Finite Group Databasebeta

← #41012589

Subgroups — #41012589

Order 27008.

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

normal subgroups only

This group has 250 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#27213494C211yesmaximal, characteristic, complemented
3#27213423C211yesmaximal, characteristic, complemented
4#27212889C211yesmaximal, characteristic, complemented
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7#462×C211C211yesmaximal, characteristic, complemented
8C2×#16809963—21no—
9#20269974—21no—
10#20269974C2211yescharacteristic
11#20269952—22no—
12#20269952—21no—
13C4×D844—21no—
14C4×D844C2211yescharacteristic
15#20269910—21no—
16C4⋊(C211⋊C8)C2211yescharacteristic
17#20269897—21no—
18#20269897—22no—
19C4×C211⋊C8C2211yescharacteristic
20C2×Q16×C211—21no—
21C4⋊Q8×C211C2211yescharacteristic
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24#103×C211—21no—
25#16809963—42no—
26#16809963—42no—
27C2×D844—22no—
28C2×D844—22no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844D411yescharacteristic
32C2×D844D412yes—
33C2×C4×D211—22no—
34C2×C4×D211—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C4⋊Dic211D411yescharacteristic
38(C2×D211)⋊C4—22no—
39(C2×D211)⋊C4—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211D411yescharacteristic
42Q16×C211—42no—
43C2×C8×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C42×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51C22×D211—22no—
52D844—21no—
53D844—21no—
54D844—42no—
55D844—44no—
56D844—41no—
57D844—41no—
58D844—41no—
59D844—41no—
60D844—82no—
61C2×Dic211—21no—
62C4×D211—44no—
63C4×D211—41no—
64C4×D211—41no—
65C211⋊C8—41no—
66C211⋊C8—41no—
67Q8×C211—21no—
68Q8×C211—41no—
69Q8×C211—41no—
70Q8×C211D812yes—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—41no—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C8×C211—41no—
78C22×C211#12011yescharacteristic
79C2×D211—42no—
80C2×D211—41no—
81C2×D211—41no—
82C2×D211—42no—
83C2×D211—42no—
84C2×D211—41no—
85C4×C211—22no—
86C4×C211—21no—
87C4×C211—41no—
88C4×C211—81no—
89C4×C211C2×D811yescharacteristic
90C4×C211#13511yescharacteristic
91C4×C211AGL(1,ℤ/8)11yescharacteristic
92C4×C211#13711yescharacteristic
93Dic211—41no—
94C2×C211#44711yescharacteristic
95C2×C211#45211yescharacteristic
96C2×C211#45511yescharacteristic
97D211—81no—
98D211—82no—
99C211#164111yescharacteristic, complemented, Sylow
100#1641—2111nomaximal, Sylow
101#550—2111no—
102#486—2111no—
103#474—2111no—
104#462D21111yescharacteristic, complemented
105#331—2111no—
106Q8⋊C8—2111no—
107D4⋊C8—2111no—
108C2×#41—4221no—
109C2×Q16—21no—
110C4⋊Q8C2×D21111yescharacteristic
111#127—4221no—
112#127—2111no—
113#121—4222no—
114#121—4221no—
115C4×Q8C2×D21111yescharacteristic
116C4×D4—4221no—
117C4×D4—2111no—
118#107—4221no—
119C4⋊C8—2111no—
120C4⋊C8C2×D21111yescharacteristic
121#103—21no—
122#102—4221no—
123#102—4222no—
124C4×C8—2111no—
125#41—8442no—
126#41—8442no—
127C2×Q8—21no—
128C2×Q8C22⋊D21111yescharacteristic
129C2×D4—4222no—
130C2×D4—4222no—
131C2×D4—4221no—
132C2×D4—4221no—
133C2×D4—2112no—
134C2×D4—2111no—
135C22×C4—4222no—
136C22×C4—4221no—
137Q16—42no—
138C2×C8—21no—
139C2×C8—4221no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—2111no—
145C4⋊C4C22⋊D21111yescharacteristic
146C22⋊C4—4222no—
147C22⋊C4—4221no—
148C42—21no—
149C42C22×D21111yescharacteristic
150C23—4222no—
151C23—4221no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1680994712yes—
156D4—8442no—
157D4—8444no—
158D4—8441no—
159D4—8441no—
160D4—8441no—
161D4—8441no—
162D4—16882no—
163D4—4221no—
164D4—4221no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—8444no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—4221no—
172C2×C4D4×D21111yescharacteristic
173C2×C4D4×D21111yescharacteristic
174C2×C4C2×C22⋊D21111yescharacteristic
175C8—41no—
176C8—8441no—
177C8—8441no—
178C22—8442no—
179C22—8441no—
180C22—8442no—
181C22—8441no—
182C22—8442no—
183C22—8441no—
184C22#2026994511yescharacteristic
185C4—21no—
186C4—22no—
187C4—41no—
188C4—81no—
189C4—8441no—
190C4C2×#1680994711yescharacteristic
191C4#2027000011yescharacteristic
192C4#2027000211yescharacteristic
193C4#2027000911yescharacteristic
194C2—16882no—
195C2—16881no—
196C2#2721327911yescharacteristic
197C2#2721330711yescharacteristic
198C2#2721332211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#462×C211C2
socleC22×C211#120
prime radical#27213818C2
hypercenter#462D211
hypocenterC211#1641
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #27213494C2semidirect
3. #27213423C2semidirect
4. #27212889C2semidirect
5. Q8⋊(C211⋊C8)C2semidirect
6. #27212873C2semidirect
7. #462×C211C2semidirect
99. C211#1641semidirect
104. #462D211semidirect

Maximal subgroups

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

subgroupindex
#272138182
#272134942
#272134232
#272128892
Q8⋊(C211⋊C8)2
#272128732
#462×C2112
#1641211