Finite Group Databasebeta

← #41011813

Subgroups — #41011813

Order 27008.

262 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213693C211yesmaximal, characteristic
2#27213198C211yesmaximal, characteristic, complemented
3C4×#16809945C211yesmaximal, characteristic, complemented
4#27212875C211yesmaximal, characteristic, complemented
5D4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#16809962—21no—
9C2×#16809945—21no—
10#20269976C411yescharacteristic
11#20269973C411yescharacteristic
12#20269963—21no—
13#20269952C411yescharacteristic
14#20269950—21no—
15#20269948C411yescharacteristic
16C4×D844C2211yescharacteristic
17C4×C22⋊D211—21no—
18C42×D211C2211yescharacteristic
19C4⋊(C211⋊C8)C2211yescharacteristic
20(C2×D211)⋊C8—21no—
21#20269887—21no—
22C4×C211⋊C8C2211yescharacteristic
23C8×Dic211—21no—
24C4×D4×C211C2211yescharacteristic
25C4⋊C8×C211C2211yescharacteristic
26C4×C8×C211C2211yescharacteristic
27#16809962—42no—
28#16809962—42no—
29C2×Dic422—21no—
30C2×D844—21no—
31C2×D844C2×C411yescharacteristic
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C2×C22⋊D211—21no—
35C2×C4×D211—21no—
36C2×C4×D211C2×C411yescharacteristic
37C2×C4×D211D411yescharacteristic
38C2×C4×D211D411yescharacteristic
39#16809945—24no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8—21no—
42C4⋊Dic211C2×C411yescharacteristic
43Dic211⋊C4—21no—
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47C4×Dic211C2×C411yescharacteristic
48C4×Dic211D411yescharacteristic
49C4×Dic211D411yescharacteristic
50C2×D4×C211C2×C411yescharacteristic
51C22×C4×C211—21no—
52C2×C8×C211—21no—
53C2×C8×C211—21no—
54C4⋊C4×C211C2×C411yescharacteristic
55C22⋊C4×C211—21no—
56C42×C211C2311yescharacteristic
57C22×D211—21no—
58C22×D211C22⋊C411yescharacteristic
59Dic422—42no—
60D844—21no—
61D844—21no—
62D844—42no—
63C22⋊D211—42no—
64C22⋊D211—42no—
65C22⋊D211—42no—
66C2×Dic211—21no—
67C2×Dic211C22⋊C411yescharacteristic
68C2×Dic211C22⋊C411yescharacteristic
69C2×Dic211C22⋊C411yescharacteristic
70C4×D211—21no—
71C4×D211—21no—
72C4×D211—21no—
73C4×D211—21no—
74C4×D211—22no—
75C4×D211—41no—
76C4×D211—41no—
77C211⋊C8—22no—
78C211⋊C8—41no—
79C23×C211—21no—
80D4×C211—21no—
81D4×C211—21no—
82C2×C4×C211—21no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211C22×C411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C8×C211—22no—
89C8×C211—41no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211C2×C22⋊C411yescharacteristic
94C2×D211—21no—
95C2×D211—21no—
96C2×D211—21no—
97C2×D211—41no—
98C2×D211—41no—
99C2×D211—41no—
100C4×C211—21no—
101C4×C211—41no—
102C4×C211#13111yescharacteristic
103C4×C211#13111yescharacteristic
104C4×C211AGL(1,ℤ/8)11yescharacteristic
105C4×C211AGL(1,ℤ/8)11yescharacteristic
106Dic211—21no—
107Dic211—21no—
108Dic211—41no—
109Dic211#10412yes—
110C2×C211—81no—
111C2×C211#40811yescharacteristic
112C2×C211#41811yescharacteristic
113C2×C211C2×#10411yescharacteristic
114D211—41no—
115D211—81no—
116C211#148411yescharacteristic, complemented, Sylow
117#1484—2111nomaximal, Sylow
118#532—2111no—
119#431—2111no—
120C4×M16—2111no—
121D4⋊C8—2111no—
122D4⋊C8—2111no—
123D4⋊C8—2111no—
124D4⋊C8D21111yescharacteristic, complemented
125C2×#41—4221no—
126C2×M16—4221no—
127C4⋊Q8—2111no—
128#127—2111no—
129#124—4221no—
130#121—4221no—
131#121—2111no—
132#121—2111no—
133C4×D4—4221no—
134C4×D4—2111no—
135C4×D4C2×D21111yescharacteristic
136C2×C42—2111no—
137C4⋊C8—2111no—
138C4⋊C8C2×D21111yescharacteristic
139C22⋊C8—4221no—
140#97—4221no—
141C4×C8—4221no—
142C4×C8—2111no—
143C4×C8C2×D21111yescharacteristic
144#41—8442no—
145#41—8442no—
146C2×Q8—4221no—
147C2×D4—4221no—
148C2×D4—4221no—
149C2×D4—4221no—
150C2×D4—4221no—
151C2×D4—2111no—
152C2×D4C4×D21111yescharacteristic
153C22×C4—21no—
154C22×C4—4221no—
155C22×C4—2111no—
156C22×C4—2111no—
157C22×C4—2111no—
158M16—4224no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—4221no—
162C2×C8—4221no—
163C4⋊C4—4221no—
164C4⋊C4—2111no—
165C4⋊C4C4×D21111yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—4221no—
168C22⋊C4—4221no—
169C22⋊C4—4221no—
170C42—2111no—
171C42—2111no—
172C42—2111no—
173C42C22×D21111yescharacteristic
174C23—21no—
175C23—4221no—
176C23—2111no—
177Q8—8442no—
178D4—21no—
179D4—21no—
180D4—8442no—
181D4—8442no—
182D4—8442no—
183D4—8442no—
184D4—4221no—
185D4—4221no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—4222no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4—4221no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4—2111no—
198C2×C4—2111no—
199C2×C4—2111no—
200C2×C4C2×C4×D21111yescharacteristic
201C2×C4D4×D21111yescharacteristic
202C2×C4D4×D21111yescharacteristic
203C8—22no—
204C8—41no—
205C8—8441no—
206C8—4222no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—8441no—
211C22—8441no—
212C22—8441no—
213C22—4221no—
214C22—4221no—
215C22—4221no—
216C22C22⋊C4×D21111yescharacteristic
217C4—21no—
218C4—41no—
219C4—8441no—
220C4—4221no—
221C4—4221no—
222C4—2112no—
223C4#2026998311yescharacteristic
224C4#2026998511yescharacteristic
225C4#2027000111yescharacteristic
226C4#2027000411yescharacteristic
227C2—81no—
228C2—8441no—
229C2—16881no—
230C2#2721311111yescharacteristic
231C2#2721314911yescharacteristic
232C2#104×D21111yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
FittingD4⋊C8×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213693C2
cosocleC42×C211C23
hypercenterD4⋊C8D211
hypocenterC211#1484
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27213198C2semidirect
3. C4×#16809945C2semidirect
4. #27212875C2semidirect
5. D4⋊(C211⋊C8)C2semidirect
6. #27212873C2semidirect
7. D4⋊C8×C211C2semidirect
116. C211#1484semidirect
124. D4⋊C8D211semidirect

Maximal subgroups

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

subgroupindex
#272136932
#272131982
C4×#168099452
#272128752
D4⋊(C211⋊C8)2
#272128732
D4⋊C8×C2112
#1484211