Finite Group Databasebeta

← #41011968

Subgroups — #41011968

Order 27008.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C22⋊C8×D211C211yesmaximal, characteristic
3#27213005C211yesmaximal, 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×C8×D211—21no—
11#20269957C411yescharacteristic, complemented
12#20269956C2211yescharacteristic
13#20269955—21no—
14#20269954—21no—
15#20269953C411yescharacteristic, complemented
16C2×Dic211⋊C4—21no—
17C4⋊C4×D211—21no—
18C4⋊C4×D211C411yescharacteristic, complemented
19C2×(C2×D211)⋊C4—21no—
20C22⋊C4×D211—21no—
21C22⋊(C211⋊C8)C2211yescharacteristic
22(C2×D211)⋊C8—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
40C2×C211⋊C8—21no—
41C8×D211—42no—
42C4⋊Dic211—21no—
43C4⋊Dic211C2×C411yescharacteristic
44Dic211⋊C4—21no—
45Dic211⋊C4—21no—
46Dic211⋊C4—21no—
47Dic211⋊C4—42no—
48C22⋊Dic211—21no—
49(C2×D211)⋊C4—21no—
50(C2×D211)⋊C4—21no—
51(C2×D211)⋊C4—42no—
52C2×Q8×C211C2×C411yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55C2×C8×C211—21no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211C2×C411yescharacteristic
58C22⋊C4×C211—21no—
59C22×D211—41no—
60C22×D211—41no—
61C22×D211C22⋊C411yescharacteristic
62C22×D211C22⋊C411yescharacteristic
63C22×D211SD1612yes—
64C22×D211Q1612yes—
65Dic422—42no—
66Dic422—41no—
67C2×Dic211—22no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211C22⋊C411yescharacteristic
74C2×Dic211C22⋊C411yescharacteristic
75C4×D211—22no—
76C4×D211—22no—
77C4×D211—41no—
78C4×D211—41no—
79C4×D211—42no—
80C4×D211—42no—
81C211⋊C8—41no—
82C23×C211C2×D411yescharacteristic
83Q8×C211—41no—
84C2×C4×C211—21no—
85C2×C4×C211—21no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211C22×C411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C8×C211—41no—
93C22×C211—21no—
94C22×C211—21no—
95C22×C211C2×C22⋊C411yescharacteristic
96C22×C211C2×SD1611yescharacteristic
97C22×C211C2×Q1611yescharacteristic
98C2×D211—21no—
99C2×D211—21no—
100C2×D211—21no—
101C2×D211—21no—
102C2×D211—42no—
103C2×D211—42no—
104C2×D211#10312yes—
105C2×D211#10312yes—
106C4×C211—21no—
107C4×C211—21no—
108C4×C211—41no—
109C4×C211—41no—
110C4×C211—81no—
111Dic211—21no—
112Dic211—41no—
113Dic211—41no—
114Dic211—81no—
115Dic211#9912yes—
116C2×C211—21no—
117C2×C211—21no—
118C2×C211C2×#9911yescharacteristic
119C2×C211C2×#10311yescharacteristic
120C2×C211#42111yescharacteristic
121D211—22no—
122D211—22no—
123C211#152011yescharacteristic, complemented, Sylow
124#1520—2111nomaximal, Sylow
125C2×C22⋊Q8—2111no—
126C2×C22⋊C8—2111no—
127#380—2111no—
128#328—2111no—
129#328—2111no—
130#328—2111no—
131#328D21111yescharacteristic, complemented
132C22×Q8—2111no—
133C23×C4—2111no—
134C22×C8—4221no—
135C22⋊Q8—4221no—
136C22⋊Q8—4221no—
137C22⋊Q8—2111no—
138C22⋊Q8—2111no—
139C22⋊Q8—2111no—
140C22⋊Q8C2×D21111yescharacteristic
141C2×C4⋊C4—4221no—
142C2×C4⋊C4—4221no—
143C2×C4⋊C4—2111no—
144C2×C22⋊C4—4221no—
145C2×C22⋊C4—4221no—
146C22⋊C8—4221no—
147C22⋊C8—2111no—
148C22⋊C8C2×D21111yescharacteristic
149#95—2111no—
150#95C2×D21111yescharacteristic
151C24—2111no—
152C2×Q8—8442no—
153C2×Q8—4221no—
154C2×Q8—2111no—
155C2×Q8C4×D21111yescharacteristic
156C22×C4—21no—
157C22×C4—8442no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—4221no—
161C22×C4—2111no—
162C22×C4—2111no—
163C22×C4—2111no—
164C22×C4—2111no—
165C22×C4—2111no—
166C22×C4C22×D21111yescharacteristic
167C2×C8—21no—
168C2×C8—8442no—
169C2×C8—4221no—
170C4⋊C4—21no—
171C4⋊C4—8442no—
172C4⋊C4—4221no—
173C4⋊C4—4221no—
174C4⋊C4—4221no—
175C4⋊C4—4221no—
176C4⋊C4—2111no—
177C4⋊C4C4×D21111yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—8442no—
180C22⋊C4—4221no—
181C22⋊C4—4221no—
182C22⋊C4—4221no—
183C23—8441no—
184C23—8441no—
185C23—2112no—
186C23—2112no—
187C23—2111no—
188C23—2111no—
189C23D4×D21111yescharacteristic
190Q8—41no—
191Q8—8441no—
192Q8—8442no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—8441no—
200C2×C4—8441no—
201C2×C4—8442no—
202C2×C4—8442no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—8441no—
206C2×C4—4222no—
207C2×C4—4222no—
208C2×C4—4221no—
209C2×C4—4221no—
210C2×C4—4222no—
211C2×C4—2111no—
212C2×C4—2111no—
213C2×C4C2×C4×D21111yescharacteristic
214C2×C4D4×D21111yescharacteristic
215C8—41no—
216C8—8441no—
217C22—21no—
218C22—21no—
219C22—8442no—
220C22—8442no—
221C22—4221no—
222C22—4221no—
223C22—4221no—
224C22—4221no—
225C22—2112no—
226C22—2112no—
227C22C22⋊C4×D21111yescharacteristic
228C22SD16×D21111yescharacteristic
229C22Q16×D21111yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—8441no—
236C4—8441no—
237C4—16881no—
238C4—4221no—
239C4—2112no—
240C2—21no—
241C2—21no—
242C2—4222no—
243C2—4222no—
244C2#99×D21111yescharacteristic
245C2#103×D21111yescharacteristic
246C2#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#1520
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
18. C4⋊C4×D211C4semidirect
123. C211#1520semidirect
131. #328D211semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2112
C22⋊C8×D2112
#272130052
#272128842
#272128832
#272128822
#328×C2112
#1520211