Finite Group Databasebeta

← #41014303

Subgroups — #41014303

Order 27008.

296 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 97 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×#20269913C211yesmaximal, characteristic, complemented
2#27213155C211yesmaximal, characteristic, complemented
3#27213151C211yesmaximal, characteristic
4C2×#20269896C211yesmaximal, characteristic, complemented
5C2×#20269894C211yesmaximal, characteristic
6#27213101C211yesmaximal, characteristic, complemented
7C22×SD16×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269913—22no—
13#20269901—22no—
14#20269898—22no—
15#20269896—22no—
16#20269894—22no—
17#20269887—21no—
18C8×Dic211—21no—
19C22×Q8×C211C2211yescharacteristic
20C22×D4×C211C2211yescharacteristic
21C2×SD16×C211—21no—
22C2×SD16×C211—21no—
23C2×SD16×C211—22no—
24C2×SD16×C211C414yescomplemented
25C22×C8×C211C2211yescharacteristic
26#16809946—21no—
27#16809946—21no—
28#16809946—21no—
29#16809946—21no—
30#16809946—21no—
31#16809946D412yes—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211D412yes—
36C22⋊Dic211—21no—
37C4×Dic211—21no—
38C24×C211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211C2×C412yes—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211C2×C412yes—
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53SD16×C211—24no—
54SD16×C211—24no—
55C2×C8×C211C2×C412yes—
56C2×C8×C211D411yescharacteristic
57C2×C8×C211D411yescharacteristic
58C2×C8×C211D412yes—
59C2×Dic211—22no—
60C211⋊C8—22no—
61C211⋊C8—41no—
62C211⋊C8—41no—
63C23×C211—22no—
64C23×C211—41no—
65C23×C211—41no—
66C23×C211—82no—
67C23×C211#4111yescharacteristic
68Q8×C211—22no—
69Q8×C211—22no—
70Q8×C211—41no—
71Q8×C211—41no—
72D4×C211—22no—
73D4×C211—22no—
74D4×C211—41no—
75D4×C211—41no—
76C2×C4×C211—22no—
77C2×C4×C211—41no—
78C2×C4×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×C211C22⋊C412yes—
86C8×C211C22⋊C412yes—
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×C211—42no—
96C4×C211C2×C22⋊C411yescharacteristic
97C4×C211C4×D411yescharacteristic
98C4×C211#12111yescharacteristic
99C4×C211#12711yescharacteristic
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×SD16D21111yescharacteristic
109C2×#108—2111no—
110#419—2111no—
111#418—2111no—
112C2×#101—2111no—
113C2×#100—2111no—
114#405—2111no—
115C22×Q8C2×D21111yescharacteristic
116C22×D4C2×D21111yescharacteristic
117C2×SD16—21no—
118C2×SD16—21no—
119C2×SD16—22no—
120C2×SD16Dic21114yescomplemented
121C2×M16—2111no—
122C2×M16—2111no—
123C2×M16—2111no—
124C22×C8C2×D21111yescharacteristic
125#117—2111no—
126#108—4222no—
127#103—4222no—
128#102—4222no—
129#101—4222no—
130#100—4222no—
131#97—4221no—
132C4×C8—4221no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C2×Dic21112yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C2×Dic21112yes—
146C22×C4—21no—
147C22×C4C22×D21111yescharacteristic
148SD16—24no—
149SD16—24no—
150M16—4221no—
151M16—4221no—
152M16—4221no—
153M16—4221no—
154M16—4221no—
155M16—2112no—
156C2×C8—4221no—
157C2×C8—4221no—
158C2×C8—4221no—
159C2×C8C2×Dic21112yes—
160C2×C8C22⋊D21111yescharacteristic
161C2×C8C22⋊D21111yescharacteristic
162C2×C8C22⋊D21112yes—
163C4⋊C4—2112no—
164C22⋊C4—4221no—
165C42—4221no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1680996111yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—4222no—
183C2×C4C22×Dic21111yescharacteristic
184C2×C4D4×D21111yescharacteristic
185C2×C4D4×D21111yescharacteristic
186C2×C4C2×C22⋊D21111yescharacteristic
187C2×C4C2×C22⋊D21111yescharacteristic
188C2×C4#1680996111yescharacteristic
189C8—8441no—
190C8—8441no—
191C8—4222no—
192C8C22⋊Dic21112yes—
193C8C22⋊Dic21112yes—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22D4×Dic21111yescharacteristic
200C22#2026994811yescharacteristic
201C22#2026996411yescharacteristic
202C4—42no—
203C4—8442no—
204C4C2×C22⋊Dic21111yescharacteristic
205C4D4×Dic21111yescharacteristic
206C4#2026994811yescharacteristic
207C4#2026997311yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2721304911yescharacteristic
212C2#2721340612yes—
subgroupG/H
centerC22#20269948
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×SD16×C211C2
socleC22×C211#121
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC22×SD16D211
hypocenterC211#1977
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#20269913C2semidirect
2. #27213155C2semidirect
4. C2×#20269896C2semidirect
6. #27213101C2semidirect
24. C2×SD16×C211C4semidirect
106. C211#1977semidirect
120. C2×SD16Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699132
#272131552
#272131512
C2×#202698962
C2×#202698942
#272131012
C22×SD16×C2112
#1977211