Finite Group Databasebeta

← #41012368

Subgroups — #41012368

Order 27008.

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

normal subgroups only

This group has 282 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
2#27213287C211yesmaximal, characteristic, complemented
3#27213284C211yesmaximal, characteristic, complemented
4#27213067C211yesmaximal, characteristic, complemented
5#27212879C211yesmaximal, characteristic, complemented
6#27212871C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C2×#16809961—21no—
9C22×Dic422C2211yescharacteristic
10C2×#16809951—21no—
11C2×#16809950—21no—
12#20269955—21no—
13#20269954—21no—
14#20269951C2211yescharacteristic
15#20269950—21no—
16#20269949—21no—
17#20269949—21no—
18C2×C4⋊Dic211—21no—
19C2×Dic211⋊C4C2211yescharacteristic
20#20269902—21no—
21#20269900—21no—
22C22⋊(C211⋊C8)C2211yescharacteristic
23C211⋊#95—21no—
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27#16809961—42no—
28#16809961—42no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—42no—
33C2×D844D411yescharacteristic
34C2×C22⋊D211—21no—
35C2×C22⋊D211—21no—
36C2×C22⋊D211—21no—
37C22×Dic211—21no—
38C22×Dic211—21no—
39C22×Dic211D411yescharacteristic
40C22×Dic211D411yescharacteristic
41C2×C4×D211—21no—
42#16809951—42no—
43#16809950—42no—
44C2×C211⋊C8—21no—
45C4⋊Dic211—42no—
46C4⋊Dic211D411yescharacteristic
47Dic211⋊C4—21no—
48Dic211⋊C4—21no—
49Dic211⋊C4—21no—
50C22⋊Dic211—21no—
51(C2×D211)⋊C4—21no—
52(C2×D211)⋊C4—21no—
53C2×D4×C211—21no—
54C2×D4×C211D411yescharacteristic
55C22×C4×C211—21no—
56C22×C4×C211C2311yescharacteristic
57C2×C8×C211—21no—
58C4⋊C4×C211D411yescharacteristic
59C22⋊C4×C211—21no—
60C22×D211—21no—
61Dic422—22no—
62Dic422—22no—
63Dic422—42no—
64Dic422—41no—
65Dic422—41no—
66D844—41no—
67C22⋊D211—42no—
68C22⋊D211—41no—
69C22⋊D211—41no—
70C22⋊D211—81no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—22no—
74C2×Dic211—21no—
75C2×Dic211—21no—
76C2×Dic211—42no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C2×Dic211SD1612yescomplemented
81C4×D211—42no—
82C211⋊C8—41no—
83C23×C211—21no—
84C23×C211C2×D411yescharacteristic
85D4×C211—41no—
86D4×C211—41no—
87D4×C211—41no—
88C2×C4×C211—22no—
89C2×C4×C211—21no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C8×C211—41no—
96C22×C211—21no—
97C22×C211—21no—
98C22×C211—41no—
99C22×C211—41no—
100C22×C211—41no—
101C22×C211#12011yescharacteristic
102C22×C211C2×SD1611yescharacteristic
103C22×C211AGL(1,ℤ/8)11yescharacteristic
104C2×D211—41no—
105C2×D211—41no—
106C2×D211—41no—
107C4×C211—21no—
108C4×C211—21no—
109C4×C211—42no—
110C4×C211—41no—
111Dic211—22no—
112Dic211—41no—
113Dic211—42no—
114Dic211—41no—
115C2×C211—21no—
116C2×C211—21no—
117C2×C211—81no—
118C2×C211#45011yescharacteristic
119C2×C211#45611yescharacteristic
120C2×C211B(4,𝔽2)11yescharacteristic
121D211—81no—
122C211#160411yescharacteristic, complemented, Sylow
123#1604—2111nomaximal, Sylow
124#537—2111no—
125#448—2111no—
126#448—2111no—
127#395—2111no—
128#327—2111no—
129#327D21111yescharacteristic, complemented
130#324—2111no—
131C2×#41—4221no—
132C22×Q8—2111no—
133C2×SD16—4221no—
134C2×SD16—4221no—
135C22⋊Q8—4221no—
136C22⋊Q8—4221no—
137#121—4221no—
138#121—4221no—
139#121—4221no—
140#121—2111no—
141#121C2×D21111yescharacteristic
142C2×C4⋊C4—4221no—
143C2×C4⋊C4—2111no—
144#103—4221no—
145#103—4221no—
146C22⋊C8—2111no—
147C22⋊C8C2×D21111yescharacteristic
148#95—4221no—
149#95C2×D21111yescharacteristic
150#41—8442no—
151#41—8442no—
152C2×Q8—8442no—
153C2×Q8—4221no—
154C2×Q8—4221no—
155C2×Q8—4221no—
156C2×D4—21no—
157C2×D4—4221no—
158C2×D4—4221no—
159C2×D4—4221no—
160C2×D4—2111no—
161C2×D4D84411yescharacteristic
162C22×C4—21no—
163C22×C4—4221no—
164C22×C4—4221no—
165C22×C4—4221no—
166C22×C4—2111no—
167C22×C4—2111no—
168C22×C4C22×D21111yescharacteristic
169SD16—8442no—
170SD16—8442no—
171C2×C8—21no—
172C2×C8—4221no—
173C4⋊C4—8442no—
174C4⋊C4—4221no—
175C4⋊C4—4221no—
176C4⋊C4—4221no—
177C4⋊C4—2111no—
178C4⋊C4D84411yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—4221no—
181C22⋊C4—4221no—
182C22⋊C4—4221no—
183C23—21no—
184C23—4221no—
185C23D4×D21111yescharacteristic
186Q8—8441no—
187Q8—8441no—
188Q8—8442no—
189Q8—4222no—
190Q8—4222no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—8441no—
195D4—8441no—
196D4—8441no—
197D4—8442no—
198D4—16881no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—8442no—
205C2×C4—8441no—
206C2×C4—8441no—
207C2×C4—8442no—
208C2×C4—8441no—
209C2×C4—4222no—
210C2×C4—4221no—
211C2×C4—4221no—
212C2×C4—4221no—
213C2×C4—4221no—
214C2×C4—2112no—
215C2×C4D4×D21111yescharacteristic
216C2×C4C2×D84411yescharacteristic
217C8—41no—
218C8—8441no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—8441no—
225C22—8441no—
226C22—8441no—
227C22#2026994711yescharacteristic
228C22SD16×D21111yescharacteristic
229C22#2027000111yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—8442no—
235C4—8441no—
236C4—8441no—
237C4—4222no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—16881no—
242C2#2721329811yescharacteristic
243C2#2721333011yescharacteristic
244C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#27213723C2
hypercenter#327D211
hypocenterC211#1604
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
2. #27213287C2semidirect
3. #27213284C2semidirect
4. #27213067C2semidirect
5. #27212879C2semidirect
6. #27212871C2semidirect
7. #327×C211C2semidirect
80. C2×Dic211SD16semidirect
122. C211#1604semidirect
129. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272137232
#272132872
#272132842
#272130672
#272128792
#272128712
#327×C2112
#1604211