Finite Group Databasebeta

← #41013633

Subgroups — #41013633

Order 27008.

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

normal subgroups only

This group has 286 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
1#27213629C211yesmaximal, characteristic
2#27213188C211yesmaximal, characteristic, complemented
3#27213147C211yesmaximal, characteristic
4#27213012C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9Q8×Dic211—21no—
10D4×Dic211—22no—
11D4×Dic211—21no—
12#20269936—21no—
13#20269936C2211yescharacteristic
14C2×C4⋊Dic211—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×C4⋊Dic211C2211yescharacteristic
17#20269911—21no—
18#20269909—21no—
19#20269901—21no—
20#20269898—21no—
21C211⋊#95—22no—
22C2×#41×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×C42×C211C2211yescharacteristic
25#104×C211C414yescomplemented
26C22×Dic211—22no—
27C22×Dic211—21no—
28C22×Dic211D412yes—
29#16809946—21no—
30#16809946—21no—
31C2×C211⋊C8—21no—
32C4⋊Dic211—22no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211D411yescharacteristic
38C4⋊Dic211D411yescharacteristic
39C22⋊Dic211—22no—
40C22⋊Dic211—21no—
41C4×Dic211—22no—
42C4×Dic211—21no—
43#41×C211—21no—
44#41×C211—21no—
45#41×C211C2×C412yes—
46#41×C211Q812yes—
47C2×Q8×C211D411yescharacteristic
48C2×D4×C211—21no—
49C2×D4×C211D411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53M16×C211—21no—
54M16×C211C2×C412yescomplemented
55C2×C8×C211—21no—
56C42×C211—21no—
57C42×C211C2×C412yescomplemented
58C2×Dic211—21no—
59C2×Dic211—22no—
60C2×Dic211—21no—
61C2×Dic211—42no—
62C2×Dic211—42no—
63C2×Dic211—41no—
64C2×Dic211—82no—
65C211⋊C8—41no—
66C23×C211—21no—
67C23×C211C2×D411yescharacteristic
68Q8×C211—21no—
69Q8×C211C4⋊C412yes—
70D4×C211—22no—
71D4×C211—22no—
72D4×C211—21no—
73D4×C211C4⋊C412yes—
74C2×C4×C211—21no—
75C2×C4×C211—22no—
76C2×C4×C211—22no—
77C2×C4×C211—22no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×Q811yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—22no—
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—42no—
97C4×C211C2×C4⋊C411yescharacteristic
98C4×C211C4×D411yescharacteristic
99C4×C211C22⋊Q811yescharacteristic
100C4×C211#12311yescharacteristic
101Dic211—41no—
102Dic211—42no—
103Dic211—41no—
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#520—2111no—
114#428—2111no—
115C2×#104D21111yescharacteristic
116#417—2111no—
117#382—2111no—
118#343—2111no—
119#339—2111no—
120C2×#41C2×D21111yescharacteristic
121C2×M16—2111no—
122C2×M16C2×D21111yescharacteristic
123C4×Q8—4221no—
124C4×D4—4221no—
125C4×D4—4222no—
126#117—4221no—
127#117—2111no—
128C2×C4⋊C4—4221no—
129C2×C4⋊C4—2111no—
130C2×C4⋊C4—2111no—
131C2×C42C2×D21111yescharacteristic
132#107—4221no—
133#106—4221no—
134#104Dic21114yescomplemented
135#103—4221no—
136#102—4221no—
137#95—4222no—
138#41—21no—
139#41—21no—
140#41C2×Dic21112yes—
141#41Dic42212yes—
142C2×Q8D84411yescharacteristic
143C2×D4—21no—
144C2×D4D84411yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—4221no—
148C22×C4—4222no—
149C22×C4—2112no—
150C22×C4C22×D21111yescharacteristic
151M16—21no—
152M16—4221no—
153M16—4221no—
154M16C2×Dic21112yescomplemented
155C2×C8—21no—
156C2×C8—4221no—
157C4⋊C4—4222no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C4⋊C4—4221no—
161C4⋊C4—4221no—
162C4⋊C4—2111no—
163C4⋊C4—2111no—
164C22⋊C4—4222no—
165C22⋊C4—4221no—
166C42—21no—
167C42—4222no—
168C42—4221no—
169C42C2×Dic21112yescomplemented
170C23—21no—
171C23D4×D21111yescharacteristic
172Q8—21no—
173Q8C4⋊Dic21112yes—
174D4—22no—
175D4—22no—
176D4—21no—
177D4C4⋊Dic21112yes—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—8442no—
185C2×C4—8442no—
186C2×C4—8441no—
187C2×C4—16882no—
188C2×C4—4221no—
189C2×C4—4222no—
190C2×C4—4221no—
191C2×C4C22×Dic21111yescharacteristic
192C2×C4D4×D21111yescharacteristic
193C2×C4C2×D84411yescharacteristic
194C2×C4C2×Dic42211yescharacteristic
195C2×C4#1680996111yescharacteristic
196C2×C4#1680996111yescharacteristic
197C8—22no—
198C8—8441no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—41no—
203C22—41no—
204C22D4×Dic21111yescharacteristic
205C22#2026994711yescharacteristic
206C22#2026995411yescharacteristic
207C4—22no—
208C4—42no—
209C4—8441no—
210C4—8442no—
211C4—8441no—
212C4—16881no—
213C4C2×C4⋊Dic21111yescharacteristic
214C4D4×Dic21111yescharacteristic
215C4#2026995411yescharacteristic
216C4#2026996111yescharacteristic
217C2—21no—
218C2—21no—
219C2—42no—
220C2#2721302111yescharacteristic
221C2#2721332011yescharacteristic
222C2#2721332511yescharacteristic
subgroupG/H
centerC22#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211#120
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1860
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #27213188C2semidirect
4. #27213012C2semidirect
5. #27212914C2semidirect
6. #27212906C2semidirect
25. #104×C211C4semidirect
54. M16×C211C2×C4semidirect
57. C42×C211C2×C4semidirect
111. C211#1860semidirect
134. #104Dic211semidirect
154. M16C2×Dic211semidirect
169. C42C2×Dic211semidirect

Maximal subgroups

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

subgroupindex
#272136292
#272131882
#272131472
#272130122
#272129142
#272129062
C2×#104×C2112
#1860211