Finite Group Databasebeta

← #41012334

Subgroups — #41012334

Order 27008.

310 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213723C211yesmaximal, characteristic, complemented
2#27213282C211yesmaximal, characteristic, complemented
3#27213279C211yesmaximal, characteristic, complemented
4#27213064C211yesmaximal, characteristic, complemented
5#27212879C211yesmaximal, characteristic, complemented
6#27212871C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C2×#16809961—21no—
9C22×D844C2211yescharacteristic
10C2×D1688—21no—
11C2×#16809947—21no—
12#20269955—21no—
13#20269954—21no—
14#20269951C2211yescharacteristic
15#20269950—21no—
16#20269949—21no—
17#20269949—21no—
18C2×Dic211⋊C4C2211yescharacteristic
19C2×(C2×D211)⋊C4—21no—
20C2×(C2×D211)⋊C4—21no—
21#20269899—21no—
22#20269897—21no—
23C22⋊(C211⋊C8)C2211yescharacteristic
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211—21no—
28#16809961—42no—
29#16809961—42no—
30C2×Dic422—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—42no—
34C2×D844D411yescharacteristic
35C2×C22⋊D211—21no—
36C2×C22⋊D211—21no—
37C2×C22⋊D211—21no—
38C22×Dic211—21no—
39C22×Dic211D411yescharacteristic
40C22×Dic211D411yescharacteristic
41C2×C4×D211—21no—
42D1688—42no—
43#16809947—42no—
44C2×C211⋊C8—21no—
45C4⋊Dic211D411yescharacteristic
46Dic211⋊C4—21no—
47Dic211⋊C4—21no—
48Dic211⋊C4—21no—
49C22⋊Dic211—21no—
50(C2×D211)⋊C4—21no—
51(C2×D211)⋊C4—21no—
52(C2×D211)⋊C4—42no—
53(C2×D211)⋊C4—42no—
54C2×D4×C211—21no—
55C2×D4×C211D411yescharacteristic
56C22×C4×C211—21no—
57C22×C4×C211C2311yescharacteristic
58C2×C8×C211—21no—
59C4⋊C4×C211D411yescharacteristic
60C22⋊C4×C211—21no—
61C22×D211—22no—
62C22×D211—21no—
63C22×D211—41no—
64C22×D211—41no—
65C22×D211—41no—
66C22×D211—41no—
67C22×D211—41no—
68C22×D211—41no—
69Dic422—42no—
70D844—22no—
71D844—22no—
72D844—41no—
73D844—41no—
74D844—41no—
75C22⋊D211—42no—
76C22⋊D211—41no—
77C22⋊D211—41no—
78C22⋊D211—81no—
79C2×Dic211—21no—
80C2×Dic211—21no—
81C2×Dic211—21no—
82C2×Dic211—21no—
83C2×Dic211—42no—
84C2×Dic211—41no—
85C2×Dic211D812yescomplemented
86C4×D211—42no—
87C211⋊C8—41no—
88C23×C211—21no—
89C23×C211C2×D411yescharacteristic
90D4×C211—41no—
91D4×C211—41no—
92D4×C211—41no—
93C2×C4×C211—21no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C8×C211—41no—
101C22×C211—21no—
102C22×C211—21no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211—41no—
106C22×C211#12011yescharacteristic
107C22×C211C2×D811yescharacteristic
108C22×C211AGL(1,ℤ/8)11yescharacteristic
109C2×D211—42no—
110C2×D211—42no—
111C2×D211—41no—
112C2×D211—41no—
113C2×D211—41no—
114C2×D211—42no—
115C2×D211—81no—
116C2×D211—81no—
117C2×D211—81no—
118C2×D211—81no—
119C4×C211—21no—
120C4×C211—21no—
121C4×C211—41no—
122C4×C211—81no—
123Dic211—22no—
124Dic211—41no—
125Dic211—41no—
126C2×C211—21no—
127C2×C211—21no—
128C2×C211—81no—
129C2×C211#44711yescharacteristic
130C2×C211#45411yescharacteristic
131C2×C211B(4,𝔽2)11yescharacteristic
