Finite Group Databasebeta

← #41014298

Subgroups — #41014298

Order 27008.

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

normal subgroups only

This group has 296 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
1C22×#16809950C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213156C211yesmaximal, characteristic, complemented
4#27213151C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic, complemented
6#27213104C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809950—21no—
10C2×#16809950—21no—
11C2×#16809950—22no—
12C2×#16809950C414yescomplemented
13C2×#16809946C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15#20269936C2211yescharacteristic
16#20269912—22no—
17#20269902—22no—
18#20269898—22no—
19#20269895—22no—
20#20269887—21no—
21C8×Dic211—21no—
22C22×D4×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25#100×C211—22no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422C2×C412yes—
32C22×Dic211—21no—
33#16809950—24no—
34#16809950—24no—
35#16809946—21no—
36#16809946—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8C2×C412yes—
39C2×C211⋊C8D411yescharacteristic
40C2×C211⋊C8D411yescharacteristic
41C2×C211⋊C8D412yes—
42C4⋊Dic211D412yes—
43C22⋊Dic211—21no—
44C4×Dic211—21no—
45C24×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211—21no—
51C2×D4×C211C2×C412yes—
52C22×C4×C211C2311yescharacteristic
53M16×C211—21no—
54M16×C211—21no—
55M16×C211—21no—
56M16×C211D412yes—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59Dic422—22no—
60Dic422—22no—
61Dic422—41no—
62Dic422—41no—
63C2×Dic211—22no—
64C2×Dic211—22no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C211⋊C8—41no—
68C211⋊C8C22⋊C412yes—
69C211⋊C8C22⋊C412yes—
70C23×C211—22no—
71C23×C211—41no—
72C23×C211—41no—
73C23×C211—82no—
74C23×C211#4111yescharacteristic
75D4×C211—22no—
76D4×C211—22no—
77D4×C211—41no—
78D4×C211—41no—
79C2×C4×C211C22×C411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—22no—
86C8×C211—41no—
87C22×C211—22no—
88C22×C211—42no—
89C22×C211—82no—
90C22×C211—82no—
91C22×C211—82no—
92C22×C211C4×D411yescharacteristic
93C22×C211#12111yescharacteristic
94C22×C211#12411yescharacteristic
95C4×C211C2×C22⋊C411yescharacteristic
96C4×C211C4×D411yescharacteristic
97C4×C211#12111yescharacteristic
98C4×C211#12711yescharacteristic
99Dic211—42no—
100Dic211—42no—
101C2×C211—21no—
102C2×C211—21no—
103C2×C211—82no—
104C2×C211#39011yescharacteristic
105C2×C211#47112yes—
106C211#197711yescharacteristic, complemented, Sylow
107#1977—2111nomaximal, Sylow
108C22×SD16—2111no—
109C2×#108—2111no—
110#419—2111no—
111#418—2111no—
112C2×#101—2111no—
113C2×#100D21111yescharacteristic
114#405—2111no—
115C22×Q8—2111no—
116C22×D4C2×D21111yescharacteristic
117C2×SD16—4221no—
118C2×SD16—4221no—
119C2×SD16—4222no—
120C2×SD16—2114no—
121C2×M16—2111no—
122C2×M16C2×D21111yescharacteristic
123C2×M16C2×D21111yescharacteristic
124C22×C8—2111no—
125#117—2111no—
126#108—4222no—
127#103—4222no—
128#102—4222no—
129#101—4222no—
130#100—22no—
131#97—4221no—
132C4×C8—4221no—
133C24—21no—
134C2×Q8—4221no—
135C2×Q8—4221no—
136C2×Q8—4221no—
137C2×Q8—4221no—
138C2×Q8—4221no—
139C2×Q8—2112no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C4×D21112yes—
146C22×C4—4221no—
147C22×C4C22×D21111yescharacteristic
148SD16—4224no—
149SD16—4224no—
150M16—21no—
151M16—21no—
152M16—21no—
153M16—4221no—
154M16—4221no—
155M16C22⋊D21112yes—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—4221no—
159C2×C8—2112no—
160C2×C8—2111no—
161C2×C8—2111no—
162C2×C8—2112no—
163C4⋊C4—2112no—
164C22⋊C4—4221no—
165C42—4221no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1680996211yescharacteristic
171Q8—8441no—
172Q8—8441no—
173Q8—4222no—
174Q8—4222no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—8441no—
180C2×C4—8441no—
181C2×C4—4222no—
182C2×C4—4222no—
183C2×C4C2×C4×D21111yescharacteristic
184C2×C4D4×D21111yescharacteristic
185C2×C4D4×D21111yescharacteristic
186C2×C4D4×D21111yescharacteristic
187C2×C4C2×C22⋊D21111yescharacteristic
188C2×C4#1680996111yescharacteristic
189C8—22no—
190C8—41no—
191C8—8441no—
192C8—2112no—
193C8—2112no—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22C4×C22⋊D21111yescharacteristic
200C22#2026995011yescharacteristic
201C22#2026996311yescharacteristic
202C4—8442no—
203C4—8442no—
204C4C22⋊C4×D21111yescharacteristic
205C4(C22⋊D211)⋊C411yescharacteristic
206C4#2026994911yescharacteristic
207C4#2026997311yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2721304611yescharacteristic
212C2#2721340812yes—
subgroupG/H
centerC22#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211#121
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#100D211
hypocenterC211#1977
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#20269912C2semidirect
3. #27213156C2semidirect
5. C2×#20269895C2semidirect
6. #27213104C2semidirect
12. C2×#16809950C4semidirect
106. C211#1977semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
C2×#202699122
#272131562
#272131512
C2×#202698952
#272131042
C2×#100×C2112
#1977211