Finite Group Databasebeta

← #41013614

Subgroups — #41013614

Order 27008.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#27213623C211yesmaximal, characteristic
2#27213189C211yesmaximal, characteristic
3#27213188C211yesmaximal, characteristic
4C2×#20269895C211yesmaximal, characteristic
5#27213052C211yesmaximal, characteristic
6#27212906C211yesmaximal, characteristic
7#339×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C4×Dic422—21no—
11C4×Dic422—21no—
12#20269942—21no—
13#20269942—21no—
14#20269936C2211yescharacteristic
15#20269933—22no—
16C2×C4⋊Dic211C2211yescharacteristic
17#20269911—21no—
18#20269910—21no—
19#20269909—21no—
20#20269908—21no—
21#20269895C414yescomplemented
22C211⋊#95—22no—
23C2×M16×C211C2211yescharacteristic
24#117×C211C2211yescharacteristic
25C2×C42×C211C2211yescharacteristic
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422C2×C412yes—
31C2×Dic422Q812yes—
32C22×Dic211—21no—
33C22×Dic211D412yes—
34#16809946—21no—
35#16809946C2×C412yes—
36C2×C211⋊C8—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211D411yescharacteristic
40C4⋊Dic211D411yescharacteristic
41Dic211⋊C4—22no—
42Dic211⋊C4—22no—
43C22⋊Dic211—21no—
44C4×Dic211—22no—
45C4×Dic211—22no—
46C4×Dic211—21no—
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49M16×C211—21no—
50M16×C211C2×C412yes—
51C2×C8×C211—21no—
52C4⋊C4×C211D411yescharacteristic
53C4⋊C4×C211D411yescharacteristic
54C22⋊C4×C211—21no—
55C42×C211—21no—
56C42×C211—21no—
57C42×C211—21no—
58Dic422—22no—
59Dic422—22no—
60Dic422—22no—
61Dic422—22no—
62C2×Dic211—22no—
63C2×Dic211—22no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—42no—
67C2×Dic211—41no—
68C2×Dic211C4⋊C414yes—
69C211⋊C8—22no—
70C23×C211C2×D411yescharacteristic
71C2×C4×C211—22no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211C22×C411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×Q811yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C2×C4×C211#4111yescharacteristic
82C8×C211—22no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211C2×C4⋊C411yescharacteristic
86C22×C211#12011yescharacteristic
87C22×C211#12311yescharacteristic
88C4×C211—42no—
89C4×C211—41no—
90C4×C211—41no—
91C4×C211C4×D411yescharacteristic
92C4×C211C4×D411yescharacteristic
93C4×C211C22⋊Q811yescharacteristic
94C4×C211C22⋊Q811yescharacteristic
95Dic211—24no—
96Dic211—41no—
97Dic211—41no—
98Dic211—81no—
99C2×C211—21no—
100C2×C211—21no—
101C2×C211#38511yescharacteristic
102C2×C211#45411yescharacteristic
103C2×C211#45611yescharacteristic
104C211#185711yescharacteristic, complemented, Sylow
105#1857—2111nomaximal, Sylow
106#519—2111no—
107#428—2111no—
108#428—2111no—
109C2×#101—2111no—
110#391—2111no—
111#339—2111no—
112#339D21111yescharacteristic
113C22×Q8—2111no—
114C2×M16—2111no—
115C2×M16C2×D21111yescharacteristic
116C4×Q8—4221no—
117C4×Q8—4221no—
118C4×Q8—4221no—
119C4×Q8—4221no—
120#117—4222no—
121#117—2111no—
122#117C2×D21111yescharacteristic
123C2×C4⋊C4—2111no—
124C2×C42C2×D21111yescharacteristic
125#107—4221no—
126#107—4221no—
127#106—4221no—
128#106—4221no—
129#101—2114no—
130#95—4222no—
131C2×Q8—4221no—
132C2×Q8—4221no—
133C2×Q8—4221no—
134C2×Q8—4221no—
135C2×Q8—2112no—
136C2×Q8—2112no—
137C22×C4—21no—
138C22×C4—4221no—
139C22×C4—2112no—
140C22×C4C22×D21111yescharacteristic
141M16—21no—
142M16—4221no—
143M16—2112no—
144M16C4×D21112yes—
145C2×C8—21no—
146C2×C8—4221no—
147C4⋊C4—4222no—
148C4⋊C4—4222no—
149C4⋊C4—4221no—
150C4⋊C4—4221no—
151C4⋊C4—2111no—
152C4⋊C4—2111no—
153C4⋊C4D84411yescharacteristic
154C4⋊C4D84411yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—4221no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—4222no—
161C42—4222no—
162C42—4221no—
163C23D4×D21111yescharacteristic
164Q8—4222no—
165Q8—4222no—
166Q8—4222no—
167Q8—4222no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—8442no—
174C2×C4—8441no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—4222no—
178C2×C4—4222no—
179C2×C4—2114no—
180C2×C4C2×C4×D21111yescharacteristic
181C2×C4D4×D21111yescharacteristic
182C2×C4C2×D84411yescharacteristic
183C2×C4Q8×D21111yescharacteristic
184C2×C4#1680996111yescharacteristic
185C2×C4#1680996211yescharacteristic
186C8—22no—
187C8—4222no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D21111yescharacteristic
191C22#2026994711yescharacteristic
192C22#2026995911yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—8441no—
197C4—8441no—
198C4—16881no—
199C4—4224no—
200C4(C22⋊D211)⋊C411yescharacteristic
201C4C4×D84411yescharacteristic
202C4#2026995511yescharacteristic
203C4#2026995711yescharacteristic
204C2—21no—
205C2—21no—
206C2#2721302011yescharacteristic
207C2#2721332011yescharacteristic
208C2#2721333011yescharacteristic
subgroupG/H
centerC22#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1857
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
21. #20269895C4semidirect
104. C211#1857semidirect

Maximal subgroups

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

subgroupindex
#272136232
#272131892
#272131882
C2×#202698952
#272130522
#272129062
#339×C2112
#1857211