Finite Group Databasebeta

← #41014056

Subgroups — #41014056

Order 27008.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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#27213189C211yesmaximal, characteristic
2C2×#20269901C211yesmaximal, characteristic
3C2×#20269896C211yesmaximal, characteristic
4#27213102C211yesmaximal, characteristic
5#27212914C211yesmaximal, characteristic
6#27212910C211yesmaximal, characteristic
7#419×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269910—21no—
13#20269908—21no—
14#20269901—21no—
15#20269901—21no—
16#20269896C414yescomplemented
17#20269888—21no—
18C4×C211⋊C8—21no—
19C22×Q8×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22#103×C211—21no—
23#103×C211—21no—
24C22×Dic211—21no—
25#16809946—21no—
26#16809946—21no—
27#16809946C2×C412yes—
28#16809946C2×C412yes—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8D411yescharacteristic
34C2×C211⋊C8D411yescharacteristic
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C2×Q8×C211—21no—
38C2×Q8×C211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211C2×C412yes—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45M16×C211—21no—
46M16×C211—21no—
47C2×C8×C211—21no—
48C4⋊C4×C211D411yescharacteristic
49C4⋊C4×C211D411yescharacteristic
50C22⋊C4×C211—21no—
51C42×C211—21no—
52C2×Dic211—41no—
53C2×Dic211—81no—
54C211⋊C8—22no—
55C211⋊C8—22no—
56C211⋊C8—21no—
57C211⋊C8—21no—
58C23×C211#4111yescharacteristic
59Q8×C211—22no—
60Q8×C211—22no—
61Q8×C211—41no—
62Q8×C211—41no—
63C2×C4×C211—22no—
64C2×C4×C211—21no—
65C2×C4×C211—21no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211C22×C411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C2×C4×C211#4111yescharacteristic
74C8×C211—41no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211#11711yescharacteristic
78C22×C211#12111yescharacteristic
79C22×C211#12611yescharacteristic
80C4×C211—42no—
81C4×C211—41no—
82C4×C211—41no—
83C4×C211C4×D411yescharacteristic
84C4×C211C4×D411yescharacteristic
85C4×C211#12311yescharacteristic
86C4×C211#12411yescharacteristic
87Dic211—81no—
88C2×C211—21no—
89C2×C211—21no—
90C2×C211#38811yescharacteristic
91C2×C211#47111yescharacteristic
92C2×C211#47311yescharacteristic
93C211#193011yescharacteristic, complemented, Sylow
94#1930—2111nomaximal, Sylow
95#428—2111no—
96#419D21111yescharacteristic
97C2×#103—2111no—
98C2×#101—2111no—
99#405—2111no—
100#343—2111no—
101#341—2111no—
102C22×Q8C2×D21111yescharacteristic
103C2×M16—2111no—
104C2×M16—2111no—
105C2×M16C2×D21111yescharacteristic
106C22×C8—2111no—
107#117C2×D21111yescharacteristic
108C2×C4⋊C4—2111no—
109#107—4221no—
110#106—4221no—
111#103—21no—
112#103—21no—
113#103—4221no—
114#103—4221no—
115#101—2114no—
116#97—4221no—
117C4×C8—4221no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8C4×D21112yes—
124C22×C4—21no—
125C22×C4—4221no—
126C22×C4C22×D21111yescharacteristic
127M16—21no—
128M16—21no—
129M16—4221no—
130M16—4221no—
131M16—2112no—
132M16—2112no—
133C2×C8—21no—
134C2×C8—4221no—
135C2×C8—4221no—
136C2×C8—4221no—
137C2×C8—4221no—
138C2×C8—2111no—
139C2×C8—2111no—
140C4⋊C4—4221no—
141C4⋊C4—4221no—
142C4⋊C4C22⋊D21111yescharacteristic
143C4⋊C4C22⋊D21111yescharacteristic
144C22⋊C4—21no—
145C42—21no—
146C23#1680996111yescharacteristic
147Q8—22no—
148Q8—22no—
149Q8—41no—
150Q8—41no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—8441no—
157C2×C4—16881no—
158C2×C4C2×C4×D21111yescharacteristic
159C2×C4D4×D21111yescharacteristic
160C2×C4C2×C22⋊D21111yescharacteristic
161C2×C4#1680996111yescharacteristic
162C2×C4#1680996111yescharacteristic
163C2×C4#1680996211yescharacteristic
164C8—41no—
165C8—4222no—
166C8—4222no—
167C8—4221no—
168C8—4221no—
169C22—21no—
170C22—21no—
171C22#2026993311yescharacteristic
172C22#2026994911yescharacteristic
173C22#2026997011yescharacteristic
174C4—41no—
175C4—41no—
176C4—42no—
177C4—16881no—
178C4(C22⋊D211)⋊C411yescharacteristic
179C4C4×C22⋊D21111yescharacteristic
180C4#2026995911yescharacteristic
181C4#2026996411yescharacteristic
182C2—21no—
183C2—21no—
184C2#2721304011yescharacteristic
185C2#2721341111yescharacteristic
186C2#2721341911yescharacteristic
subgroupG/H
centerC22#20269949
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC22×C211#121
prime radicalC23×C211#41
cosocleC22×C4×C211C23
hypercenter#419D211
hypocenterC211#1930
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
16. #20269896C4semidirect
93. C211#1930semidirect

Maximal subgroups

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

subgroupindex
#272131892
C2×#202699012
C2×#202698962
#272131022
#272129142
#272129102
#419×C2112
#1930211