Finite Group Databasebeta

← #41012117

Subgroups — #41012117

Order 27008.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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#27213694C211yesmaximal, characteristic
2#27213202C211yesmaximal, characteristic, complemented
3#27213168C211yesmaximal, characteristic, complemented
4#27212893C211yesmaximal, characteristic, complemented
5#27212891C211yesmaximal, characteristic, complemented
6#27212890C211yesmaximal, characteristic, complemented
7#331×C211C211yesmaximal, characteristic, complemented
8C2×#16809961—21no—
9C2×#16809945—21no—
10#20269977C2211yescharacteristic
11#20269964—21no—
12#20269948—21no—
13#20269948C412yes—
14D4×Dic211—21no—
15D4×Dic211C412yes—
16C42×D211C2211yescharacteristic
17C4⋊(C211⋊C8)C2211yescharacteristic
18C4⋊(C211⋊C8)C2211yescharacteristic
19Dic211⋊C8—21no—
20(C2×D211)⋊C8—21no—
21#20269887—21no—
22#127×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4⋊C8×C211C2211yescharacteristic
25#16809961—42no—
26#16809961—42no—
27C2×Dic422—21no—
28C2×C22⋊D211—22no—
29C22×Dic211—22no—
30C2×C4×D211C2×C411yescharacteristic
31C2×C4×D211D411yescharacteristic
32C2×C4×D211D411yescharacteristic
33#16809945—42no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211C2×C412yes—
38C22⋊Dic211—22no—
39C22⋊Dic211—22no—
40C4×Dic211C2×C411yescharacteristic
41C4×Dic211D411yescharacteristic
42C4×Dic211D411yescharacteristic
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211C2×C412yes—
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C42×C211C2311yescharacteristic
49C22×D211C22⋊C411yescharacteristic
50Dic422—41no—
51Dic422—41no—
52C22⋊D211—44no—
53C2×Dic211—22no—
54C2×Dic211—44no—
55C2×Dic211C22⋊C411yescharacteristic
56C2×Dic211C22⋊C411yescharacteristic
57C2×Dic211C22⋊C411yescharacteristic
58C4×D211—21no—
59C4×D211—21no—
60C4×D211—21no—
61C4×D211—21no—
62C4×D211—41no—
63C211⋊C8—41no—
64C211⋊C8—41no—
65C23×C211—22no—
66D4×C211—42no—
67D4×C211—41no—
68D4×C211—41no—
69D4×C211—41no—
70D4×C211—41no—
71C2×C4×C211C22×C411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C8×C211—41no—
75C8×C211—41no—
76C22×C211—42no—
77C22×C211—42no—
78C22×C211—42no—
79C22×C211C2×C22⋊C411yescharacteristic
80C2×D211—21no—
81C2×D211—21no—
82C2×D211—21no—
83C4×C211—21no—
84C4×C211AGL(1,ℤ/8)11yescharacteristic
85C4×C211AGL(1,ℤ/8)11yescharacteristic
86C4×C211AGL(1,ℤ/8)11yescharacteristic
87C4×C211#13711yescharacteristic
88Dic211—21no—
89Dic211—21no—
90Dic211—21no—
91Dic211—42no—
92C2×C211—82no—
93C2×C211#41311yescharacteristic
94C2×C211#41711yescharacteristic
95C2×C211#41811yescharacteristic
96D211—41no—
97C211#155711yescharacteristic, complemented, Sylow
98#1557—2111nomaximal, Sylow
99#532—2111no—
100#432—2111no—
101#423—2111no—
102#332—2111no—
103#332—2111no—
104#331—2111no—
105#331D21111yescharacteristic, complemented
106C2×#41—4221no—
107C2×M16—4221no—
108C4⋊Q8—2111no—
109#127C2×D21111yescharacteristic
110#124—4221no—
111#121—4221no—
112#121—2112no—
113C4×D4—4221no—
114C4×D4—2112no—
115C2×C42—2111no—
116C4⋊C8—4221no—
117C4⋊C8—2111no—
118C4⋊C8—2111no—
119C4⋊C8C2×D21111yescharacteristic
120C4⋊C8C2×D21111yescharacteristic
121C22⋊C8—4221no—
122#97—4221no—
123#41—8442no—
124#41—8442no—
125C2×Q8—4221no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—4222no—
129C2×D4C4×D21112yes—
130C22×C4—4222no—
131C22×C4—2111no—
132C22×C4—2111no—
133C22×C4—2111no—
134M16—8442no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—4221no—
138C2×C8—4221no—
139C4⋊C4—4221no—
140C4⋊C4—2112no—
141C22⋊C4—4222no—
142C22⋊C4—4222no—
143C42—2111no—
144C42—2111no—
145C42—2111no—
146C42C22×D21111yescharacteristic
147C23—22no—
148C23—2111no—
149Q8—8441no—
150Q8—8441no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—8444no—
157C2×C4—8441no—
158C2×C4—8444no—
159C2×C4—4222no—
160C2×C4—4221no—
161C2×C4—4221no—
162C2×C4—4221no—
163C2×C4—4221no—
164C2×C4—2111no—
165C2×C4—2111no—
166C2×C4—2111no—
167C2×C4C2×C4×D21111yescharacteristic
168C2×C4D4×D21111yescharacteristic
169C2×C4D4×D21111yescharacteristic
170C8—41no—
171C8—41no—
172C8—8441no—
173C8—8441no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—4221no—
178C22—4221no—
179C22—4221no—
180C22C22⋊C4×D21111yescharacteristic
181C4—21no—
182C4—8442no—
183C4—4221no—
184C4—4221no—
185C4—4221no—
186C4#2027000111yescharacteristic
187C4#2027000111yescharacteristic
188C4#2027000411yescharacteristic
189C4#2027000611yescharacteristic
190C2—82no—
191C2—8441no—
192C2#2721312711yescharacteristic
193C2#2721314411yescharacteristic
194C2#2721314911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213694C2
cosocleC42×C211C23
hypercenter#331D211
hypocenterC211#1557
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27213202C2semidirect
3. #27213168C2semidirect
4. #27212893C2semidirect
5. #27212891C2semidirect
6. #27212890C2semidirect
7. #331×C211C2semidirect
97. C211#1557semidirect
105. #331D211semidirect

Maximal subgroups

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

subgroupindex
#272136942
#272132022
#272131682
#272128932
#272128912
#272128902
#331×C2112
#1557211