Finite Group Databasebeta

← #41011951

Subgroups — #41011951

Order 27008.

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

normal subgroups only

This group has 284 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#27213656C211yesmaximal, characteristic
2C22⋊C8×D211C211yesmaximal, characteristic, complemented
3#27213008C211yesmaximal, characteristic, complemented
4#27212884C211yesmaximal, characteristic, complemented
5#27212882C211yesmaximal, characteristic, complemented
6#27212880C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C2×#16809961C411yescharacteristic, complemented
9C22×C4×D211C2211yescharacteristic
10C2×C8×D211—21no—
11#20269959—21no—
12#20269957C411yescharacteristic, complemented
13#20269956C2211yescharacteristic
14#20269948C411yescharacteristic, complemented
15#20269945—21no—
16D4×Dic211—21no—
17(C22⋊D211)⋊C4—21no—
18#20269935C411yescharacteristic, complemented
19C2×(C2×D211)⋊C4—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21(C2×D211)⋊C8—21no—
22C211⋊#95—21no—
23C211⋊#95C2211yescharacteristic
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28#16809961—42no—
29C2×Dic422C2×C411yescharacteristic
30C2×C22⋊D211—21no—
31C2×C22⋊D211—21no—
32C22×Dic211—21no—
33C22×Dic211—21no—
34C22×Dic211D411yescharacteristic
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—
45C22⋊Dic211—21no—
46C22⋊Dic211—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—42no—
50C4×Dic211—21no—
51C2×D4×C211—21no—
52C2×D4×C211C2×C411yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55C2×C8×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—41no—
65C22⋊D211—42no—
66C22⋊D211—42no—
67C2×Dic211—21no—
68C2×Dic211—21no—
69C2×Dic211—41no—
70C2×Dic211—42no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C2×Dic211C22⋊C411yescharacteristic
75C2×Dic211C22⋊C411yescharacteristic
76C4×D211—22no—
77C4×D211—22no—
78C4×D211—41no—
79C4×D211—41no—
80C211⋊C8—41no—
81C23×C211—21no—
82C23×C211C2×D411yescharacteristic
83D4×C211—41no—
84D4×C211—41no—
85D4×C211—41no—
86C2×C4×C211—21no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211C22×C411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C8×C211—41no—
94C22×C211—21no—
95C22×C211—21no—
96C22×C211—41no—
97C22×C211—41no—
98C22×C211—41no—
99C22×C211C2×C22⋊C411yescharacteristic
100C22×C211#13511yescharacteristic
101C22×C211#13511yescharacteristic
102C2×D211—21no—
103C2×D211—21no—
104C2×D211—21no—
105C2×D211—21no—
106C2×D211—21no—
107C2×D211—42no—
108C2×D211—42no—
109C2×D211#9912yes—
110C4×C211—21no—
111C4×C211—21no—
112C4×C211—41no—
113C4×C211—81no—
114Dic211—21no—
115Dic211—21no—
116Dic211—41no—
117Dic211—41no—
118Dic211—81no—
119C2×C211—21no—
120C2×C211—21no—
121C2×C211—81no—
122C2×C211C2×#9911yescharacteristic
123C2×C211#41611yescharacteristic
124C2×C211#42111yescharacteristic
125D211—22no—
126D211—41no—
127C211#151611yescharacteristic, complemented, Sylow
128#1516—2111nomaximal, Sylow
129#525—2111no—
130C2×C22⋊C8—2111no—
131#381—2111no—
132#328—2111no—
133#328—2111no—
134#327—2111no—
135#327D21111yescharacteristic, complemented
136C2×#41—2111no—
137C23×C4—2111no—
138C22×C8—4221no—
139#123—4221no—
140C22⋊Q8—2111no—
141C22⋊Q8—2111no—
142#121—2111no—
143#121C2×D21111yescharacteristic
144#120—4221no—
145C4×D4—4221no—
146C4×D4—4221no—
147#117—2111no—
148C2×C22⋊C4—4221no—
149C22⋊C8—4221no—
150C22⋊C8—2111no—
151C22⋊C8C2×D21111yescharacteristic
152#95—4221no—
153#95—2111no—
154#95C2×D21111yescharacteristic
155C24—2111no—
156#41—8442no—
157C2×Q8—2111no—
158C2×D4—21no—
159C2×D4—4221no—
160C2×D4—4221no—
161C2×D4C4×D21111yescharacteristic
162C22×C4—21no—
163C22×C4—8442no—
164C22×C4—4221no—
165C22×C4—4221no—
166C22×C4—2111no—
167C22×C4—2111no—
168C22×C4—2111no—
169C22×C4—2111no—
170C22×C4—2111no—
171C22×C4C22×D21111yescharacteristic
172C2×C8—21no—
173C2×C8—8442no—
174C2×C8—4221no—
175C4⋊C4—4221no—
176C4⋊C4—4221no—
177C4⋊C4—2111no—
178C4⋊C4C4×D21111yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—8442no—
181C22⋊C4—4221no—
182C22⋊C4—4221no—
183C22⋊C4—4221no—
184C22⋊C4—4221no—
185C42—4221no—
186C23—21no—
187C23—8441no—
188C23—4222no—
189C23—4221no—
190C23—4221no—
191C23—2111no—
192C23—2111no—
193C23D4×D21111yescharacteristic
194Q8—8441no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—8442no—
199D4—8442no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—8441no—
206C2×C4—8441no—
207C2×C4—8442no—
208C2×C4—8441no—
209C2×C4—8441no—
210C2×C4—8441no—
211C2×C4—8441no—
212C2×C4—4222no—
213C2×C4—4222no—
214C2×C4—4221no—
215C2×C4—4221no—
216C2×C4—2111no—
217C2×C4—2111no—
218C2×C4C2×C4×D21111yescharacteristic
219C2×C4D4×D21111yescharacteristic
220C8—41no—
221C8—8441no—
222C22—21no—
223C22—21no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—8442no—
228C22—8442no—
229C22—4221no—
230C22—4221no—
231C22—4221no—
232C22—4221no—
233C22—4221no—
234C22—2112no—
235C22C22⋊C4×D21111yescharacteristic
236C22#2026999611yescharacteristic
237C22#2026999811yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—81no—
242C4—8441no—
243C4—8441no—
244C4—16881no—
245C4—4221no—
246C4—4221no—
247C2—21no—
248C2—21no—
249C2—81no—
250C2—8441no—
251C2—4222no—
252C2#99×D21111yescharacteristic
253C2#2721313911yescharacteristic
254C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213656C2
cosocleC22×C4×C211C23
hypercenter#327D211
hypocenterC211#1516
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D211C2semidirect
3. #27213008C2semidirect
4. #27212884C2semidirect
5. #27212882C2semidirect
6. #27212880C2semidirect
7. #327×C211C2semidirect
8. C2×#16809961C4semidirect
12. #20269957C4semidirect
14. #20269948C4semidirect
18. #20269935C4semidirect
127. C211#1516semidirect
135. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272136562
C22⋊C8×D2112
#272130082
#272128842
#272128822
#272128802
#327×C2112
#1516211