Finite Group Databasebeta

← #27142871

Subgroups — #27142871

Order 12416.

326 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15690809C211yesmaximal, 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×Dic194—21no—
9C22×D388C2211yescharacteristic
10C22×#11715895C2211yescharacteristic
11#15690809—22no—
12#15690809—22no—
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×Dic194—24no—
25C2×Dic194—42no—
26C2×Dic194—42no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388—21no—
32C2×D388Q812yescomplemented
33C22×Dic97—21no—
34C22×Dic97D411yescharacteristic
35C22×Dic97D411yescharacteristic
36C2×#11715895—21no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97D412yes—
41#13095490—24no—
42#13095488—42no—
43#13095488—42no—
44C4×Dic97—21no—
45C4×Dic97—21no—
46C22×C4×C97—21no—
47C22×C4×C97—21no—
48C22×C4×C97C2311yescharacteristic
49C2×C8×C97—21no—
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97—21no—
55C4⋊C4×C97D412yes—
56C23×C97C2×D411yescharacteristic
57C22×D97—22no—
58C22×D97—41no—
59C22×D97—41no—
60C22×D97—41no—
61C22×D97—41no—
62C22×D97—41no—
63C22×D97—41no—
64C2×C4×C97—22no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×Q811yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×C4×C97#4111yescharacteristic
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97—22no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97D812yes—
84C2×Dic97SD1612yes—
85D388—22no—
86D388—22no—
87D388—41no—
88D388—41no—
89Dic194—48no—
90C8×C97—41no—
91#11715895—41no—
92C22×C97#12011yescharacteristic
93C22×C97C22⋊Q811yescharacteristic
94C22×C97#12411yescharacteristic
95C22×C97C2×D811yescharacteristic
96C22×C97C2×SD1611yescharacteristic
97C22×C97AGL(1,ℤ/8)11yescharacteristic
98C22×C97#13711yescharacteristic
99C2×D97—42no—
100C2×D97—42no—
101C2×D97—42no—
102C2×D97—81no—
103C2×D97—81no—
104C2×D97—81no—
105C2×D97—81no—
106C4×C97—42no—
107C4×C97—81no—
108C4×C97C22⋊Q811yescharacteristic
109C4×C97C22⋊Q811yescharacteristic
110C4×C97#12411yescharacteristic
111C4×C97#12411yescharacteristic
112Dic97—22no—
113Dic97—22no—
114Dic97—42no—
115C2×C97#39311yescharacteristic
116C2×C97#44711yescharacteristic
117C2×C97#44811yescharacteristic
118C2×C97#47411yescharacteristic
119C2×C97#47611yescharacteristic
120C2×C97#48611yescharacteristic
121C2×C97#48911yescharacteristic
122D97—82no—
123#2023—971nomaximal, Sylow
124C97#202311yescharacteristic, complemented, Sylow
125C2×C4⋊Q8—971no—
126C2×#102—971no—
127C2×#102—971no—
128#390—971no—
129#340—971no—
130#340D9711yescharacteristic, complemented
131#336—971no—
132C22×Q8—1941no—
133C22×D4—971no—
134C22×C8—971no—
135C22×C8C2×D9711yescharacteristic
136C4⋊Q8—1942no—
137C4⋊Q8—1942no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—1941no—
140C2×C4⋊C4C2×D9711yescharacteristic
141C2×C4⋊C4C2×D9711yescharacteristic
142C2×C22⋊C4—1941no—
143C2×C22⋊C4—1941no—
144C2×C42—971no—
145#102—1942no—
146#102—1942no—
147C24—1941no—
148C2×Q8—3882no—
149C2×Q8—3882no—
150C2×Q8—1944no—
151C2×D4—972no—
152C2×D4—1941no—
153C2×D4—1941no—
154C2×D4—1941no—
155C2×D4—1941no—
156C2×D4—1941no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—971no—
160C22×C4—971no—
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—972no—
173C4⋊C4—1944no—
174C4⋊C4—1941no—
175C4⋊C4D38812yes—
176C22⋊C4—3882no—
177C22⋊C4—3882no—
178C42—1941no—
179C42—1941no—
180C23—3881no—
181C23—3881no—
182C23—3881no—
183C23—3881no—
184C23—3881no—
185C23—3881no—
186C23—1942no—
187C23D4×D9711yescharacteristic
188Q8—3888no—
189D4—3881no—
190D4—3881no—
191D4—1942no—
192D4—1942no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—972no—
200C2×C4—972no—
201C2×C4—3881no—
202C2×C4—3881no—
203C2×C4—1942no—
204C2×C4—1941no—
205C2×C4—1941no—
206C2×C4—1941no—
207C2×C4—1941no—
208C2×C4D4×D9711yescharacteristic
209C2×C4C2×D38811yescharacteristic
210C2×C4Q8×D9711yescharacteristic
211C2×C4#1309551011yescharacteristic
212C2×C4#1309551111yescharacteristic
213C2×C4#1309551111yescharacteristic
214C8—41no—
215C8—3881no—
216C22—7761no—
217C22—7761no—
218C22—7761no—
219C22—7761no—
220C22—3882no—
221C22—3882no—
222C22—3882no—
223C22#1569078011yescharacteristic
224C22#1569078811yescharacteristic
225C22#1569079611yescharacteristic
226C22D8×D9711yescharacteristic
227C22SD16×D9711yescharacteristic
228C22#1569083711yescharacteristic
229C22#1569084211yescharacteristic
230C4—42no—
231C4—81no—
232C4—3882no—
233C4—1942no—
234C4—1942no—
235C4#1569078811yescharacteristic
236C4#1569079011yescharacteristic
237C4#1569079611yescharacteristic
238C4#1569079811yescharacteristic
239C2—7762no—
240C2#2026064811yescharacteristic
241C2#2026087011yescharacteristic
242C2#2026087511yescharacteristic
243C2#2026101411yescharacteristic
244C2#2026102411yescharacteristic
245C2#2026108011yescharacteristic
246C2#2026109611yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
hypercenter#340D97
hypocenterC97#2023
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×#15690809C2semidirect
5. #20260497C2semidirect
6. #20260489C2semidirect
7. #340×C97C2semidirect
32. C2×D388Q8semidirect
124. C97#2023semidirect
130. #340D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908092
C2×#156907322
C2×#156907302
#202606362
#202604972
#202604892
#340×C972
#202397