Finite Group Databasebeta

← #41012527

Subgroups — #41012527

Order 27008.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213818C211yesmaximal, characteristic
2#27213519C211yesmaximal, characteristic, complemented
3#27213441C211yesmaximal, characteristic, complemented
4#27213437C211yesmaximal, characteristic, complemented
5#27212889C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic, complemented
7Q8⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#16809963—21no—
9C2×#16809951—22no—
10#20269977C2211yescharacteristic
11#20269974—21no—
12#20269974C2211yescharacteristic
13#20269952—22no—
14#20269952—21no—
15C4×D844—21no—
16C4×D844C2211yescharacteristic
17#20269910—21no—
18#20269909—21no—
19C4⋊(C211⋊C8)C2211yescharacteristic
20#20269901—21no—
21#20269899—21no—
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25#16809963—42no—
26#16809963—42no—
27C2×Dic422—21no—
28C2×D844—22no—
29C2×D844—22no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844D411yescharacteristic
33C2×D844D412yes—
34C2×C4×D211—22no—
35C2×C4×D211—21no—
36#16809951—44no—
37C2×C211⋊C8—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211D411yescharacteristic
41(C2×D211)⋊C4—22no—
42(C2×D211)⋊C4—21no—
43C2×Q8×C211D411yescharacteristic
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C42×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51C22×D211—22no—
52Dic422—42no—
53D844—21no—
54D844—21no—
55D844—42no—
56D844—44no—
57D844—41no—
58D844—41no—
59D844—41no—
60D844—41no—
61D844—82no—
62C2×Dic211—21no—
63C2×Dic211—41no—
64C4×D211—44no—
65C4×D211—41no—
66C4×D211—41no—
67C211⋊C8—41no—
68Q8×C211—21no—
69Q8×C211SD1612yescomplemented
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C8×C211—22no—
76C8×C211—41no—
77C22×C211#12011yescharacteristic
78C2×D211—42no—
79C2×D211—41no—
80C2×D211—41no—
81C2×D211—42no—
82C2×D211—42no—
83C2×D211—41no—
84C4×C211—22no—
85C4×C211—21no—
86C4×C211—41no—
87C4×C211C2×SD1611yescharacteristic
88C4×C211#13511yescharacteristic
89C4×C211AGL(1,ℤ/8)11yescharacteristic
90C4×C211#13711yescharacteristic
91Dic211—41no—
92Dic211—81no—
93C2×C211#45011yescharacteristic
94C2×C211#45211yescharacteristic
95C2×C211#45311yescharacteristic
96D211—81no—
97D211—82no—
98C211#163111yescharacteristic, complemented, Sylow
99#1631—2111nomaximal, Sylow
100#550—2111no—
101#492—2111no—
102#477—2111no—
103#476—2111no—
104#331—2111no—
105Q8⋊C8D21111yescharacteristic, complemented
106D4⋊C8—2111no—
107C2×#41—4221no—
108C2×SD16—4222no—
109C4⋊Q8—2111no—
110#127—4221no—
111#127—2111no—
112#121—4222no—
113#121—4221no—
114C4×Q8C2×D21111yescharacteristic
115C4×D4—4221no—
116C4×D4—2111no—
117#107—4221no—
118#106—4221no—
119C4⋊C8—2111no—
120C4⋊C8C2×D21111yescharacteristic
121#103—4221no—
122#102—4221no—
123C4×C8C2×D21111yescharacteristic
124#41—8442no—
125#41—8442no—
126C2×Q8—4221no—
127C2×Q8D84411yescharacteristic
128C2×D4—4222no—
129C2×D4—4222no—
130C2×D4—4221no—
131C2×D4—4221no—
132C2×D4—2112no—
133C2×D4—2111no—
134C22×C4—4222no—
135C22×C4—4221no—
136SD16—8444no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—4221no—
140C4⋊C4—21no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C4⋊C4—2111no—
144C4⋊C4D84411yescharacteristic
145C22⋊C4—4222no—
146C22⋊C4—4221no—
147C42—21no—
148C42C22×D21111yescharacteristic
149C23—4222no—
150C23—4221no—
151Q8—21no—
152Q8—8442no—
153Q8#1680995112yescomplemented
154D4—8442no—
155D4—8444no—
156D4—8441no—
157D4—8441no—
158D4—8441no—
159D4—8441no—
160D4—16882no—
161D4—4221no—
162D4—4221no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—8444no—
166C2×C4—8441no—
167C2×C4—8441no—
168C2×C4—8441no—
169C2×C4—4221no—
170C2×C4D4×D21111yescharacteristic
171C2×C4D4×D21111yescharacteristic
172C2×C4C2×D84411yescharacteristic
173C8—22no—
174C8—41no—
175C8—8441no—
176C22—8442no—
177C22—8441no—
178C22—8442no—
179C22—8441no—
180C22—8442no—
181C22—8441no—
182C22#2026994711yescharacteristic
183C4—21no—
184C4—22no—
185C4—41no—
186C4—8441no—
187C4—16881no—
188C4C2×#1680995111yescharacteristic
189C4#2026999811yescharacteristic
190C4#2027000511yescharacteristic
191C4#2027000911yescharacteristic
192C2—16882no—
193C2—16881no—
194C2#2721329911yescharacteristic
195C2#2721331111yescharacteristic
196C2#2721331511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingQ8⋊C8×C211C2
socleC22×C211#120
prime radical#27213818C2
hypercenterQ8⋊C8D211
hypocenterC211#1631
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #27213519C2semidirect
3. #27213441C2semidirect
4. #27213437C2semidirect
5. #27212889C2semidirect
6. #27212875C2semidirect
7. Q8⋊C8×C211C2semidirect
69. Q8×C211SD16semidirect
98. C211#1631semidirect
105. Q8⋊C8D211semidirect
153. Q8#16809951semidirect

Maximal subgroups

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

subgroupindex
#272138182
#272135192
#272134412
#272134372
#272128892
#272128752
Q8⋊C8×C2112
#1631211