Finite Group Databasebeta

← #27142881

Subgroups — #27142881

Order 12416.

294 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15690801C211yesmaximal, characteristic, complemented
2C2×#15690732C211yesmaximal, characteristic
3C2×#15690730C211yesmaximal, characteristic
4#20260636C211yesmaximal, characteristic
5#20260497C211yesmaximal, characteristic, complemented
6#20260489C211yesmaximal, characteristic, complemented
7#340×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10#15690801—22no—
11#15690801—22no—
12C2×C4⋊Dic97—21no—
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095490—21no—
15C2×#13095488—21no—
16C2×#13095488—21no—
17C2×C4×Dic97C2211yescharacteristic
18#15690732—22no—
19#15690730—22no—
20C22×C8×C97C2211yescharacteristic
21C2×C4⋊C4×C97C2211yescharacteristic
22C2×C4⋊C4×C97C2211yescharacteristic
23C23×D97—21no—
24C2×D388—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388Q812yes—
30C22×Dic97—21no—
31C22×Dic97D411yescharacteristic
32C22×Dic97D411yescharacteristic
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895—21no—
36C4⋊Dic97—24no—
37C4⋊Dic97—21no—
38C4⋊Dic97D412yes—
39#13095490—24no—
40#13095488—42no—
41#13095488—42no—
42C4×Dic97—21no—
43C4×Dic97—21no—
44C22×C4×C97—21no—
45C22×C4×C97—21no—
46C22×C4×C97C2311yescharacteristic
47C2×C8×C97—21no—
48C2×C8×C97—21no—
49C2×C8×C97—21no—
50C4⋊C4×C97—21no—
51C4⋊C4×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97D412yes—
54C23×C97C2×D411yescharacteristic
55C22×D97—22no—
56C22×D97—41no—
57C22×D97—41no—
58C22×D97—41no—
59C22×D97—41no—
60C22×D97—41no—
61C22×D97—41no—
62C2×C4×C97—22no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×Q811yescharacteristic
71C2×C4×C97#4111yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—22no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97—21no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83D388—22no—
84D388—22no—
85D388—41no—
86D388—41no—
87C8×C97—41no—
88#11715895—41no—
89C22×C97#12011yescharacteristic
90C22×C97C22⋊Q811yescharacteristic
91C22×C97#12411yescharacteristic
92C22×C97#13511yescharacteristic
93C22×C97#13511yescharacteristic
94C22×C97AGL(1,ℤ/8)11yescharacteristic
95C22×C97AGL(1,ℤ/8)11yescharacteristic
96C2×D97—42no—
97C2×D97—42no—
98C2×D97—42no—
99C2×D97—81no—
100C2×D97—81no—
101C2×D97—81no—
102C2×D97—81no—
103C4×C97—42no—
104C4×C97—81no—
105C4×C97C22⋊Q811yescharacteristic
106C4×C97C22⋊Q811yescharacteristic
107C4×C97#12411yescharacteristic
108C4×C97#12411yescharacteristic
109Dic97—41no—
110Dic97—41no—
111Dic97—42no—
112C2×C97#39311yescharacteristic
113C2×C97#44911yescharacteristic
114C2×C97#44911yescharacteristic
115C2×C97#47811yescharacteristic
116C2×C97#47811yescharacteristic
117C2×C97#48811yescharacteristic
118C2×C97#49011yescharacteristic
119D97—82no—
120#2025—971nomaximal, Sylow
121C97#202511yescharacteristic, complemented, Sylow
122C2×#125—971no—
123C2×#102—971no—
124C2×#102—971no—
125#390—971no—
126#340—971no—
127#340D9711yescharacteristic, complemented
128#336—971no—
129C22×D4—971no—
130C22×C8—971no—
131C22×C8C2×D9711yescharacteristic
132#125—1942no—
133#125—1942no—
134C2×C4⋊C4—971no—
135C2×C4⋊C4—1941no—
136C2×C4⋊C4—1941no—
137C2×C4⋊C4C2×D9711yescharacteristic
138C2×C4⋊C4C2×D9711yescharacteristic
139C2×C22⋊C4—1941no—
140C2×C22⋊C4—1941no—
141C2×C42—971no—
142#102—1942no—
143#102—1942no—
144C24—1941no—
145C2×D4—972no—
146C2×D4—1941no—
147C2×D4—1941no—
148C2×D4—1941no—
149C2×D4—1941no—
150C2×D4—1941no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—971no—
154C22×C4—971no—
155C22×C4—1941no—
156C22×C4C22×D9711yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—1941no—
161C2×C8—1941no—
162C2×C8—1941no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—972no—
167C4⋊C4—1944no—
168C4⋊C4—1944no—
169C4⋊C4—1941no—
170C4⋊C4D38812yes—
171C22⋊C4—3882no—
172C22⋊C4—3882no—
173C42—1941no—
174C42—1941no—
175C23—3881no—
176C23—3881no—
177C23—3881no—
178C23—3881no—
179C23—3881no—
180C23—3881no—
181C23—1942no—
182C23D4×D9711yescharacteristic
183D4—3881no—
184D4—3881no—
185D4—1942no—
186D4—1942no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—3881no—
194C2×C4—3881no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4—1942no—
198C2×C4—1941no—
199C2×C4—1941no—
200C2×C4—1941no—
201C2×C4—1941no—
202C2×C4D4×D9711yescharacteristic
203C2×C4C2×D38811yescharacteristic
204C2×C4Q8×D9711yescharacteristic
205C2×C4#1309551011yescharacteristic
206C2×C4#1309551111yescharacteristic
207C2×C4#1309551111yescharacteristic
208C8—41no—
209C8—3881no—
210C22—7761no—
211C22—7761no—
212C22—7761no—
213C22—7761no—
214C22—3882no—
215C22—3882no—
216C22—3882no—
217C22#1569078011yescharacteristic
218C22#1569078811yescharacteristic
219C22#1569079611yescharacteristic
220C22#1569083111yescharacteristic
221C22#1569083311yescharacteristic
222C22#1569083411yescharacteristic
223C22#1569083711yescharacteristic
224C4—42no—
225C4—81no—
226C4—3881no—
227C4—3882no—
228C4—3881no—
229C4#1569078811yescharacteristic
230C4#1569079011yescharacteristic
231C4#1569079611yescharacteristic
232C4#1569079811yescharacteristic
233C2—7762no—
234C2#2026064811yescharacteristic
235C2#2026088111yescharacteristic
236C2#2026088311yescharacteristic
237C2#2026103411yescharacteristic
238C2#2026103811yescharacteristic
239C2#2026109111yescharacteristic
240C2#2026109811yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
hypercenter#340D97
hypocenterC97#2025
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#15690801C2semidirect
5. #20260497C2semidirect
6. #20260489C2semidirect
7. #340×C97C2semidirect
121. C97#2025semidirect
127. #340D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908012
C2×#156907322
C2×#156907302
#202606362
#202604972
#202604892
#340×C972
#202597