Finite Group Databasebeta

← #41013631

Subgroups — #41013631

Order 27008.

286 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 93 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#27213189C211yesmaximal, characteristic, complemented
2C2×#20269904C211yesmaximal, characteristic
3#27213147C211yesmaximal, characteristic
4#27213011C211yesmaximal, characteristic, complemented
5#27212915C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#520×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4×Dic211C2211yescharacteristic
12#20269910—21no—
13#20269908—21no—
14#20269904C414yescomplemented
15#20269901—21no—
16#20269898—21no—
17C211⋊#95—22no—
18C2×#41×C211C2211yescharacteristic
19C4×Q8×C211—21no—
20C4×D4×C211—21no—
21C4×D4×C211—22no—
22#117×C211—21no—
23#117×C211C2211yescharacteristic
24C2×C4⋊C4×C211—21no—
25C2×C4⋊C4×C211C2211yescharacteristic
26C22×Dic211—21no—
27C22×Dic211—21no—
28#16809946—21no—
29#16809946—21no—
30#16809946—21no—
31#16809946C2×C412yescomplemented
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4×Dic211—21no—
37C4×Dic211C2×C412yescomplemented
38#41×C211—21no—
39#41×C211—21no—
40#41×C211C2×C412yes—
41#41×C211Q812yes—
42C2×Q8×C211D411yescharacteristic
43C2×D4×C211—21no—
44C2×D4×C211D411yescharacteristic
45C22×C4×C211—21no—
46C22×C4×C211—22no—
47C22×C4×C211D412yes—
48C22×C4×C211C2311yescharacteristic
49C4⋊C4×C211—22no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211D411yescharacteristic
53C4⋊C4×C211D411yescharacteristic
54C22⋊C4×C211—22no—
55C22⋊C4×C211—21no—
56C42×C211—22no—
57C42×C211—21no—
58C2×Dic211—22no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—42no—
62C2×Dic211—41no—
63C211⋊C8—22no—
64C211⋊C8—41no—
65C23×C211—21no—
66C23×C211C2×D411yescharacteristic
67Q8×C211—21no—
68Q8×C211C4⋊C412yes—
69D4×C211—22no—
70D4×C211—22no—
71D4×C211—21no—
72D4×C211C4⋊C412yes—
73C2×C4×C211—21no—
74C2×C4×C211—22no—
75C2×C4×C211—22no—
76C2×C4×C211—21no—
77C2×C4×C211—22no—
78C2×C4×C211—21no—
79C2×C4×C211—42no—
80C2×C4×C211—82no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×Q811yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—22no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211C4×D411yescharacteristic
93C22×C211#12011yescharacteristic
94C22×C211C22⋊Q811yescharacteristic
95C4×C211—22no—
96C4×C211—41no—
97C4×C211—42no—
98C4×C211—41no—
99C4×C211C2×C4⋊C411yescharacteristic
100C4×C211C4×D411yescharacteristic
101C4×C211C22⋊Q811yescharacteristic
102C4×C211#12311yescharacteristic
103Dic211—42no—
104Dic211—81no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—42no—
108C2×C211#38511yescharacteristic
109C2×C211#45411yescharacteristic
110C2×C211#45511yescharacteristic
111C211#186011yescharacteristic, complemented, Sylow
112#1860—2111nomaximal, Sylow
113#520D21111yescharacteristic
114#428—2111no—
115C2×#104—2111no—
116#417—2111no—
117#382—2111no—
118#343—2111no—
119#339—2111no—
120C2×#41C2×D21111yescharacteristic
121C2×M16—2111no—
122C2×M16—2111no—
123C4×Q8—21no—
124C4×D4—21no—
125C4×D4—22no—
126#117—21no—
127#117C2×D21111yescharacteristic
128C2×C4⋊C4—21no—
129C2×C4⋊C4—2111no—
130C2×C4⋊C4C2×D21111yescharacteristic
131C2×C42—2111no—
132#107—4221no—
133#106—4221no—
134#104—2114no—
135#103—4221no—
136#102—4221no—
137#95—4222no—
138#41—21no—
139#41—21no—
140#41C4×D21112yes—
141#41Dic42212yes—
142C2×Q8C22⋊D21111yescharacteristic
143C2×D4—21no—
144C2×D4C22⋊D21111yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—4221no—
148C22×C4—4221no—
149C22×C4C22⋊D21112yes—
150C22×C4C22×D21111yescharacteristic
151M16—4221no—
152M16—4221no—
153M16—4221no—
154M16—2112no—
155C2×C8—4221no—
156C2×C8—4221no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—4221no—
161C4⋊C4—4221no—
162C4⋊C4C22⋊D21111yescharacteristic
163C4⋊C4C22⋊D21111yescharacteristic
164C22⋊C4—22no—
165C22⋊C4—21no—
166C42—22no—
167C42—21no—
168C42—4221no—
169C42—2112no—
170C23—21no—
171C23C2×C22⋊D21111yescharacteristic
172Q8—21no—
173Q8Dic211⋊C412yes—
174D4—22no—
175D4—22no—
176D4—21no—
177D4Dic211⋊C412yes—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—8442no—
189C2×C4—8441no—
190C2×C4—4222no—
191C2×C4C2×C4×D21111yescharacteristic
192C2×C4C2×C22⋊D21111yescharacteristic
193C2×C4C2×C22⋊D21111yescharacteristic
194C2×C4C2×Dic42211yescharacteristic
195C2×C4#1680996211yescharacteristic
196C2×C4#1680996211yescharacteristic
197C8—8441no—
198C8—4222no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—41no—
203C22—41no—
204C22C4×C22⋊D21111yescharacteristic
205C22#2026994611yescharacteristic
206C22#2026995611yescharacteristic
207C4—22no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—8442no—
212C4—16881no—
213C4C2×Dic211⋊C411yescharacteristic
214C4C4×C22⋊D21111yescharacteristic
215C4#2026995611yescharacteristic
216C4#2026996011yescharacteristic
217C2—21no—
218C2—21no—
219C2—42no—
220C2#2721301911yescharacteristic
221C2#2721331911yescharacteristic
222C2#2721332411yescharacteristic
subgroupG/H
centerC22#20269946
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#520×C211C2
socleC22×C211#120
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenter#520D211
hypocenterC211#1860
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #27213189C2semidirect
4. #27213011C2semidirect
5. #27212915C2semidirect
6. #27212905C2semidirect
14. #20269904C4semidirect
31. #16809946C2×C4semidirect
37. C4×Dic211C2×C4semidirect
111. C211#1860semidirect

Maximal subgroups

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

subgroupindex
#272131892
C2×#202699042
#272131472
#272130112
#272129152
#272129052
#520×C2112
#1860211