Finite Group Databasebeta

← #41014064

Subgroups — #41014064

Order 27008.

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

normal subgroups only

This group has 216 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#27213194C211yesmaximal, characteristic, complemented
2#27213163C211yesmaximal, characteristic
3#27213130C211yesmaximal, characteristic
4C4×#16809946C211yesmaximal, characteristic, complemented
5#27212917C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269912—22no—
13#20269904—22no—
14#20269896C412yescomplemented
15#20269894C412yescomplemented
16#20269888—21no—
17C4×C211⋊C8—21no—
18C2×#41×C211C2211yescharacteristic
19C2×M16×C211C2211yescharacteristic
20C2×C42×C211C2211yescharacteristic
21#104×C211—21no—
22#104×C211—21no—
23#16809946—21no—
24#16809946—21no—
25#16809946—21no—
26#16809946C2×C412yes—
27#16809946C2×C412yes—
28#16809946D412yes—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C4⋊Dic211—21no—
33C22⋊Dic211—21no—
34C4×Dic211—21no—
35#41×C211—21no—
36#41×C211—21no—
37#41×C211—21no—
38#41×C211—21no—
39C2×Q8×C211C2×C411yescharacteristic
40C2×D4×C211—21no—
41C2×D4×C211C2×C411yescharacteristic
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45M16×C211—21no—
46M16×C211—21no—
47C2×C8×C211—21no—
48C42×C211—21no—
49C42×C211D411yescharacteristic
50C42×C211D411yescharacteristic
51C2×Dic211—41no—
52C211⋊C8—22no—
53C211⋊C8—22no—
54C211⋊C8—22no—
55C23×C211—21no—
56C23×C211#4111yescharacteristic
57Q8×C211—21no—
58Q8×C211—21no—
59D4×C211—21no—
60D4×C211—21no—
61D4×C211—41no—
62D4×C211—41no—
63C2×C4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—21no—
66C2×C4×C211—42no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211C22×C411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C2×C4×C211#4111yescharacteristic
74C2×C4×C211#4111yescharacteristic
75C8×C211—41no—
76C22×C211—41no—
77C22×C211—41no—
78C22×C211—81no—
79C22×C211#11711yescharacteristic
80C22×C211#12311yescharacteristic
81C22×C211#12411yescharacteristic
82C4×C211—22no—
83C4×C211—41no—
84C4×C211—41no—
85C4×C211C4×D411yescharacteristic
86C4×C211C4×D411yescharacteristic
87C4×C211#12111yescharacteristic
88C4×C211#12611yescharacteristic
89Dic211—81no—
90C2×C211—21no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211—81no—
94C2×C211#38811yescharacteristic
95C211#193211yescharacteristic, complemented, Sylow
96#1932—2111nomaximal, Sylow
97#430—2111no—
98#421—2111no—
99C2×#104D21111yescharacteristic
100#413—2111no—
101C4×M16—2111no—
102#344—2111no—
103#339—2111no—
104C2×#41C2×D21111yescharacteristic
105C2×M16—2111no—
106C2×M16—2111no—
107C2×M16—2111no—
108C2×M16C2×D21111yescharacteristic
109#117—2111no—
110C2×C42C2×D21111yescharacteristic
111#108—4222no—
112#104—21no—
113#104—21no—
114#104—4222no—
115#101—2112no—
116#100—2112no—
117#97—4221no—
118C4×C8—4221no—
119#41—21no—
120#41—21no—
121#41—21no—
122#41—21no—
123C2×Q8C4×D21111yescharacteristic
124C2×D4—21no—
125C2×D4C4×D21111yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4C22×D21111yescharacteristic
129M16—21no—
130M16—21no—
131M16—4221no—
132M16—4221no—
133M16—4221no—
134M16—2112no—
135M16—2112no—
136M16—2112no—
137C2×C8—21no—
138C2×C8—4221no—
139C2×C8—4221no—
140C2×C8—4221no—
141C4⋊C4—4221no—
142C22⋊C4—4221no—
143C42—21no—
144C42—4221no—
145C42C22⋊D21111yescharacteristic
146C42C22⋊D21111yescharacteristic
147C23—21no—
148C23#1680996111yescharacteristic
149Q8—21no—
150Q8—21no—
151D4—21no—
152D4—21no—
153D4—41no—
154D4—41no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—8441no—
162C2×C4C2×C4×D21111yescharacteristic
163C2×C4D4×D21111yescharacteristic
164C2×C4C2×C22⋊D21111yescharacteristic
165C2×C4#1680996111yescharacteristic
166C2×C4#1680996111yescharacteristic
167C2×C4#1680996211yescharacteristic
168C8—41no—
169C8—4222no—
170C8—4222no—
171C8—4222no—
172C22—41no—
173C22—41no—
174C22—81no—
175C22#2026993311yescharacteristic
176C22#2026995811yescharacteristic
177C22#2026996311yescharacteristic
178C4—22no—
179C4—41no—
180C4—41no—
181C4—16881no—
182C4(C22⋊D211)⋊C411yescharacteristic
183C4C4×C22⋊D21111yescharacteristic
184C4#2026994811yescharacteristic
185C4#2026997011yescharacteristic
186C2—21no—
187C2—21no—
188C2—21no—
189C2—81no—
190C2#2721303411yescharacteristic
subgroupG/H
centerC4#20269948
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC2×C211#388
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1932
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. #27213194C2semidirect
4. C4×#16809946C2semidirect
5. #27212917C2semidirect
6. #27212906C2semidirect
14. #20269896C4semidirect
15. #20269894C4semidirect
95. C211#1932semidirect

Maximal subgroups

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

subgroupindex
#272131942
#272131632
#272131302
C4×#168099462
#272129172
#272129062
C2×#104×C2112
#1932211