Finite Group Databasebeta

← #27141745

Subgroups — #27141745

Order 12416.

356 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15690772C211yesmaximal, characteristic
2C2×#15690739C211yesmaximal, characteristic, complemented
3C2×#15690732C211yesmaximal, characteristic
4C2×#15690730C211yesmaximal, characteristic
5#20260600C211yesmaximal, characteristic, complemented
6#20260497C211yesmaximal, characteristic, complemented
7#340×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C22×C4×D97—21no—
10C22×#11715895C2211yescharacteristic
11#15690772—22no—
12#15690772C414yescomplemented
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095488—21no—
15C2×C4×Dic97C2211yescharacteristic
16#15690739—22no—
17#15690732—22no—
18#15690730—22no—
19#15690717—21no—
20#15690717—21no—
21C22×C8×C97C2211yescharacteristic
22C2×C4⋊C4×C97C2211yescharacteristic
23C2×C4⋊C4×C97C2211yescharacteristic
24C23×D97—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388C2×C412yes—
29C2×D388D412yes—
30C2×D388Q812yes—
31C22×Dic97—21no—
32C22×Dic97D411yescharacteristic
33C22×Dic97D411yescharacteristic
34C2×C4×D97—24no—
35C2×C4×D97—42no—
36C2×C4×D97—42no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C2×#11715895—21no—
40C4⋊Dic97—21no—
41C4⋊Dic97—21no—
42#13095488—24no—
43C4×Dic97C2×C412yes—
44C4×Dic97D412yes—
45C22×C4×C97—21no—
46C22×C4×C97—21no—
47C22×C4×C97C2311yescharacteristic
48C2×C8×C97—21no—
49C2×C8×C97—21no—
50C2×C8×C97—21no—
51C4⋊C4×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97C2×C412yes—
55C23×C97C2×D411yescharacteristic
56C22×D97—22no—
57C22×D97—22no—
58C22×D97—22no—
59C22×D97—41no—
60C22×D97—41no—
61C22×D97—41no—
62C22×D97—41no—
63C2×C4×C97—22no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97C22×C411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×Q811yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97C22⋊C412yes—
83C2×Dic97C22⋊C412yes—
84D388—22no—
85D388—22no—
86D388—22no—
87D388C4⋊C414yes—
88C4×D97—48no—
89C8×C97—41no—
90#11715895—41no—
91C22×C97C2×C22⋊C411yescharacteristic
92C22×C97#12111yescharacteristic
93C22×C97C22⋊Q811yescharacteristic
94C22×C97#13511yescharacteristic
95C22×C97#13511yescharacteristic
96C22×C97AGL(1,ℤ/8)11yescharacteristic
97C22×C97AGL(1,ℤ/8)11yescharacteristic
98C2×D97—24no—
99C2×D97—42no—
100C2×D97—42no—
101C2×D97—42no—
102C2×D97—42no—
103C2×D97—42no—
104C2×D97—42no—
105C4×C97—42no—
106C4×C97—81no—
107C4×C97C2×C4⋊C411yescharacteristic
108C4×C97#11711yescharacteristic
109C4×C97#12111yescharacteristic
110C4×C97C22⋊Q811yescharacteristic
111Dic97—22no—
112Dic97—22no—
113Dic97—81no—
114C2×C97#38011yescharacteristic
115C2×C97#41611yescharacteristic
116C2×C97#41811yescharacteristic
117C2×C97#46311yescharacteristic
118C2×C97#46311yescharacteristic
119C2×C97#47811yescharacteristic
120C2×C97#47811yescharacteristic
121D97—44no—
122#1804—971nomaximal, Sylow
123C97#180411yescharacteristic, complemented, Sylow
124C2×C4×D4—971no—
125C2×C4⋊C8—971no—
126C2×#102—971no—
127C2×#102—971no—
128#382—971no—
129#340—971no—
130#340D9711yescharacteristic, complemented
131C22×D4—971no—
132C23×C4—1941no—
133C22×C8—971no—
134C22×C8C2×D9711yescharacteristic
135C4×D4—974no—
136C4×D4—1942no—
137C2×C4⋊C4—971no—
138C2×C4⋊C4C2×D9711yescharacteristic
139C2×C4⋊C4C2×D9711yescharacteristic
140C2×C22⋊C4—1941no—
141C2×C42—971no—
142C4⋊C8—1942no—
143#102—1942no—
144#102—1942no—
145#95—1941no—
146#95—1941no—
147C24—1941no—
148C2×D4—972no—
149C2×D4—972no—
150C2×D4—972no—
151C2×D4—1941no—
152C2×D4—1941no—
153C2×D4—1941no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—3882no—
159C22×C4—3882no—
160C22×C4—1944no—
161C22×C4—1941no—
162C22×C4C22×D9711yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—1941no—
167C2×C8—1941no—
168C2×C8—1941no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C4⋊C4C4×D9712yes—
175C22⋊C4—1944no—
176C42—972no—
177C42—972no—
178C23—3881no—
179C23—3881no—
180C23—3881no—
181C23—3881no—
182C23—1942no—
183C23—1942no—
184C23—1942no—
185C23D4×D9711yescharacteristic
186D4—974no—
187D4—1942no—
188D4—1942no—
189D4—1942no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—972no—
197C2×C4—972no—
198C2×C4—3888no—
199C2×C4—3881no—
200C2×C4—3881no—
201C2×C4—3881no—
202C2×C4—1941no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4—1941no—
206C2×C4C2×C4×D9711yescharacteristic
207C2×C4D4×D9711yescharacteristic
208C2×C4D4×D9711yescharacteristic
209C2×C4Q8×D9711yescharacteristic
210C2×C4#1309551111yescharacteristic
211C2×C4#1309551111yescharacteristic
212C8—41no—
213C8—3881no—
214C22—3882no—
215C22—3882no—
216C22—3882no—
217C22—3882no—
218C22—3882no—
219C22—3882no—
220C22—1944no—
221C22C22⋊C4×D9711yescharacteristic
222C22#1569078311yescharacteristic
223C22#1569078811yescharacteristic
224C22#1569083111yescharacteristic
225C22#1569083311yescharacteristic
226C22#1569083411yescharacteristic
227C22#1569083711yescharacteristic
228C4—42no—
229C4—81no—
230C4—7761no—
231C4—1942no—
232C4—1942no—
233C4C4⋊C4×D9711yescharacteristic
234C4#1569076711yescharacteristic
235C4#1569078311yescharacteristic
236C4#1569078811yescharacteristic
237C2—3884no—
238C2#2026059411yescharacteristic
239C2#2026073011yescharacteristic
240C2#2026073811yescharacteristic
241C2#2026094911yescharacteristic
242C2#2026095311yescharacteristic
243C2#2026103411yescharacteristic
244C2#2026103811yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenter#340D97
hypocenterC97#1804
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#15690739C2semidirect
5. #20260600C2semidirect
6. #20260497C2semidirect
7. #340×C97C2semidirect
12. #15690772C4semidirect
123. C97#1804semidirect
130. #340D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907722
C2×#156907392
C2×#156907322
C2×#156907302
#202606002
#202604972
#340×C972
#180497