Finite Group Databasebeta

← #41014462

Subgroups — #41014462

Order 27008.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#27213653C212yesmaximal, complemented
2#27213163C212yesmaximal, complemented
3#27213109C211yesmaximal, characteristic, complemented
4#27212903C211yesmaximal, characteristic, complemented
5#575×C211C211yesmaximal, characteristic, complemented
6C22×C4×D211C2211yescharacteristic
7C2×#16809946C2211yescharacteristic
8#20269958—22no—
9#20269953—22no—
10#20269948—22no—
11#20269945—22no—
12C4×C22⋊D211—24no—
13#20269936C2212yes—
14#20269904—24no—
15(C2×D211)⋊C8—22no—
16C2×#41×C211C2212yes—
17#137×C211—22no—
18AGL(1,ℤ/8)×C211—22no—
19#135×C211—22no—
20C2×M16×C211C2211yescharacteristic
21C23×D211D411yescharacteristic
22C2×C22⋊D211—24no—
23C22×Dic211D411yescharacteristic
24C2×C4×D211—21no—
25C2×C4×D211—22no—
26C2×C4×D211—22no—
27C2×C4×D211—21no—
28#16809946—22no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—22no—
31Dic211⋊C4—24no—
32C22⋊Dic211—24no—
33C22⋊Dic211—22no—
34(C2×D211)⋊C4—24no—
35C4×Dic211D414yescomplemented
36#41×C211—22no—
37#41×C211—22no—
38#41×C211—22no—
39#41×C211—22no—
40C2×Q8×C211D412yescomplemented
41C2×D4×C211—22no—
42C2×D4×C211D412yescomplemented
43C22×C4×C211—22no—
44C22×C4×C211C2311yescharacteristic
45Q16×C211—22no—
46SD16×C211—24no—
47D8×C211—22no—
48M16×C211—22no—
49C2×C8×C211D412yescomplemented
50C22×D211—22no—
51C22×D211—21no—
52C22×D211—42no—
53C22⋊D211—48no—
54C2×Dic211—24no—
55C2×Dic211—22no—
56C2×Dic211—21no—
57C4×D211—41no—
58C4×D211—42no—
59C4×D211—41no—
60C211⋊C8—41no—
61C23×C211—22no—
62C23×C211C2×D411yescharacteristic
63Q8×C211—22no—
64Q8×C211—22no—
65D4×C211—22no—
66D4×C211—22no—
67D4×C211—42no—
68D4×C211—42no—
69C2×C4×C211—22no—
70C2×C4×C211—42no—
71C2×C4×C211—42no—
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D412yes—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211#4111yescharacteristic
76C8×C211—22no—
77C22×C211—41no—
78C22×C211—44no—
79C22×C211—42no—
80C22×C211#12012yes—
81C22×C211#12111yescharacteristic
82C2×D211—24no—
83C2×D211—22no—
84C2×D211—42no—
85C2×D211—41no—
86C2×D211—41no—
87C4×C211—42no—
88C4×C211#12011yescharacteristic
89C4×C211#12112yes—
90C4×C211#12711yescharacteristic
91Dic211—44no—
92Dic211—41no—
93C2×C211—22no—
94C2×C211—21no—
95C2×C211—82no—
96C2×C211#39211yescharacteristic
97D211—42no—
98C211#201011yescharacteristic, complemented, Sylow
99#2010—2111nomaximal, Sylow
100#575D21111yescharacteristic, complemented
101#525—2112no—
102#421—2112no—
103#407—2111no—
104H(3,ℤ/4)—2111no—
105C2×#41C2×D21112yes—
106C23×C4—2111no—
107#137—22no—
108AGL(1,ℤ/8)—22no—
109#135—22no—
110C2×M16—2111no—
111C2×M16C2×D21111yescharacteristic
112#123—4222no—
113C22⋊Q8—4222no—
114#121—4222no—
115#120—4222no—
116C4×D4—4224no—
117#117—2112no—
118#104—4224no—
119C22⋊C8—4222no—
120C24—2111no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8C22⋊D21112yescomplemented
126C2×D4—22no—
127C2×D4—4224no—
128C2×D4C22⋊D21112yescomplemented
129C22×C4—22no—
130C22×C4—4222no—
131C22×C4—4221no—
132C22×C4—4222no—
133C22×C4—4221no—
134C22×C4—2111no—
135C22×C4C22×D21111yescharacteristic
136Q16—22no—
137SD16—24no—
138D8—22no—
139M16—22no—
140M16—4222no—
141C2×C8—4221no—
142C2×C8C22⋊D21112yescomplemented
143C4⋊C4—4224no—
144C4⋊C4—4222no—
145C22⋊C4—4224no—
146C22⋊C4—4224no—
147C22⋊C4—4222no—
148C42—2114no—
149C23—22no—
150C23—8442no—
151C23—4222no—
152C23—4221no—
153C23D4×D21111yescharacteristic
154Q8—22no—
155Q8—22no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160D4—8448no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—8441no—
165C2×C4—8442no—
166C2×C4—8441no—
167C2×C4—4224no—
168C2×C4—4222no—
169C2×C4—4221no—
170C2×C4D4×D21112yes—
171C2×C4C2×C22⋊D21112yes—
172C2×C4C2×C22⋊D21111yescharacteristic
173C2×C4#1680996111yescharacteristic
174C8—22no—
175C8—8441no—
176C22—41no—
177C22—44no—
178C22—42no—
179C22—8442no—
180C22—8441no—
181C22—8441no—
182C22—4224no—
183C22—4222no—
184C22#2026994512yes—
185C22#2026994811yescharacteristic
186C4—42no—
187C4—8444no—
188C4—8441no—
189C4#2026994611yescharacteristic
190C4#2026994912yes—
191C4#2026997311yescharacteristic
192C2—22no—
193C2—21no—
194C2—82no—
195C2—8442no—
196C2#2721305411yescharacteristic
subgroupG/H
centerC4#20269973
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#575×C211C2
socleC2×C211#392
hypercenter#575D211
hypocenterC211#2010
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #27213653C2semidirect
2. #27213163C2semidirect
3. #27213109C2semidirect
4. #27212903C2semidirect
5. #575×C211C2semidirect
35. C4×Dic211D4semidirect
40. C2×Q8×C211D4semidirect
42. C2×D4×C211D4semidirect
49. C2×C8×C211D4semidirect
98. C211#2010semidirect
100. #575D211semidirect
125. C2×Q8C22⋊D211semidirect
128. C2×D4C22⋊D211semidirect
142. C2×C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272136532
#272131632
#272131092
#272129032
#575×C2112
#2010211