Finite Group Databasebeta

← #41013616

Subgroups — #41013616

Order 27008.

252 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 79 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#27213912C211yesmaximal, characteristic
2#27213194C211yesmaximal, characteristic, complemented
3#27213128C211yesmaximal, characteristic
4#27213114C211yesmaximal, characteristic
5#27212910C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7#430×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269985—22no—
10#20269983—22no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2211yescharacteristic
13M16×D211—22no—
14C22×C211⋊C8C2211yescharacteristic
15#20269912—22no—
16#20269895C412yes—
17#20269893C412yes—
18(C2×D211)⋊C8—22no—
19C2×M16×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#108×C211—22no—
23#16809962—21no—
24#16809962—21no—
25#16809962—21no—
26#16809962—21no—
27C2×Dic422C2×C411yescharacteristic
28C2×D844C2×C411yescharacteristic
29C2×C22⋊D211Q812yes—
30C2×C4×D211D412yes—
31#16809946—21no—
32#16809946—21no—
33#16809946C2×C412yes—
34#16809946D412yes—
35#16809945—24no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C8×D211—24no—
42C22×C4×C211C2311yescharacteristic
43M16×C211—21no—
44M16×C211—21no—
45M16×C211—21no—
46M16×C211—21no—
47M16×C211C2×C412yes—
48M16×C211D412yes—
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C22×D211C4⋊C412yes—
53Dic422—21no—
54Dic422—21no—
55D844—21no—
56D844—21no—
57C22⋊D211—22no—
58C22⋊D211—22no—
59C2×Dic211C4⋊C412yes—
60C4×D211—22no—
61C4×D211—22no—
62C211⋊C8—22no—
63C211⋊C8—22no—
64C211⋊C8—41no—
65C23×C211C2×Q811yescharacteristic
66C2×C4×C211C22×C411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C8×C211—22no—
73C8×C211—22no—
74C8×C211—41no—
75C22×C211—41no—
76C22×C211C2×C4⋊C411yescharacteristic
77C22×C211C22⋊Q811yescharacteristic
78C22×C211C22⋊Q811yescharacteristic
79C2×D211—22no—
80C2×D211—42no—
81C4×C211C4×D411yescharacteristic
82C4×C211C4×D411yescharacteristic
83C4×C211#12011yescharacteristic
84C4×C211#12311yescharacteristic
85Dic211—22no—
86C2×C211—21no—
87C2×C211—21no—
88C2×C211—21no—
89C2×C211#38511yescharacteristic
90D211—42no—
91C211#185811yescharacteristic, complemented, Sylow
92#1858—2111nomaximal, Sylow
93#568—2111no—
94#430—2111no—
95#430D21111yescharacteristic, complemented
96#413—2111no—
97#408—2111no—
98#341—2111no—
99#341—2111no—
100C2×#41—2111no—
101#131—4222no—
102#131—4222no—
103C2×M16—4222no—
104C2×M16—2111no—
105C2×M16—2111no—
106C2×M16C2×D21111yescharacteristic
107C2×M16C2×D21111yescharacteristic
108C2×M16C2×D21111yescharacteristic
109C22×C8—2111no—
110#108—22no—
111#108—4222no—
112#101—2112no—
113#100—2112no—
114C22⋊C8—4222no—
115#41—4221no—
116#41—4221no—
117#41—4221no—
118#41—4221no—
119C2×Q8—2111no—
120C2×D4—2112no—
121C2×D4—2111no—
122C22×C4—2112no—
123C22×C4C22×D21111yescharacteristic
124M16—21no—
125M16—21no—
126M16—21no—
127M16—21no—
128M16—4224no—
129M16—4221no—
130M16—4221no—
131M16—2112no—
132M16—2112no—
133M16C4×D21112yes—
134M16C22⋊D21112yes—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—4224no—
139C2×C8—4221no—
140C2×C8—4221no—
141C2×C8—4221no—
142C2×C8—4221no—
143C2×C8—4221no—
144C23—2112no—
145C23Q8×D21111yescharacteristic
146Q8—4221no—
147Q8—4221no—
148D4—4222no—
149D4—4222no—
150D4—4221no—
151D4—4221no—
152C2×C4—4222no—
153C2×C4—4222no—
154C2×C4—2112no—
155C2×C4C2×C4×D21111yescharacteristic
156C2×C4D4×D21111yescharacteristic
157C2×C4D4×D21111yescharacteristic
158C2×C4C2×C22⋊D21111yescharacteristic
159C2×C4#1680996211yescharacteristic
160C2×C4#1680996311yescharacteristic
161C8—22no—
162C8—22no—
163C8—41no—
164C8—8441no—
165C8—4222no—
166C8—4222no—
167C22—41no—
168C22—8442no—
169C22—4222no—
170C22C4⋊C4×D21111yescharacteristic
171C22#2026995311yescharacteristic
172C22#2026995511yescharacteristic
173C4—4222no—
174C4C4×C22⋊D21111yescharacteristic
175C4#2026993911yescharacteristic
176C4#2026994511yescharacteristic
177C4#2026996211yescharacteristic
178C2—21no—
179C2—21no—
180C2—21no—
181C2—8442no—
182C2#2721301711yescharacteristic
subgroupG/H
centerC4#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#430×C211C2
socleC2×C211#385
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#430D211
hypocenterC211#1858
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #430

Complements

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

NH ≅ G/Ntype
2. #27213194C2semidirect
5. #27212910C2semidirect
6. #27212910C2semidirect
7. #430×C211C2semidirect
91. C211#1858semidirect
95. #430D211semidirect

Maximal subgroups

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

subgroupindex
#272139122
#272131942
#272131282
#272131142
#272129102
#430×C2112
#1858211