Finite Group Databasebeta

← #41013622

Subgroups — #41013622

Order 27008.

302 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 89 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#27213186C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic
4#27213015C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7#417×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×Dic211⋊C4—22no—
17C2×C4×Dic211C2211yescharacteristic
18#20269910—21no—
19#20269908—21no—
20#20269904C414yescomplemented
21C2×#41×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#103×C211—21no—
25#102×C211—21no—
26C22×Dic211—22no—
27C22×Dic211—21no—
28C22×Dic211D412yes—
29#16809946—21no—
30#16809946C2×C412yescomplemented
31C2×C211⋊C8—21no—
32C4⋊Dic211—22no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211D411yescharacteristic
36C4⋊Dic211D411yescharacteristic
37Dic211⋊C4—44no—
38C22⋊Dic211—22no—
39C22⋊Dic211—21no—
40C4×Dic211—22no—
41C4×Dic211—21no—
42C4×Dic211—21no—
43C4×Dic211C2×C412yescomplemented
44#41×C211—21no—
45#41×C211—21no—
46#41×C211C2×C412yes—
47#41×C211Q812yes—
48C2×Q8×C211D411yescharacteristic
49C2×D4×C211—21no—
50C2×D4×C211D411yescharacteristic
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54M16×C211—21no—
55M16×C211—21no—
56C2×C8×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59C2×Dic211—22no—
60C2×Dic211—22no—
61C2×Dic211—22no—
62C2×Dic211—21no—
63C2×Dic211—22no—
64C2×Dic211—21no—
65C2×Dic211—42no—
66C2×Dic211—82no—
67C211⋊C8—22no—
68C23×C211—21no—
69C23×C211C2×D411yescharacteristic
70Q8×C211—21no—
71Q8×C211C4⋊C412yescomplemented
72D4×C211—22no—
73D4×C211—22no—
74D4×C211—21no—
75D4×C211C4⋊C412yescomplemented
76C2×C4×C211—21no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—42no—
80C2×C4×C211—41no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×Q811yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C8×C211—41no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211—22no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211C4×D411yescharacteristic
94C22×C211#12011yescharacteristic
95C22×C211C22⋊Q811yescharacteristic
96C4×C211—22no—
97C4×C211—81no—
98C4×C211C2×C4⋊C411yescharacteristic
99C4×C211C4×D411yescharacteristic
100C4×C211C22⋊Q811yescharacteristic
101C4×C211#12311yescharacteristic
102Dic211—24no—
103Dic211—42no—
104Dic211—41no—
105Dic211—41no—
106C2×C211—21no—
107C2×C211—21no—
108C2×C211—42no—
109C2×C211#38511yescharacteristic
110C2×C211#45311yescharacteristic
111C2×C211#45611yescharacteristic
112C211#185911yescharacteristic, complemented, Sylow
113#1859—2111nomaximal, Sylow
114#520—2111no—
115#428—2111no—
116C2×#104—2111no—
117#417D21111yescharacteristic
118#384—2111no—
119#343—2111no—
120#339—2111no—
121C2×#41C2×D21111yescharacteristic
122C2×M16—2111no—
123C2×M16C2×D21111yescharacteristic
124C4×Q8—4221no—
125C4×D4—4221no—
126C4×D4—4222no—
127#117—4221no—
128#117—2111no—
129C2×C4⋊C4—4222no—
130C2×C4⋊C4—4221no—
131C2×C4⋊C4—2111no—
132C2×C4⋊C4C2×D21111yescharacteristic
133C2×C42—2111no—
134#107—4221no—
135#106—4221no—
136#104—2114no—
137#103—21no—
138#102—21no—
139#41—21no—
140#41—21no—
141#41C4×D21112yes—
142#41Dic42212yes—
143C2×Q8C22⋊D21111yescharacteristic
144C2×D4—21no—
145C2×D4C22⋊D21111yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—4222no—
149C22×C4—4221no—
150C22×C4—2112no—
151C22×C4C22×D21111yescharacteristic
152M16—21no—
153M16—21no—
154M16—4221no—
155M16—2112no—
156C2×C8—21no—
157C2×C8—4221no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—8444no—
161C4⋊C4—4222no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C4⋊C4—2111no—
165C4⋊C4—2111no—
166C22⋊C4—4222no—
167C22⋊C4—4221no—
168C42—4222no—
169C42—4221no—
170C42—4221no—
171C42—2112no—
172C23—21no—
173C23D4×D21111yescharacteristic
174Q8—21no—
175Q8Dic211⋊C412yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4Dic211⋊C412yescomplemented
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—8442no—
186C2×C4—16882no—
187C2×C4—4221no—
188C2×C4—4222no—
189C2×C4—4221no—
190C2×C4—4222no—
191C2×C4—4222no—
192C2×C4—4222no—
193C2×C4C2×C4×D21111yescharacteristic
194C2×C4D4×D21111yescharacteristic
195C2×C4C2×C22⋊D21111yescharacteristic
196C2×C4C2×Dic42211yescharacteristic
197C2×C4#1680996111yescharacteristic
198C2×C4#1680996111yescharacteristic
199C8—41no—
200C8—4222no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22(C22⋊D211)⋊C411yescharacteristic
207C22#2026994511yescharacteristic
208C22#2026995411yescharacteristic
209C4—22no—
210C4—81no—
211C4—8442no—
212C4—8441no—
213C4—8441no—
214C4—4224no—
215C4C2×Dic211⋊C411yescharacteristic
216C4(C22⋊D211)⋊C411yescharacteristic
217C4#2026995411yescharacteristic
218C4#2026995811yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#2721301811yescharacteristic
223C2#2721331211yescharacteristic
224C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#417×C211C2
socleC22×C211#120
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenter#417D211
hypocenterC211#1859
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #27213186C2semidirect
4. #27213015C2semidirect
5. #27212914C2semidirect
6. #27212907C2semidirect
20. #20269904C4semidirect
30. #16809946C2×C4semidirect
43. C4×Dic211C2×C4semidirect
71. Q8×C211C4⋊C4semidirect
75. D4×C211C4⋊C4semidirect
112. C211#1859semidirect
175. Q8Dic211⋊C4semidirect
179. D4Dic211⋊C4semidirect

Maximal subgroups

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

subgroupindex
#272136292
#272131862
C2×#202699042
#272130152
#272129142
#272129072
#417×C2112
#1859211