Finite Group Databasebeta

← #41014211

Subgroups — #41014211

Order 27008.

344 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20269966C211yesmaximal, characteristic
2C2×#20269901C211yesmaximal, characteristic, complemented
3C2×#20269899C211yesmaximal, characteristic
4C2×#20269897C211yesmaximal, characteristic
5C2×#20269887C211yesmaximal, characteristic, complemented
6#27213015C211yesmaximal, characteristic, complemented
7C2×#103×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269966—22no—
11#20269966C414yescomplemented
12C2×C4⋊Dic211C2211yescharacteristic
13C2×Dic211⋊C4—22no—
14C2×(C2×D211)⋊C4—22no—
15C2×C4×Dic211C2211yescharacteristic
16#20269901—22no—
17#20269899—22no—
18#20269897—22no—
19#20269887—22no—
20C22×Q8×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#103×C211—22no—
24C23×D211—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844C2×C412yes—
31C22×Dic211—21no—
32C22×Dic211D412yes—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8D412yes—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38Dic211⋊C4—44no—
39(C2×D211)⋊C4—28no—
40C4×Dic211C2×C412yes—
41C4×Dic211D412yes—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211C2×C412yes—
48C22×C4×C211—21no—
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C2×C8×C211—21no—
53C2×C8×C211D412yes—
54C4⋊C4×C211—21no—
55C4⋊C4×C211—21no—
56C22×D211—22no—
57C22×D211—41no—
58C22×D211—41no—
59C22×D211—42no—
60C22×D211—42no—
61D844—22no—
62D844—22no—
63D844—41no—
64D844—41no—
65C2×Dic211—22no—
66C2×Dic211—22no—
67C2×Dic211—41no—
68C2×Dic211—42no—
69C2×Dic211C22⋊C414yes—
70C211⋊C8—22no—
71C23×C211C2×D411yescharacteristic
72Q8×C211—22no—
73Q8×C211—22no—
74Q8×C211—41no—
75Q8×C211—41no—
76C2×C4×C211—22no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—42no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C8×C211—22no—
88C22×C211C2×C22⋊C411yescharacteristic
89C22×C211#12411yescharacteristic
90C22×C211#12711yescharacteristic
91C22×C211AGL(1,ℤ/8)12yes—
92C22×C211#13712yes—
93C2×D211—42no—
94C2×D211—44no—
95C2×D211—82no—
96C2×D211—81no—
97C2×D211—81no—
98C4×C211—42no—
99C4×C211—81no—
100C4×C211C4×D411yescharacteristic
101C4×C211C4×D411yescharacteristic
102C4×C211#12111yescharacteristic
103C4×C211#12111yescharacteristic
104Dic211—24no—
105Dic211—81no—
106C2×C211#39011yescharacteristic
107C2×C211#41712yes—
108C2×C211#48912yes—
109C2×C211#49611yescharacteristic
110C2×C211#49711yescharacteristic
111D211—82no—
112C211#195911yescharacteristic, complemented, Sylow
113#1959—2111nomaximal, Sylow
114C2×#124—2111no—
115C2×#103—2111no—
116C2×#103D21111yescharacteristic, complemented
117C2×#102—2111no—
118C2×#102—2111no—
119C2×#97—2111no—
120#384—2111no—
121C22×Q8C2×D21111yescharacteristic
122C22×D4—2111no—
123C22×C8—2111no—
124C22×C8C2×D21111yescharacteristic
125#124—4222no—
126#124—2114no—
127C2×C4⋊C4—4222no—
128C2×C4⋊C4—2111no—
129C2×C4⋊C4C2×D21111yescharacteristic
130C2×C22⋊C4—4222no—
131C2×C42—2111no—
132#103—22no—
133#103—4222no—
134#102—4222no—
135#102—4222no—
136#97—4222no—
137C24—4221no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C4×D21112yes—
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—4221no—
148C2×D4—4221no—
149C2×D4—2112no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—4221no—
153C22×C4—2112no—
154C22×C4C22×D21111yescharacteristic
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—4221no—
158C2×C8—4221no—
159C2×C8—2112no—
160C2×C8C22⋊D21112yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—8444no—
164C4⋊C4—4221no—
165C4⋊C4—4221no—
166C22⋊C4—4228no—
167C42—2112no—
168C42—2112no—
169C23—8441no—
170C23—8441no—
171C23—8442no—
172C23—8442no—
173C23—4222no—
174C23D4×D21111yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—41no—
178Q8—41no—
179D4—8441no—
180D4—8441no—
181D4—4222no—
182D4—4222no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—8441no—
189C2×C4—8442no—
190C2×C4—4222no—
191C2×C4—4222no—
192C2×C4—2114no—
193C2×C4C2×C4×D21111yescharacteristic
194C2×C4D4×D21111yescharacteristic
195C2×C4D4×D21111yescharacteristic
196C2×C4C2×C22⋊D21111yescharacteristic
197C2×C4#1680996111yescharacteristic
198C2×C4#1680996211yescharacteristic
199C8—22no—
200C8—4222no—
201C22—8444no—
202C22—8442no—
203C22—16882no—
204C22—16881no—
205C22—16881no—
206C22C22⋊C4×D21111yescharacteristic
207C22#2026996311yescharacteristic
208C22#2026997311yescharacteristic
209C22#2027000412yes—
210C22#2027000912yes—
211C4—42no—
212C4—81no—
213C4—16881no—
214C4—4224no—
215C4(C22⋊D211)⋊C411yescharacteristic
216C4C4×C22⋊D21111yescharacteristic
217C4#2026994911yescharacteristic
218C4#2026995011yescharacteristic
219C2—16882no—
220C2#2721304611yescharacteristic
221C2#2721314612yes—
222C2#2721350712yes—
223C2#2721353211yescharacteristic
224C2#2721353911yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
cosocleC22×C4×C211C23
hypercenterC2×#103D211
hypocenterC211#1959
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#20269901C2semidirect
5. C2×#20269887C2semidirect
6. #27213015C2semidirect
7. C2×#103×C211C2semidirect
11. #20269966C4semidirect
112. C211#1959semidirect
116. C2×#103D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699662
C2×#202699012
C2×#202698992
C2×#202698972
C2×#202698872
#272130152
C2×#103×C2112
#1959211