Finite Group Databasebeta

← #41012007

Subgroups — #41012007

Order 27008.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#27213695C211yesmaximal, characteristic
2#27213201C211yesmaximal, characteristic, complemented
3#97×D211C211yesmaximal, characteristic, complemented
4#27212886C211yesmaximal, characteristic, complemented
5#27212885C211yesmaximal, characteristic, complemented
6#27212885C211yesmaximal, characteristic, complemented
7#329×C211C211yesmaximal, characteristic, complemented
8C2×#16809963C411yescharacteristic
9C2×#16809961C411yescharacteristic
10C2×C8×D211—21no—
11#20269975C411yescharacteristic
12#20269973C411yescharacteristic
13#20269965C2211yescharacteristic
14#20269950—21no—
15#20269950—21no—
16(C22⋊D211)⋊C4—21no—
17(C22⋊D211)⋊C4—21no—
18C42×D211C2211yescharacteristic
19Dic211⋊C8—21no—
20(C2×D211)⋊C8—21no—
21#20269888C2211yescharacteristic
22#20269888C2211yescharacteristic
23#20269887—21no—
24#124×C211C2211yescharacteristic
25#97×C211C2211yescharacteristic
26#97×C211C2211yescharacteristic
27#16809963—42no—
28#16809961—42no—
29C2×Dic422C2×C411yescharacteristic
30C2×D844—21no—
31C2×D844C2×C411yescharacteristic
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C2×C22⋊D211—21no—
35C22×Dic211—21no—
36C2×C4×D211—21no—
37C2×C4×D211C2×C411yescharacteristic
38C2×C4×D211D411yescharacteristic
39C2×C4×D211D411yescharacteristic
40C2×C211⋊C8—21no—
41C2×C211⋊C8—21no—
42C8×D211—24no—
43Dic211⋊C4—21no—
44Dic211⋊C4—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47C4×Dic211C2×C411yescharacteristic
48C4×Dic211D411yescharacteristic
49C4×Dic211D411yescharacteristic
50C2×Q8×C211C2×C411yescharacteristic
51C2×D4×C211C2×C411yescharacteristic
52C2×C8×C211—21no—
53C2×C8×C211—21no—
54C22⋊C4×C211—21no—
55C22⋊C4×C211—21no—
56C42×C211C2311yescharacteristic
57C22×D211—21no—
58C22×D211C22⋊C411yescharacteristic
59Dic422—41no—
60D844—41no—
61D844—42no—
62C22⋊D211—42no—
63C22⋊D211—42no—
64C22⋊D211—42no—
65C2×Dic211—21no—
66C2×Dic211—42no—
67C2×Dic211C22⋊C411yescharacteristic
68C2×Dic211C22⋊C411yescharacteristic
69C2×Dic211C22⋊C411yescharacteristic
70C4×D211—22no—
71C4×D211—22no—
72C4×D211—41no—
73C4×D211—42no—
74C211⋊C8—22no—
75C211⋊C8—41no—
76C23×C211—21no—
77Q8×C211—41no—
78D4×C211—41no—
79C2×C4×C211—21no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C8×C211—22no—
84C8×C211—41no—
85C22×C211—41no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211C2×C22⋊C411yescharacteristic
89C2×D211—21no—
90C2×D211—21no—
91C2×D211—41no—
92C2×D211—41no—
93C2×D211—41no—
94C2×D211#10012yes—
95C4×C211—21no—
96C4×C211—21no—
97C4×C211—21no—
98C4×C211—41no—
99Dic211—21no—
100Dic211—21no—
101Dic211—41no—
102Dic211#10412yes—
103C2×C211—81no—
104C2×C211C2×#10011yescharacteristic
105C2×C211C2×#10411yescharacteristic
106C2×C211#42111yescharacteristic
107D211—22no—
108D211—81no—
109C211#153011yescharacteristic, complemented, Sylow
110#1530—2111nomaximal, Sylow
111#532—2111no—
112#432—2111no—
113C2×#97—2111no—
114#329—2111no—
115#329—2111no—
116#329—2111no—
117#329D21111yescharacteristic, complemented
118C2×#41—2111no—
119C2×#41—2111no—
120C22×C8—4221no—
121C4⋊Q8—2111no—
122#127—2111no—
123#124—2111no—
124#124C2×D21111yescharacteristic
125#121—4221no—
126#121—4221no—
127C4×D4—4221no—
128C4×D4—4221no—
129C2×C42—2111no—
130C4⋊C8—4221no—
131C22⋊C8—4221no—
132#97—4221no—
133#97—2111no—
134#97—2111no—
135#97C2×D21111yescharacteristic
136#97C2×D21111yescharacteristic
137#41—8442no—
138#41—8442no—
139C2×Q8—2111no—
140C2×Q8C4×D21111yescharacteristic
141C2×D4—4221no—
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—2111no—
146C2×D4C4×D21111yescharacteristic
147C22×C4—4221no—
148C22×C4—4221no—
149C22×C4—2111no—
150C22×C4—2111no—
151C22×C4—2111no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—4224no—
155C2×C8—4221no—
156C2×C8—4221no—
157C4⋊C4—4221no—
158C4⋊C4—4221no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—4221no—
162C22⋊C4—4221no—
163C42—2111no—
164C42—2111no—
165C42—2111no—
166C42C22×D21111yescharacteristic
167C23—21no—
168C23—4221no—
169C23—2111no—
170Q8—41no—
171Q8—8441no—
172D4—41no—
173D4—8441no—
174D4—8442no—
175D4—8442no—
176D4—8442no—
177D4—8442no—
178C2×C4—21no—
179C2×C4—8441no—
180C2×C4—8442no—
181C2×C4—8442no—
182C2×C4—4222no—
183C2×C4—4222no—
184C2×C4—4221no—
185C2×C4—2111no—
186C2×C4—2111no—
187C2×C4—2111no—
188C2×C4C2×C4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C8—22no—
192C8—41no—
193C8—8441no—
194C8—4222no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—8441no—
199C22—8441no—
200C22—8441no—
201C22—4221no—
202C22—4221no—
203C22—2112no—
204C22C22⋊C4×D21111yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—41no—
209C4—8441no—
210C4—4221no—
211C4—4221no—
212C4—2112no—
213C2—81no—
214C2—16881no—
215C2—4222no—
216C2#100×D21111yescharacteristic
217C2#104×D21111yescharacteristic
218C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#329×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213695C2
cosocleC42×C211C23
hypercenter#329D211
hypocenterC211#1530
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. #97×D211C2semidirect
4. #27212886C2semidirect
5. #27212885C2semidirect
6. #27212885C2semidirect
7. #329×C211C2semidirect
109. C211#1530semidirect
117. #329D211semidirect

Maximal subgroups

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

subgroupindex
#272136952
#272132012
#97×D2112
#272128862
#272128852
#329×C2112
#1530211