Finite Group Databasebeta

← #41012002

Subgroups — #41012002

Order 27008.

216 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 59 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#27213675C211yesmaximal, characteristic
2#27213201C211yesmaximal, characteristic, complemented
3#27213098C211yesmaximal, characteristic, complemented
4#27212887C211yesmaximal, characteristic, complemented
5#27212887C211yesmaximal, characteristic, complemented
6#27212886C211yesmaximal, characteristic, complemented
7#329×C211C211yesmaximal, characteristic, complemented
8C2×#16809945—21no—
9#20269970—21no—
10#20269969C2211yescharacteristic
11#20269959—21no—
12#20269953C411yescharacteristic
13#20269948C411yescharacteristic
14Q8×Dic211C411yescharacteristic
15D4×Dic211C411yescharacteristic
16(C22⋊D211)⋊C4—21no—
17#20269933—21no—
18C42×D211C2211yescharacteristic
19Dic211⋊C8—21no—
20(C2×D211)⋊C8—21no—
21#20269888C2211yescharacteristic
22#20269888C2211yescharacteristic
23C8×Dic211—21no—
24#124×C211C2211yescharacteristic
25#97×C211C2211yescharacteristic
26#97×C211C2211yescharacteristic
27C2×C22⋊D211—21no—
28C22×Dic211—21no—
29C2×C4×D211C2×C411yescharacteristic
30C2×C4×D211D411yescharacteristic
31C2×C4×D211D411yescharacteristic
32#16809945—24no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211C2×C411yescharacteristic
37C4⋊Dic211C2×C411yescharacteristic
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41C22⋊Dic211—21no—
42C22⋊Dic211—21no—
43(C2×D211)⋊C4—21no—
44C4×Dic211—21no—
45C4×Dic211C2×C411yescharacteristic
46C4×Dic211D411yescharacteristic
47C4×Dic211D411yescharacteristic
48C2×Q8×C211C2×C411yescharacteristic
49C2×D4×C211C2×C411yescharacteristic
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C22⋊C4×C211—21no—
53C22⋊C4×C211—21no—
54C42×C211C2311yescharacteristic
55C22×D211C22⋊C411yescharacteristic
56C22⋊D211—42no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—42no—
60C2×Dic211C22⋊C411yescharacteristic
61C2×Dic211C22⋊C411yescharacteristic
62C2×Dic211C22⋊C411yescharacteristic
63C4×D211—22no—
64C4×D211—41no—
65C4×D211—41no—
66C211⋊C8—22no—
67C211⋊C8—41no—
68C23×C211—21no—
69Q8×C211—41no—
70D4×C211—41no—
71C2×C4×C211—21no—
72C2×C4×C211C22×C411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C8×C211—22no—
76C8×C211—41no—
77C22×C211—41no—
78C22×C211—41no—
79C22×C211—41no—
80C22×C211C2×C22⋊C411yescharacteristic
81C2×D211—21no—
82C2×D211—21no—
83C2×D211—21no—
84C4×C211—21no—
85C4×C211—21no—
86C4×C211—21no—
87C4×C211—41no—
88Dic211—21no—
89Dic211—21no—
90Dic211—41no—
91Dic211—41no—
92Dic211#10412yes—
93C2×C211—81no—
94C2×C211#41311yescharacteristic
95C2×C211C2×#10411yescharacteristic
96C2×C211#42111yescharacteristic
97D211—41no—
98C211#152911yescharacteristic, complemented, Sylow
99#1529—2111nomaximal, Sylow
100#529—2111no—
101#432—2111no—
102C4×M16—2111no—
103#330—2111no—
104#330—2111no—
105#329—2111no—
106#329D21111yescharacteristic, complemented
107C2×M16—4221no—
108#126—4221no—
109#125—2111no—
110#124C2×D21111yescharacteristic
111#123—4221no—
112C22⋊Q8—2111no—
113#121—2111no—
114C4×Q8—2111no—
115C4×D4—4221no—
116C4×D4—2111no—
117#117—4221no—
118C2×C42—2111no—
119C4⋊C8—4221no—
120C22⋊C8—4221no—
121#97—2111no—
122#97—2111no—
123#97C2×D21111yescharacteristic
124#97C2×D21111yescharacteristic
125C4×C8—4221no—
126C2×Q8C4×D21111yescharacteristic
127C2×D4—4221no—
128C2×D4C4×D21111yescharacteristic
129C22×C4—4221no—
130C22×C4—2111no—
131C22×C4—2111no—
132C22×C4—2111no—
133M16—4224no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—4221no—
137C2×C8—4221no—
138C4⋊C4—4221no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—4221no—
142C4⋊C4—2111no—
143C4⋊C4—2111no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—4221no—
147C22⋊C4—4221no—
148C22⋊C4—4221no—
149C42—4221no—
150C42—2111no—
151C42—2111no—
152C42—2111no—
153C42C22×D21111yescharacteristic
154C23—21no—
155C23—2111no—
156Q8—41no—
157D4—41no—
158D4—8442no—
159C2×C4—21no—
160C2×C4—8441no—
161C2×C4—8441no—
162C2×C4—8442no—
163C2×C4—4222no—
164C2×C4—4221no—
165C2×C4—4221no—
166C2×C4—2111no—
167C2×C4—2111no—
168C2×C4—2111no—
169C2×C4C2×C4×D21111yescharacteristic
170C2×C4D4×D21111yescharacteristic
171C2×C4D4×D21111yescharacteristic
172C8—22no—
173C8—41no—
174C8—8441no—
175C8—4222no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—4221no—
180C22—4221no—
181C22—4221no—
182C22C22⋊C4×D21111yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—8441no—
188C4—8441no—
189C4—4221no—
190C4—4221no—
191C4—2112no—
192C2—81no—
193C2—8441no—
194C2#2721312711yescharacteristic
195C2#104×D21111yescharacteristic
196C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#329×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269948C4
cosocleC42×C211C23
hypercenter#329D211
hypocenterC211#1529
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27213201C2semidirect
3. #27213098C2semidirect
4. #27212887C2semidirect
5. #27212887C2semidirect
6. #27212886C2semidirect
7. #329×C211C2semidirect
98. C211#1529semidirect
106. #329D211semidirect

Maximal subgroups

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

subgroupindex
#272136752
#272132012
#272130982
#272128872
#272128862
#329×C2112
#1529211