Finite Group Databasebeta

← #41011940

Subgroups — #41011940

Order 27008.

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

normal subgroups only

This group has 286 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
1#27213659C211yesmaximal, characteristic
2#27213108C211yesmaximal, characteristic, complemented
3#27213005C211yesmaximal, characteristic, complemented
4#27212883C211yesmaximal, characteristic, complemented
5#27212881C211yesmaximal, characteristic, complemented
6#27212879C211yesmaximal, characteristic, complemented
7#328×C211C211yesmaximal, characteristic, complemented
8C2×#16809963C411yescharacteristic, complemented
9C22×C4×D211C2211yescharacteristic
10C2×#16809945—21no—
11#20269962—21no—
12#20269953C411yescharacteristic, complemented
13#20269952C411yescharacteristic, complemented
14#20269951C2211yescharacteristic
15#20269947—21no—
16#20269939—21no—
17(C22⋊D211)⋊C4—21no—
18#20269935C411yescharacteristic, complemented
19C2×Dic211⋊C4—21no—
20C2×(C2×D211)⋊C4—21no—
21C22⋊(C211⋊C8)C2211yescharacteristic
22(C2×D211)⋊C8—21no—
23C211⋊#95C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28#16809963—42no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844C2×C411yescharacteristic
32C2×C22⋊D211—21no—
33C22×Dic211—21no—
34C22×Dic211D411yescharacteristic
35C2×C4×D211—21no—
36C2×C4×D211—42no—
37C2×C4×D211C2×C411yescharacteristic
38C2×C4×D211C2×C411yescharacteristic
39C2×C4×D211D411yescharacteristic
40C2×C4×D211D411yescharacteristic
41#16809945—42no—
42C2×C211⋊C8—21no—
43C4⋊Dic211C2×C411yescharacteristic
44Dic211⋊C4—21no—
45Dic211⋊C4—42no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—21no—
50(C2×D211)⋊C4—42no—
51C4×Dic211—21no—
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—21no—
60C22×D211—21no—
61C22×D211—21no—
62C22×D211—41no—
63C22×D211—41no—
64C22×D211C22⋊C411yescharacteristic
65C22×D211C22⋊C411yescharacteristic
66D844—41no—
67D844—42no—
68D844—42no—
69C22⋊D211—41no—
70C22⋊D211—41no—
71C2×Dic211—22no—
72C2×Dic211—21no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211C22⋊C411yescharacteristic
77C2×Dic211C22⋊C411yescharacteristic
78C4×D211—22no—
79C4×D211—22no—
80C4×D211—41no—
81C4×D211—41no—
82C4×D211—42no—
83C211⋊C8—41no—
84C23×C211C2×D411yescharacteristic
85Q8×C211—41no—
86C2×C4×C211—21no—
87C2×C4×C211—21no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211C22×C411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C8×C211—41no—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211C2×C22⋊C411yescharacteristic
98C22×C211AGL(1,ℤ/8)11yescharacteristic
99C22×C211#13711yescharacteristic
100C2×D211—21no—
101C2×D211—21no—
102C2×D211—21no—
103C2×D211—21no—
104C2×D211—21no—
105C2×D211—21no—
106C2×D211—42no—
107C2×D211—42no—
108C2×D211—41no—
109C2×D211—41no—
110C2×D211—41no—
111C4×C211—21no—
112C4×C211—21no—
113C4×C211—41no—
114C4×C211—41no—
115C4×C211—81no—
116Dic211—21no—
117Dic211—41no—
118Dic211—81no—
119Dic211#9912yes—
120C2×C211—21no—
121C2×C211—21no—
122C2×C211C2×#9911yescharacteristic
123C2×C211#41711yescharacteristic
