Finite Group Databasebeta

← #41014755

Subgroups — #41014755

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#27213173C211yesmaximal, characteristic, complemented
3#27213150C211yesmaximal, characteristic
4C2×#20269897C211yesmaximal, characteristic
5C2×#20269893C211yesmaximal, characteristic
6#27212915C211yesmaximal, characteristic, complemented
7#428×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11C22×C211⋊C8C2211yescharacteristic
12#20269912—22no—
13C4⋊(C211⋊C8)—22no—
14#20269897—22no—
15#20269897—21no—
16#20269897—21no—
17#20269893—22no—
18C2×M16×C211C2211yescharacteristic
19#117×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21#107×C211—21no—
22#106×C211—21no—
23C23×D211—21no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844Q812yes—
30#16809946—21no—
31#16809946—21no—
32#16809946—21no—
33#16809946—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8D412yes—
39C22×C4×C211—21no—
40C22×C4×C211C2311yescharacteristic
41M16×C211—21no—
42M16×C211D412yes—
43C2×C8×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211D411yescharacteristic
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C42×C211—21no—
50C22×D211—22no—
51C22×D211—41no—
52C22×D211—41no—
53C22×D211—81no—
54C22×D211—81no—
55D844—22no—
56D844—22no—
57D844—41no—
58D844—41no—
59C211⋊C8—22no—
60C211⋊C8—41no—
61C211⋊C8—41no—
62C23×C211#4111yescharacteristic
63C2×C4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—41no—
66C2×C4×C211—81no—
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—22no—
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—41no—
85C4×C211—41no—
86C4×C211—81no—
87C4×C211#12111yescharacteristic
88C4×C211#12111yescharacteristic
89C4×C211C22⋊Q811yescharacteristic
90C4×C211C22⋊Q811yescharacteristic
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×#108—2111no—
100#428D21111yescharacteristic, complemented
101#424—2111no—
102#418—2111no—
103C2×#102—2111no—
104C2×#100—2111no—
105#343—2111no—
106C22×D4—2111no—
107C2×M16—2111no—
108C2×M16—2111no—
109C2×M16C2×D21111yescharacteristic
110C22×C8—2111no—
111#117C2×D21111yescharacteristic
112C2×C4⋊C4C2×D21111yescharacteristic
113#108—4222no—
114#107—21no—
115#106—21no—
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—21no—
129C22×C4C22×D21111yescharacteristic
130M16—21no—
131M16—4221no—
132M16—4221no—
133M16—4221no—
134M16—4221no—
135M16D84412yes—
136C2×C8—21no—
137C2×C8—4221no—
138C2×C8—4221no—
139C2×C8—4221no—
140C2×C8—4221no—
141C2×C8—2112no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4C22⋊D21111yescharacteristic
145C4⋊C4C22⋊D21111yescharacteristic
146C22⋊C4—21no—
147C42—21no—
148C23—8441no—
149C23—8441no—
150C23—16881no—
151C23—16881no—
152C23—4222no—
153C23#1680996111yescharacteristic
154D4—8441no—
155D4—8441no—
156D4—4222no—
157D4—4222no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×C22⋊D21111yescharacteristic
164C2×C4C2×D84411yescharacteristic
165C2×C4Q8×D21111yescharacteristic
166C2×C4#1680996211yescharacteristic
167C2×C4#1680996311yescharacteristic
168C8—22no—
169C8—8441no—
170C8—8441no—
171C8—4222no—
172C22—21no—
173C22—21no—
174C22—8442no—
175C22—16882no—
176C22—16882no—
177C22—16881no—
178C22—16881no—
179C22#2026994911yescharacteristic
180C22#2026995711yescharacteristic
181C22#2026997211yescharacteristic
182C4—41no—
183C4—41no—
184C4—81no—
185C4#2026995111yescharacteristic
186C4#2026995211yescharacteristic
187C4#2026995311yescharacteristic
188C4#2026995511yescharacteristic
189C2—21no—
190C2—21no—
191C2—16882no—
192C2#2721306811yescharacteristic
193C2#2721341111yescharacteristic
194C2#2721341411yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#428×C211C2
socleC22×C211#121
prime radicalC22×D844C22
hypercenter#428D211
hypocenterC211#2063
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#20269912C2semidirect
2. #27213173C2semidirect
6. #27212915C2semidirect
7. #428×C211C2semidirect
97. C211#2063semidirect
100. #428D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699122
#272131732
#272131502
C2×#202698972
C2×#202698932
#272129152
#428×C2112
#2063211