152C2×Q8—1941no—
153C2×Q8—1941no—
154C2×Q8—1941no—
155C2×Q8—1941no—
156C2×Q8—1941no—
157C2×Q8—1941no—
158C22×C4—1941no—
159C22×C4—1941no—
160C22×C4—1941no—
161C22×C4—1941no—
162C22×C4D38811yescharacteristic
163C22×C4D38811yescharacteristic
164C22×C4C22×D9711yescharacteristic
165Q16—1948no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8C4×D9712yes—
172C4⋊C4—972no—
173C4⋊C4—1944no—
174C4⋊C4—1941no—
175C4⋊C4—1941no—
176C4⋊C4—1941no—
177C42—21no—
178C42—21no—
179C42—1944no—
180C23C2×D38811yescharacteristic
181Q8—974no—
182Q8—3881no—
183Q8—3881no—
184Q8—1942no—
185Q8—1942no—
186Q8—1942no—
187Q8—1942no—
188Q8—1942no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—3881no—
195C2×C4—3881no—
196C2×C4—3881no—
197C2×C4—3881no—
198C2×C4—3881no—
199C2×C4—3881no—
200C2×C4—3881no—
201C2×C4—1942no—
202C2×C4—1942no—
203C2×C4—1942no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4Dic38812yescomplemented
207C2×C4C2×C4×D9711yescharacteristic
208C2×C4D4×D9711yescharacteristic
209C2×C4D4×D9711yescharacteristic
210C2×C4D4×D9711yescharacteristic
211C2×C4#1309551111yescharacteristic
212C2×C4#1309551211yescharacteristic
213C8—22no—
214C8—41no—
215C22C4×D38811yescharacteristic
216C22#1569078011yescharacteristic
217C22#1569078511yescharacteristic
218C22C2×Dic38811yescharacteristic
219C22#1569083211yescharacteristic
220C22#1569084311yescharacteristic
221C22#1569084311yescharacteristic
222C4—22no—
223C4—41no—
224C4—7761no—
225C4—3882no—
226C4—3882no—
227C4—1944no—
228C4C22⋊C4×D9711yescharacteristic
229C4#1569077211yescharacteristic
230C4#1569078311yescharacteristic
231C4#1569079511yescharacteristic
232C2#2026061311yescharacteristic
233C2C4×Dic38811yescharacteristic
234C2#2026084311yescharacteristic
235C2#2026089311yescharacteristic
236C2#2026089911yescharacteristic
237C2#2026094611yescharacteristic
238C2#2026095811yescharacteristic
subgroupG/H
centerC23C2×D388
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#336×C97C2
socle, prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#336D97
hypocenterC97#1878
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
12. C2×Dic388C4semidirect
66. C2×C4×C97Q16semidirect
120. C97#1878semidirect
206. C2×C4Dic388semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3882
C2×C4×Dic1942
C2×#156907352
#202606412
#202604982
#336×C972
#187897