Finite Group Databasebeta

← #41014642

Subgroups — #41014642

Order 27008.

298 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20269975C211yesmaximal, characteristic
2C2×#20269902C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic
4C2×#20269900C211yesmaximal, characteristic
5#27213042C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7C2×#103×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269975—22no—
11#20269975—22no—
12C2×C4⋊Dic211C2211yescharacteristic
13C2×Dic211⋊C4—21no—
14C2×Dic211⋊C4—21no—
15C2×Dic211⋊C4—21no—
16C2×C4×Dic211C2211yescharacteristic
17#20269902—22no—
18#20269901—22no—
19#20269900—22no—
20C211⋊#95—21no—
21C22×Q8×C211C2211yescharacteristic
22C22×C8×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#103×C211—22no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422D412yes—
31C22×Dic211—21no—
32C22×Dic211—21no—
33C22×Dic211D411yescharacteristic
34C22×Dic211D411yescharacteristic
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211D412yes—
40Dic211⋊C4—24no—
41Dic211⋊C4—24no—
42Dic211⋊C4—42no—
43C4×Dic211—21no—
44C4×Dic211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211D412yes—
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C2×C8×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59Dic422—22no—
60Dic422—22no—
61Dic422—41no—
62Dic422—41no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—22no—
68C2×Dic211—22no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211SD1612yes—
74C2×Dic211Q1612yes—
75C211⋊C8—41no—
76C23×C211C2×D411yescharacteristic
77Q8×C211—22no—
78Q8×C211—22no—
79Q8×C211—41no—
80Q8×C211—41no—
81C2×C4×C211—22no—
82C2×C4×C211—41no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211#4111yescharacteristic
91C2×C4×C211#4111yescharacteristic
92C2×C4×C211#4111yescharacteristic
93C8×C211—41no—
94C22×C211#12011yescharacteristic
95C22×C211#12111yescharacteristic
96C22×C211#12411yescharacteristic
97C22×C211C2×SD1611yescharacteristic
98C22×C211C2×Q1611yescharacteristic
99C22×C211#13711yescharacteristic
100C22×C211#13711yescharacteristic
101C4×C211—42no—
102C4×C211—81no—
103C4×C211#12111yescharacteristic
104C4×C211#12111yescharacteristic
105C4×C211#12311yescharacteristic
106C4×C211#12311yescharacteristic
107Dic211—22no—
108Dic211—22no—
109Dic211—42no—
110Dic211—42no—
111C2×C211#39411yescharacteristic
112C2×C211#44811yescharacteristic
113C2×C211#45111yescharacteristic
114C2×C211#46111yescharacteristic
115C2×C211#46211yescharacteristic
116C2×C211#48711yescharacteristic
117C2×C211#49111yescharacteristic
118C211#204311yescharacteristic, complemented, Sylow
119#2043—2111nomaximal, Sylow
120C2×C4⋊Q8—2111no—
121C2×#103—2111no—
122C2×#103—2111no—
123C2×#103—2111no—
124C2×#103D21111yescharacteristic
125#389—2111no—
126#336—2111no—
127C22×Q8—2111no—
128C22×Q8C2×D21111yescharacteristic
129C22×C8—2111no—
130C22×C8C2×D21111yescharacteristic
131C4⋊Q8—4222no—
132C4⋊Q8—4222no—
133C2×C4⋊C4—4221no—
134C2×C4⋊C4—4221no—
135C2×C4⋊C4—4221no—
136C2×C4⋊C4—2111no—
137C2×C4⋊C4C2×D21111yescharacteristic
138C2×C42—2111no—
139#103—22no—
140#103—4222no—
141#103—4222no—
142#103—4222no—
143#95—4221no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—4221no—
150C2×Q8—4221no—
151C2×Q8—4221no—
152C2×Q8—4221no—
153C2×Q8—4221no—
154C2×Q8—2112no—
155C2×Q8C22⋊D21112yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4—2111no—
162C22×C4C22×D21111yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—4221no—
167C2×C8—4221no—
168C2×C8—4221no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—8442no—
172C4⋊C4—4224no—
173C4⋊C4—4224no—
174C4⋊C4—4221no—
175C4⋊C4—2112no—
176C42—4221no—
177C42—4221no—
178C23D4×D21111yescharacteristic
179Q8—22no—
180Q8—22no—
181Q8—41no—
182Q8—41no—
183Q8—8441no—
184Q8—8441no—
185Q8—4222no—
186Q8—4222no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—8441no—
196C2×C4—8441no—
197C2×C4—4222no—
198C2×C4—4222no—
199C2×C4—4221no—
200C2×C4—4221no—
201C2×C4—4221no—
202C2×C4—4221no—
203C2×C4—2112no—
204C2×C4—2112no—
205C2×C4D4×D21111yescharacteristic
206C2×C4D4×D21111yescharacteristic
207C2×C4C2×C22⋊D21111yescharacteristic
208C2×C4#1680996111yescharacteristic
209C2×C4#1680996211yescharacteristic
210C2×C4#1680996211yescharacteristic
211C8—41no—
212C8—8441no—
213C22#2026994511yescharacteristic
214C22#2026995011yescharacteristic
215C22#2026996311yescharacteristic
216C22SD16×D21111yescharacteristic
217C22Q16×D21111yescharacteristic
218C22#2027000611yescharacteristic
219C22#2027000911yescharacteristic
220C4—42no—
221C4—81no—
222C4—8442no—
223C4—8442no—
224C4—4222no—
225C4—4222no—
226C4#2026994911yescharacteristic
227C4#2026995011yescharacteristic
228C4#2026995911yescharacteristic
229C4#2026996011yescharacteristic
230C2#2721306111yescharacteristic
231C2#2721328311yescharacteristic
232C2#2721330011yescharacteristic
233C2#2721335011yescharacteristic
234C2#2721335511yescharacteristic
235C2#2721349511yescharacteristic
236C2#2721351311yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC2×#103D211
hypocenterC211#2043
LCSG ▷ C22×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
118. C211#2043semidirect

Maximal subgroups

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

subgroupindex
C2×#202699752
C2×#202699022
C2×#202699012
C2×#202699002
#272130422
#272129002
C2×#103×C2112
#2043211