Finite Group Databasebeta

← #41011929

Subgroups — #41011929

Order 27008.

274 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 63 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⋊Q8×D211C211yesmaximal, characteristic
2#27213108C211yesmaximal, characteristic
3#27213008C211yesmaximal, characteristic
4#27212884C211yesmaximal, characteristic, complemented
5#27212883C211yesmaximal, characteristic, complemented
6#27212882C211yesmaximal, characteristic, complemented
7#328×C211C211yesmaximal, characteristic, complemented
8C2×Q8×D211C411yescharacteristic, complemented
9C22×C4×D211C2211yescharacteristic
10C2×#16809945—21no—
11#20269957C411yescharacteristic, complemented
12#20269956C2211yescharacteristic
13#20269955—21no—
14#20269954—21no—
15#20269953C411yescharacteristic, complemented
16C4⋊C4×D211—21no—
17C4⋊C4×D211C411yescharacteristic, complemented
18C2×(C2×D211)⋊C4—21no—
19C22⋊C4×D211—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21(C2×D211)⋊C8—21no—
22C211⋊#95—21no—
23C211⋊#95C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C2×Dic422—21no—
29C2×Dic422C2×C411yescharacteristic
30Q8×D211—42no—
31C22×Dic211—21no—
32C22×Dic211D411yescharacteristic
33C2×C4×D211—21no—
34C2×C4×D211—21no—
35C2×C4×D211—42no—
36C2×C4×D211C2×C411yescharacteristic
37C2×C4×D211C2×C411yescharacteristic
38C2×C4×D211D411yescharacteristic
39C2×C4×D211D411yescharacteristic
40#16809945—42no—
41C2×C211⋊C8—21no—
42C4⋊Dic211—21no—
43C4⋊Dic211C2×C411yescharacteristic
44Dic211⋊C4—21no—
45Dic211⋊C4—21no—
46Dic211⋊C4—21no—
47C22⋊Dic211—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—21no—
50(C2×D211)⋊C4—42no—
51C2×Q8×C211C2×C411yescharacteristic
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C4⋊C4×C211—21no—
56C4⋊C4×C211C2×C411yescharacteristic
57C22⋊C4×C211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211—22no—
61C22×D211—41no—
62C22×D211C22⋊C411yescharacteristic
63C22×D211C22⋊C411yescharacteristic
64Dic422—42no—
65Dic422—41no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C2×Dic211—41no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211C22⋊C411yescharacteristic
73C2×Dic211C22⋊C411yescharacteristic
74C4×D211—22no—
75C4×D211—22no—
76C4×D211—41no—
77C4×D211—41no—
78C4×D211—42no—
79C4×D211—42no—
80C211⋊C8—41no—
81C23×C211C2×D411yescharacteristic
82Q8×C211—41no—
83C2×C4×C211—21no—
84C2×C4×C211—21no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211C22×C411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C8×C211—41no—
92C22×C211—21no—
93C22×C211—21no—
94C22×C211C2×C22⋊C411yescharacteristic
95C22×C211#13711yescharacteristic
96C22×C211#13711yescharacteristic
97C2×D211—21no—
98C2×D211—21no—
99C2×D211—21no—
100C2×D211—21no—
101C2×D211—21no—
102C2×D211—42no—
103C2×D211—42no—
104C2×D211#9912yes—
105C4×C211—21no—
106C4×C211—21no—
107C4×C211—41no—
108C4×C211—41no—
109C4×C211—81no—
110Dic211—21no—
111Dic211—21no—
112Dic211—41no—
113Dic211—41no—
114Dic211—81no—
115C2×C211—21no—
116C2×C211—21no—
117C2×C211C2×#9911yescharacteristic
118C2×C211#41911yescharacteristic
119C2×C211#42111yescharacteristic
120D211—22no—
121D211—41no—
122C211#151111yescharacteristic, complemented, Sylow
123#1511—2111nomaximal, Sylow
124C2×C22⋊Q8—2111no—
125#407—2111no—
126#381—2111no—
127#328—2111no—
128#328—2111no—
129#328—2111no—
130#328D21111yescharacteristic, complemented
131C22×Q8—2111no—
132C23×C4—2111no—
133C2×M16—4221no—
134C22⋊Q8—4221no—
135C22⋊Q8—4221no—
136C22⋊Q8—2111no—
137C22⋊Q8—2111no—
138C22⋊Q8—2111no—
139C22⋊Q8C2×D21111yescharacteristic
140C2×C4⋊C4—4221no—
141C2×C4⋊C4—2111no—
142C2×C22⋊C4—4221no—
143C2×C22⋊C4—4221no—
144C22⋊C8—4221no—
145C22⋊C8—2111no—
146C22⋊C8C2×D21111yescharacteristic
147#95—4221no—
148#95—2111no—
149#95C2×D21111yescharacteristic
150C24—2111no—
151C2×Q8—8442no—
152C2×Q8—4221no—
153C2×Q8—2111no—
154C2×Q8C4×D21111yescharacteristic
155C22×C4—21no—
156C22×C4—8442no—
157C22×C4—4221no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4—2111no—
162C22×C4—2111no—
163C22×C4—2111no—
164C22×C4—2111no—
165C22×C4C22×D21111yescharacteristic
166M16—8442no—
167C2×C8—21no—
168C2×C8—4221no—
169C4⋊C4—21no—
170C4⋊C4—4221no—
171C4⋊C4—4221no—
172C4⋊C4—4221no—
173C4⋊C4—4221no—
174C4⋊C4—2111no—
175C4⋊C4C4×D21111yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—8442no—
178C22⋊C4—4221no—
179C22⋊C4—4221no—
180C22⋊C4—4221no—
181C23—8441no—
182C23—4222no—
183C23—4221no—
184C23—4221no—
185C23—2111no—
186C23—2111no—
187C23D4×D21111yescharacteristic
188Q8—41no—
189Q8—8441no—
190Q8—8442no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—8442no—
200C2×C4—8442no—
201C2×C4—8441no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—4222no—
206C2×C4—4222no—
207C2×C4—4221no—
208C2×C4—4221no—
209C2×C4—2111no—
210C2×C4—2111no—
211C2×C4C2×C4×D21111yescharacteristic
212C2×C4D4×D21111yescharacteristic
213C8—41no—
214C8—8441no—
215C22—21no—
216C22—21no—
217C22—8442no—
218C22—8442no—
219C22—4221no—
220C22—4221no—
221C22—4221no—
222C22—4221no—
223C22—4221no—
224C22—2112no—
225C22C22⋊C4×D21111yescharacteristic
226C22#2027000611yescharacteristic
227C22#2027000911yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—8441no—
234C4—8441no—
235C4—16881no—
236C4—4221no—
237C4—4221no—
238C2—21no—
239C2—21no—
240C2—8441no—
241C2—4222no—
242C2#99×D21111yescharacteristic
243C2#2721315311yescharacteristic
244C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×D211D4
cosocleC22×C4×C211C23
hypercenter#328D211
hypocenterC211#1511
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #27212884C2semidirect
5. #27212883C2semidirect
6. #27212882C2semidirect
7. #328×C211C2semidirect
8. C2×Q8×D211C4semidirect
11. #20269957C4semidirect
15. #20269953C4semidirect
17. C4⋊C4×D211C4semidirect
122. C211#1511semidirect
130. #328D211semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2112
#272131082
#272130082
#272128842
#272128832
#272128822
#328×C2112
#1511211