Finite Group Databasebeta

← #41014753

Subgroups — #41014753

Order 27008.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213188C211yesmaximal, characteristic, complemented
2#27213172C211yesmaximal, characteristic, complemented
3#27213152C211yesmaximal, characteristic
4C2×#20269899C211yesmaximal, characteristic
5C2×#20269893C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic, complemented
7C2×#108×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269936C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269911—21no—
13#20269909—21no—
14Dic211⋊C8—22no—
15#20269899—21no—
16#20269899—21no—
17#20269899—22no—
18#20269893—22no—
19C2×M16×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22#108×C211—22no—
23C23×D211—21no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844Q812yes—
30C22×Dic211—21no—
31#16809946—21no—
32#16809946—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D411yescharacteristic
37C4⋊Dic211D411yescharacteristic
38C22⋊Dic211—21no—
39C4×Dic211—21no—
40C22×C4×C211C2311yescharacteristic
41M16×C211—21no—
42M16×C211—21no—
43M16×C211—21no—
44M16×C211D412yes—
45C2×C8×C211—21no—
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211D412yes—
50C22×D211—22no—
51C22×D211—41no—
52C22×D211—41no—
53C22×D211—81no—
54C22×D211—81no—
55D844—22no—
56D844—22no—
57D844—41no—
58D844—41no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—41no—
62C2×Dic211—81no—
63C211⋊C8—41no—
64C23×C211#4111yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×Q811yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C8×C211—22no—
72C8×C211—22no—
73C8×C211—41no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211#12111yescharacteristic
77C22×C211C22⋊Q811yescharacteristic
78C22×C211#12611yescharacteristic
79C2×D211—42no—
80C2×D211—82no—
81C2×D211—81no—
82C2×D211—81no—
83C2×D211—82no—
84C4×C211#12111yescharacteristic
85C4×C211#12111yescharacteristic
86C4×C211C22⋊Q811yescharacteristic
87C4×C211C22⋊Q811yescharacteristic
88Dic211—41no—
89Dic211—41no—
90Dic211—81no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211#39611yescharacteristic
94C2×C211#47111yescharacteristic
95C2×C211#47211yescharacteristic
96D211—82no—
97C211#206311yescharacteristic, complemented, Sylow
98#2063—2111nomaximal, Sylow
99C2×#108D21111yescharacteristic, complemented
100#428—2111no—
101#424—2111no—
102#418—2111no—
103C2×#102—2111no—
104C2×#100—2111no—
105#343—2111no—
106C22×D4—2111no—
107C2×M16—2111no—
108C2×M16C2×D21111yescharacteristic
109C2×M16C2×D21111yescharacteristic
110C22×C8C2×D21111yescharacteristic
111#117—2111no—
112C2×C4⋊C4—2111no—
113#108—22no—
114#107—4221no—
115#106—4221no—
116C4⋊C8—4222no—
117#102—4222no—
118#102—4221no—
119#102—4221no—
120#100—4222no—
121C24—4221no—
122C2×D4—4221no—
123C2×D4—4221no—
124C2×D4—4221no—
125C2×D4—4221no—
126C2×D4—4221no—
127C2×D4—2112no—
128C22×C4—4221no—
129C22×C4C22×D21111yescharacteristic
130M16—21no—
131M16—21no—
132M16—21no—
133M16—4221no—
134M16—4221no—
135M16C22⋊D21112yes—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—4221no—
141C2×C8D84412yes—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—2111no—
145C4⋊C4—2111no—
146C22⋊C4—4221no—
147C42—4221no—
148C23—8441no—
149C23—8441no—
150C23—16881no—
151C23—16881no—
152C23—4222no—
153C23#1680996311yescharacteristic
154D4—8441no—
155D4—8441no—
156D4—4222no—
157D4—4222no—
158C2×C4—8441no—
159C2×C4—16881no—
160C2×C4—4221no—
161C2×C4—4221no—
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×C22⋊D21111yescharacteristic
164C2×C4C2×D84411yescharacteristic
165C2×C4Q8×D21111yescharacteristic
166C2×C4#1680996111yescharacteristic
167C2×C4#1680996211yescharacteristic
168C8—22no—
169C8—22no—
170C8—41no—
171C8—8441no—
172C22—21no—
173C22—21no—
174C22—8442no—
175C22—16882no—
176C22—16882no—
177C22—16881no—
178C22—16881no—
179C22#2026995211yescharacteristic
180C22#2026995311yescharacteristic
181C22#2026997211yescharacteristic
182C4—8441no—
183C4—8441no—
184C4—16881no—
185C4#2026994911yescharacteristic
186C4#2026995111yescharacteristic
187C4#2026995511yescharacteristic
188C4#2026995711yescharacteristic
189C2—21no—
190C2—21no—
191C2—16882no—
192C2#2721306811yescharacteristic
193C2#2721341011yescharacteristic
194C2#2721341711yescharacteristic
subgroupG/H
centerC22#20269952
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#108×C211C2
socleC22×C211#121
prime radicalC22×D844C22
hypercenterC2×#108D211
hypocenterC211#2063
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#108

Complements

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

NH ≅ G/Ntype
1. #27213188C2semidirect
2. #27213172C2semidirect
6. #27212914C2semidirect
7. C2×#108×C211C2semidirect
97. C211#2063semidirect
99. C2×#108D211semidirect

Maximal subgroups

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

subgroupindex
#272131882
#272131722
#272131522
C2×#202698992
C2×#202698932
#272129142
C2×#108×C2112
#2063211