124C2×C211#42111yescharacteristic
125D211—41no—
126D211—41no—
127D211—81no—
128C211#151311yescharacteristic, complemented, Sylow
129#1513—2111nomaximal, Sylow
130#525—2111no—
131#407—2111no—
132#380—2111no—
133#328—2111no—
134#328D21111yescharacteristic, complemented
135#327—2111no—
136#327—2111no—
137C2×#41—2111no—
138C23×C4—2111no—
139C2×M16—4221no—
140#123—4221no—
141C22⋊Q8—2111no—
142C22⋊Q8C2×D21111yescharacteristic
143#121—2111no—
144#121—2111no—
145#120—4221no—
146C4×D4—4221no—
147C4×D4—4221no—
148#117—2111no—
149C2×C4⋊C4—4221no—
150C2×C22⋊C4—4221no—
151C22⋊C8—4221no—
152C22⋊C8—2111no—
153C22⋊C8C2×D21111yescharacteristic
154#95—2111no—
155#95C2×D21111yescharacteristic
156C24—2111no—
157#41—8442no—
158C2×Q8C4×D21111yescharacteristic
159C2×D4—4221no—
160C2×D4—4221no—
161C2×D4—4221no—
162C2×D4—2111no—
163C22×C4—21no—
164C22×C4—8442no—
165C22×C4—4221no—
166C22×C4—4221no—
167C22×C4—2111no—
168C22×C4—2111no—
169C22×C4—2111no—
170C22×C4—2111no—
171C22×C4—2111no—
172C22×C4C22×D21111yescharacteristic
173M16—8442no—
174C2×C8—21no—
175C2×C8—4221no—
176C4⋊C4—21no—
177C4⋊C4—8442no—
178C4⋊C4—4221no—
179C4⋊C4—2111no—
180C4⋊C4C4×D21111yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—8442no—
183C22⋊C4—4221no—
184C22⋊C4—4221no—
185C22⋊C4—4221no—
186C22⋊C4—4221no—
187C42—4221no—
188C23—8441no—
189C23—8441no—
190C23—4221no—
191C23—4221no—
192C23—4221no—
193C23—2111no—
194C23—2111no—
195C23D4×D21111yescharacteristic
196Q8—41no—
197D4—8441no—
198D4—8442no—
199D4—8442no—
200D4—8441no—
201D4—8441no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—8441no—
209C2×C4—8441no—
210C2×C4—8442no—
211C2×C4—8441no—
212C2×C4—8441no—
213C2×C4—8441no—
214C2×C4—4222no—
215C2×C4—4222no—
216C2×C4—4221no—
217C2×C4—4222no—
218C2×C4—2111no—
219C2×C4—2111no—
220C2×C4C2×C4×D21111yescharacteristic
221C2×C4D4×D21111yescharacteristic
222C8—41no—
223C8—8441no—
224C22—21no—
225C22—21no—
226C22—8442no—
227C22—8442no—
228C22—8441no—
229C22—8441no—
230C22—8441no—
231C22—4221no—
232C22—4221no—
233C22—4221no—
234C22—4221no—
235C22—4221no—
236C22—4221no—
237C22C22⋊C4×D21111yescharacteristic
238C22#2027000411yescharacteristic
239C22#2027000911yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—8441no—
246C4—16881no—
247C4—4221no—
248C4—2112no—
249C2—21no—
250C2—21no—
251C2—8441no—
252C2—8441no—
253C2—16881no—
254C2#99×D21111yescharacteristic
255C2#2721314611yescharacteristic
256C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213659C2
cosocleC22×C4×C211C23
hypercenter#328D211
hypocenterC211#1513
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
2. #27213108C2semidirect
3. #27213005C2semidirect
4. #27212883C2semidirect
5. #27212881C2semidirect
6. #27212879C2semidirect
7. #328×C211C2semidirect
8. C2×#16809963C4semidirect
12. #20269953C4semidirect
13. #20269952C4semidirect
18. #20269935C4semidirect
128. C211#1513semidirect
134. #328D211semidirect

Maximal subgroups

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

subgroupindex
#272136592
#272131082
#272130052
#272128832
#272128812
#272128792
#328×C2112
#1513211