Finite Group Databasebeta

← #41014301

Subgroups — #41014301

Order 27008.

296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#16809951C211yesmaximal, characteristic
2C2×#20269913C211yesmaximal, characteristic, complemented
3#27213154C211yesmaximal, characteristic, complemented
4#27213150C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic, complemented
6C2×#20269893C211yesmaximal, characteristic
7#405×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C2×#16809951—21no—
11C2×#16809951—21no—
12C2×#16809951—22no—
13C2×#16809951C414yescomplemented
14C2×#16809946C2211yescharacteristic
15C2×#16809946C2211yescharacteristic
16#20269913—22no—
17#20269900—22no—
18#20269897—22no—
19#20269895—22no—
20#20269893—22no—
21C2×M16×C211C2211yescharacteristic
22C22×C8×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24#97×C211—21no—
25C4×C8×C211—21no—
26C23×D211—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422C2×C412yes—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844C2×C412yes—
39C22×Dic211—21no—
40#16809951—24no—
41#16809951—24no—
42#16809946—21no—
43#16809946—21no—
44#16809946—21no—
45#16809946—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8—21no—
48C22×C4×C211C2311yescharacteristic
49M16×C211—21no—
50M16×C211D412yes—
51C2×C8×C211—21no—
52C2×C8×C211C2×C412yes—
53C2×C8×C211D411yescharacteristic
54C2×C8×C211D411yescharacteristic
55C2×C8×C211D412yes—
56C4⋊C4×C211D412yes—
57C22⋊C4×C211—21no—
58C42×C211—21no—
59C22×D211—22no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—82no—
63Dic422—22no—
64Dic422—22no—
65Dic422—41no—
66Dic422—41no—
67D844—22no—
68D844—22no—
69D844—41no—
70D844—41no—
71C2×Dic211—22no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C211⋊C8—41no—
75C211⋊C8—41no—
76C23×C211#4111yescharacteristic
77C2×C4×C211—22no—
78C2×C4×C211C22×C411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C8×C211—22no—
85C8×C211C22⋊C412yes—
86C8×C211C22⋊C412yes—
87C22×C211—22no—
88C22×C211C4×D411yescharacteristic
89C22×C211#12111yescharacteristic
90C22×C211#12411yescharacteristic
91C2×D211—42no—
92C2×D211—82no—
93C2×D211—82no—
94C2×D211—82no—
95C4×C211—42no—
96C4×C211C2×C22⋊C411yescharacteristic
97C4×C211C4×D411yescharacteristic
98C4×C211#12111yescharacteristic
99C4×C211#12711yescharacteristic
100Dic211—42no—
101C2×C211—21no—
102C2×C211—21no—
103C2×C211#39011yescharacteristic
104C2×C211#47112yes—
105D211—82no—
106C211#197711yescharacteristic, complemented, Sylow
107#1977—2111nomaximal, Sylow
108C22×SD16—2111no—
109C2×#108—2111no—
110#419—2111no—
111#418—2111no—
112C2×#101—2111no—
113C2×#100—2111no—
114#405D21111yescharacteristic, complemented
115C22×Q8—2111no—
116C22×D4—2111no—
117C2×SD16—4221no—
118C2×SD16—4221no—
119C2×SD16—4222no—
120C2×SD16—2114no—
121C2×M16—2111no—
122C2×M16—2111no—
123C2×M16C2×D21111yescharacteristic
124C22×C8C2×D21111yescharacteristic
125#117C2×D21111yescharacteristic
126#108—4222no—
127#103—4222no—
128#102—4222no—
129#101—4222no—
130#100—4222no—
131#97—21no—
132C4×C8—21no—
133C24—4221no—
134C2×Q8—4221no—
135C2×Q8—4221no—
136C2×Q8—4221no—
137C2×Q8—4221no—
138C2×Q8—4221no—
139C2×Q8—2112no—
140C2×D4—4221no—
141C2×D4—4221no—
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—2112no—
146C22×C4—4221no—
147C22×C4C22×D21111yescharacteristic
148SD16—4224no—
149SD16—4224no—
150M16—21no—
151M16—4221no—
152M16—4221no—
153M16—4221no—
154M16—4221no—
155M16D84412yes—
156C2×C8—21no—
157C2×C8—4221no—
158C2×C8—4221no—
159C2×C8C4×D21112yes—
160C2×C8C22⋊D21111yescharacteristic
161C2×C8C22⋊D21111yescharacteristic
162C2×C8D84412yes—
163C4⋊C4D84412yes—
164C22⋊C4—21no—
165C42—21no—
166C23—8441no—
167C23—8441no—
168C23—16882no—
169C23—4222no—
170C23#1680996211yescharacteristic
171Q8—8441no—
172Q8—8441no—
173Q8—4222no—
174Q8—4222no—
175D4—8441no—
176D4—8441no—
177D4—4222no—
178D4—4222no—
179C2×C4—22no—
180C2×C4—8441no—
181C2×C4—8441no—
182C2×C4—4222no—
183C2×C4C2×C4×D21111yescharacteristic
184C2×C4C2×C22⋊D21111yescharacteristic
185C2×C4C2×D84411yescharacteristic
186C2×C4C2×D84411yescharacteristic
187C2×C4C2×D84411yescharacteristic
188C2×C4#1680996211yescharacteristic
189C8—22no—
190C8—8441no—
191C8—8441no—
192C8(C2×D211)⋊C412yes—
193C8(C2×D211)⋊C412yes—
194C22—22no—
195C22—8442no—
196C22—16882no—
197C22—16882no—
198C22—16882no—
199C22C4×D84411yescharacteristic
200C22#2026995111yescharacteristic
201C22#2026996511yescharacteristic
202C4—42no—
203C4—8442no—
204C4C2×(C2×D211)⋊C411yescharacteristic
205C4C4×D84411yescharacteristic
206C4#2026995111yescharacteristic
207C4#2026997411yescharacteristic
208C2—21no—
209C2—21no—
210C2—16882no—
211C2#2721304811yescharacteristic
212C2#2721340912yes—
subgroupG/H
centerC22#20269951
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#405×C211C2
socleC22×C211#121
prime radicalC22×D844C22
cosocleC22×C4×C211C23
hypercenter#405D211
hypocenterC211#1977
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405

Complements

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

NH ≅ G/Ntype
2. C2×#20269913C2semidirect
3. #27213154C2semidirect
5. C2×#20269895C2semidirect
7. #405×C211C2semidirect
13. C2×#16809951C4semidirect
106. C211#1977semidirect
114. #405D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099512
C2×#202699132
#272131542
#272131502
C2×#202698952
C2×#202698932
#405×C2112
#1977211