132D211—81no—
133D211—82no—
134C211#159811yescharacteristic, complemented, Sylow
135#1598—2111nomaximal, Sylow
136#537—2111no—
137#447—2111no—
138#447—2111no—
139#394—2111no—
140#327—2111no—
141#327D21111yescharacteristic, complemented
142#324—2111no—
143C2×#41—4221no—
144C22×D4—2111no—
145C2×D8—4221no—
146C2×D8—4221no—
147C22⋊Q8—4221no—
148C22⋊Q8—4221no—
149#121—4221no—
150#121—4221no—
151#121—4221no—
152#121—2111no—
153#121C2×D21111yescharacteristic
154C2×C4⋊C4—2111no—
155C2×C22⋊C4—4221no—
156C2×C22⋊C4—4221no—
157#102—4221no—
158#102—4221no—
159C22⋊C8—2111no—
160C22⋊C8C2×D21111yescharacteristic
161#95C2×D21111yescharacteristic
162C24—4221no—
163#41—8442no—
164#41—8442no—
165C2×Q8—4221no—
166C2×D4—21no—
167C2×D4—8442no—
168C2×D4—4221no—
169C2×D4—4221no—
170C2×D4—4221no—
171C2×D4—4221no—
172C2×D4—4221no—
173C2×D4—2111no—
174C2×D4D84411yescharacteristic
175C22×C4—21no—
176C22×C4—4221no—
177C22×C4—4221no—
178C22×C4—2111no—
179C22×C4—2111no—
180C22×C4C22×D21111yescharacteristic
181D8—8442no—
182D8—8442no—
183C2×C8—21no—
184C2×C8—4221no—
185C4⋊C4—4221no—
186C4⋊C4—4221no—
187C4⋊C4—4221no—
188C4⋊C4—2111no—
189C4⋊C4D84411yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—8442no—
192C22⋊C4—8442no—
193C22⋊C4—4221no—
194C22⋊C4—4221no—
195C22⋊C4—4221no—
196C23—21no—
197C23—8441no—
198C23—8441no—
199C23—8441no—
200C23—8441no—
201C23—8441no—
202C23—8441no—
203C23—4222no—
204C23—4221no—
205C23D4×D21111yescharacteristic
206Q8—8442no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—8441no—
211D4—8441no—
212D4—8441no—
213D4—8441no—
214D4—8441no—
215D4—8442no—
216D4—16881no—
217D4—4222no—
218D4—4222no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—8442no—
225C2×C4—8442no—
226C2×C4—8441no—
227C2×C4—4221no—
228C2×C4—4221no—
229C2×C4—4221no—
230C2×C4—4221no—
231C2×C4—2112no—
232C2×C4D4×D21111yescharacteristic
233C2×C4C2×D84411yescharacteristic
234C8—41no—
235C8—8441no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—8442no—
242C22—8442no—
243C22—8441no—
244C22—8442no—
245C22—8441no—
246C22—8441no—
247C22—16881no—
248C22—16881no—
249C22—16881no—
250C22—16881no—
251C22#2026994711yescharacteristic
252C22D8×D21111yescharacteristic
253C22#2027000411yescharacteristic
254C4—21no—
255C4—21no—
256C4—41no—
257C4—81no—
258C4—8441no—
259C4—8441no—
260C4—4222no—
261C2—21no—
262C2—21no—
263C2—81no—
264C2—16882no—
265C2—16881no—
266C2#2721328111yescharacteristic
267C2#2721332011yescharacteristic
268C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
hypercenter#327D211
hypocenterC211#1598
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #27213723C2semidirect
2. #27213282C2semidirect
3. #27213279C2semidirect
4. #27213064C2semidirect
5. #27212879C2semidirect
6. #27212871C2semidirect
7. #327×C211C2semidirect
85. C2×Dic211D8semidirect
134. C211#1598semidirect
141. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272137232
#272132822
#272132792
#272130642
#272128792
#272128712
#327×C2112
#1598211