Finite Group Databasebeta

← #41014758

Subgroups — #41014758

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
1C2×#20269912C211yesmaximal, characteristic, complemented
2#27213189C211yesmaximal, characteristic, complemented
3#27213173C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic
5C2×#20269894C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic, complemented
7#418×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269912—22no—
13#20269910—21no—
14#20269908—21no—
15C4⋊(C211⋊C8)—22no—
16#20269898—22no—
17#20269894—22no—
18C22×D4×C211C2211yescharacteristic
19C2×M16×C211C2211yescharacteristic
20#117×C211C2211yescharacteristic
21#102×C211—21no—
22#102×C211—21no—
23C22×Dic211—21no—
24#16809946—21no—
25#16809946—21no—
26#16809946—21no—
27#16809946D412yes—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8D412yes—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C24×C211—21no—
36C2×D4×C211—21no—
37C2×D4×C211—21no—
38C2×D4×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211Q812yes—
42C22×C4×C211C2311yescharacteristic
43M16×C211—21no—
44M16×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211D411yescharacteristic
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C42×C211—21no—
50C2×Dic211—41no—
51C2×Dic211—81no—
52C211⋊C8—22no—
53C211⋊C8—22no—
54C211⋊C8—41no—
55C23×C211—22no—
56C23×C211—41no—
57C23×C211—41no—
58C23×C211—81no—
59C23×C211—81no—
60C23×C211#4111yescharacteristic
61D4×C211—22no—
62D4×C211—22no—
63D4×C211—41no—
64D4×C211—41no—
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×Q811yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C8×C211—41no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211—42no—
77C22×C211—82no—
78C22×C211—82no—
79C22×C211—81no—
80C22×C211—81no—
81C22×C211#12111yescharacteristic
82C22×C211C22⋊Q811yescharacteristic
83C22×C211#12611yescharacteristic
84C4×C211—41no—
85C4×C211—41no—
86C4×C211#12111yescharacteristic
87C4×C211#12111yescharacteristic
88C4×C211C22⋊Q811yescharacteristic
89C4×C211C22⋊Q811yescharacteristic
90Dic211—81no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211—82no—
94C2×C211#39611yescharacteristic
95C2×C211#47111yescharacteristic
96C2×C211#47211yescharacteristic
97C211#206311yescharacteristic, complemented, Sylow
98#2063—2111nomaximal, Sylow
99C2×#108—2111no—
100#428—2111no—
101#424—2111no—
102#418D21111yescharacteristic
103C2×#102—2111no—
104C2×#100—2111no—
105#343—2111no—
106C22×D4C2×D21111yescharacteristic
107C2×M16—2111no—
108C2×M16—2111no—
109C2×M16C2×D21111yescharacteristic
110C22×C8—2111no—
111#117C2×D21111yescharacteristic
112C2×C4⋊C4—2111no—
113#108—4222no—
114#107—4221no—
115#106—4221no—
116C4⋊C8—4222no—
117#102—21no—
118#102—21no—
119#102—4222no—
120#100—4222no—
121C24—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4Dic42212yes—
128C22×C4—4221no—
129C22×C4C22×D21111yescharacteristic
130M16—21no—
131M16—21no—
132M16—4221no—
133M16—4221no—
134M16—4221no—
135M16—2112no—
136C2×C8—21no—
137C2×C8—4221no—
138C2×C8—4221no—
139C2×C8—4221no—
140C2×C8—4221no—
141C2×C8—2112no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4C22⋊D21111yescharacteristic
145C4⋊C4C22⋊D21111yescharacteristic
146C22⋊C4—21no—
147C42—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—81no—
152C23—81no—
153C23#1680996111yescharacteristic
154D4—22no—
155D4—22no—
156D4—41no—
157D4—41no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—8441no—
161C2×C4—16881no—
162C2×C4D4×D21111yescharacteristic
163C2×C4D4×D21111yescharacteristic
164C2×C4C2×C22⋊D21111yescharacteristic
165C2×C4C2×Dic42211yescharacteristic
166C2×C4#1680996111yescharacteristic
167C2×C4#1680996211yescharacteristic
168C8—41no—
169C8—8441no—
170C8—4222no—
171C8—4222no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—82no—
176C22—82no—
177C22—81no—
178C22—81no—
179C22#2026994911yescharacteristic
180C22#2026995411yescharacteristic
181C22#2026997011yescharacteristic
182C4—41no—
183C4—41no—
184C4—16881no—
185C4#2026994811yescharacteristic
186C4#2026995011yescharacteristic
187C4#2026995411yescharacteristic
188C4#2026995611yescharacteristic
189C2—21no—
190C2—21no—
191C2—82no—
192C2#2721306911yescharacteristic
193C2#2721340711yescharacteristic
194C2#2721341411yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#418×C211C2
socleC22×C211#121
prime radicalC22×D4×C211C22
hypercenter#418D211
hypocenterC211#2063
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#20269912C2semidirect
2. #27213189C2semidirect
3. #27213173C2semidirect
6. #27212914C2semidirect
97. C211#2063semidirect

Maximal subgroups

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

subgroupindex
C2×#202699122
#272131892
#272131732
C2×#202698982
C2×#202698942
#272129142
#418×C2112
#2063211