Finite Group Databasebeta

← #41013637

Subgroups — #41013637

Order 27008.

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

normal subgroups only

This group has 262 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#27213162C211yesmaximal, characteristic
4C2×#20269905C211yesmaximal, characteristic
5#27212917C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7#423×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269985—22no—
10#20269983—22no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2211yescharacteristic
13M16×D211—22no—
14#20269936C2211yescharacteristic
15#20269912—22no—
16#20269905C412yescomplemented
17#20269905C412yescomplemented
18#20269903—22no—
19C2×M16×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21C2×C42×C211C2211yescharacteristic
22C4⋊C8×C211—22no—
23#16809962—21no—
24#16809962—21no—
25#16809962C2×C412yes—
26#16809962D412yes—
27C2×Dic422Q811yescharacteristic
28C2×D844Q811yescharacteristic
29C2×C22⋊D211—21no—
30C2×C4×D211—21no—
31#16809946—21no—
32#16809946—21no—
33#16809946C2×C412yes—
34#16809946D412yes—
35#16809945—24no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C8×D211—24no—
39C4⋊Dic211—21no—
40C22⋊Dic211—21no—
41C4×Dic211—21no—
42C22×C4×C211—21no—
43C22×C4×C211C2311yescharacteristic
44M16×C211—21no—
45M16×C211—21no—
46M16×C211—21no—
47M16×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211D412yes—
50C42×C211—21no—
51C42×C211C2×C411yescharacteristic
52C42×C211C2×C411yescharacteristic
53C22×D211—21no—
54Dic422—21no—
55Dic422C4⋊C412yes—
56D844—21no—
57D844C4⋊C412yes—
58C22⋊D211—22no—
59C22⋊D211—22no—
60C2×Dic211—21no—
61C2×Dic211—41no—
62C4×D211—22no—
63C4×D211—22no—
64C211⋊C8—22no—
65C211⋊C8—22no—
66C23×C211#4111yescharacteristic
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—42no—
70C2×C4×C211C22×C411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×Q811yescharacteristic
75C2×C4×C211#4111yescharacteristic
76C8×C211—22no—
77C8×C211—41no—
78C22×C211—41no—
79C22×C211C4×D411yescharacteristic
80C22×C211C22⋊Q811yescharacteristic
81C22×C211#12311yescharacteristic
82C2×D211—22no—
83C2×D211—81no—
84C4×C211—22no—
85C4×C211—41no—
86C4×C211C2×C4⋊C411yescharacteristic
87C4×C211C4×D411yescharacteristic
88C4×C211#12011yescharacteristic
89C4×C211C22⋊Q811yescharacteristic
90Dic211—22no—
91Dic211—81no—
92C2×C211—21no—
93C2×C211—21no—
94C2×C211—21no—
95C2×C211#38511yescharacteristic
96D211—42no—
97C211#186111yescharacteristic, complemented, Sylow
98#1861—2111nomaximal, Sylow
99#568—2111no—
100#430—2111no—
101#423D21111yescharacteristic, complemented
102#421—2111no—
103C2×#104—2111no—
104#344—2111no—
105#339—2111no—
106C2×#41—2111no—
107#131—4222no—
108#131—4222no—
109C2×M16—4222no—
110C2×M16—2111no—
111C2×M16—2111no—
112C2×M16C2×D21111yescharacteristic
113C2×M16C2×D21111yescharacteristic
114#117—2111no—
115C2×C42C2×D21111yescharacteristic
116#108—4222no—
117C4⋊C8—22no—
118#104—4222no—
119#104—2112no—
120#104—2112no—
121#41—4221no—
122#41—4221no—
123#41—2112no—
124#41—2112no—
125C2×Q8—2111no—
126C2×D4—4221no—
127C2×D4—2111no—
128C22×C4—21no—
129C22×C4—4221no—
130C22×C4C22×D21111yescharacteristic
131M16—21no—
132M16—21no—
133M16—21no—
134M16—21no—
135M16—4224no—
136M16—4221no—
137M16—4221no—
138M16—2112no—
139M16—2112no—
140C2×C8—21no—
141C2×C8—4224no—
142C2×C8—4221no—
143C2×C8—4221no—
144C2×C8C22⋊D21112yes—
145C4⋊C4—4221no—
146C22⋊C4—4221no—
147C42—21no—
148C42—4221no—
149C42C4×D21111yescharacteristic
150C42C4×D21111yescharacteristic
151C23—4221no—
152C23#1680996211yescharacteristic
153Q8—4221no—
154Q8—2112no—
155D4—4222no—
156D4—4222no—
157D4—4221no—
158D4—2112no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—8441no—
163C2×C4—4221no—
164C2×C4—4222no—
165C2×C4—4222no—
166C2×C4C2×C4×D21111yescharacteristic
167C2×C4D4×D21111yescharacteristic
168C2×C4D4×D21111yescharacteristic
169C2×C4C2×C22⋊D21111yescharacteristic
170C2×C4Q8×D21111yescharacteristic
171C2×C4#1680996311yescharacteristic
172C8—22no—
173C8—41no—
174C8—4222no—
175C8—4222no—
176C22—41no—
177C22—16881no—
178C22—4222no—
179C22C4×C22⋊D21111yescharacteristic
180C22#2026995511yescharacteristic
181C22#2026996211yescharacteristic
182C4—22no—
183C4—41no—
184C4—16881no—
185C4—4222no—
186C4C4⋊C4×D21111yescharacteristic
187C4#2026993911yescharacteristic
188C4#2026994511yescharacteristic
189C4#2026995311yescharacteristic
190C2—21no—
191C2—21no—
192C2—21no—
193C2—8442no—
194C2#2721301711yescharacteristic
subgroupG/H
centerC4#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#423×C211C2
socleC2×C211#385
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#423D211
hypocenterC211#1861
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #423

Complements

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

NH ≅ G/Ntype
2. #27213194C2semidirect
5. #27212917C2semidirect
6. #27212906C2semidirect
7. #423×C211C2semidirect
16. #20269905C4semidirect
17. #20269905C4semidirect
97. C211#1861semidirect
101. #423D211semidirect

Maximal subgroups

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

subgroupindex
#272139122
#272131942
#272131622
C2×#202699052
#272129172
#272129062
#423×C2112
#1861211