Finite Group Databasebeta

← #41014431

Subgroups — #41014431

Order 27008.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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#27213717C211yesmaximal, characteristic
2C2×#20269975C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic
4#27213155C211yesmaximal, characteristic, complemented
5C2×#20269895C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7C2×#137×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269975—22no—
11#20269975—22no—
12#20269975—21no—
13#20269964—21no—
14#20269958—22no—
15#20269956—22no—
16#20269936C2211yescharacteristic
17C2×Dic211⋊C4—22no—
18C2×C4×Dic211C2211yescharacteristic
19#20269904—22no—
20#20269901—22no—
21#20269895—22no—
22C2×#41×C211C2211yescharacteristic
23C22×Q8×C211C2211yescharacteristic
24#137×C211—22no—
25#137×C211—22no—
26C2×Q16×C211—21no—
27C2×SD16×C211—21no—
28C2×M16×C211C2211yescharacteristic
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C22×Dic211—21no—
36C22×Dic211D412yes—
37#16809946—21no—
38#16809946—21no—
39C2×C211⋊C8—21no—
40C4⋊Dic211D412yes—
41Dic211⋊C4—28no—
42Dic211⋊C4—24no—
43C22⋊Dic211—24no—
44C22⋊Dic211—21no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C4×Dic211D412yes—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211D411yescharacteristic
58C2×Q8×C211D412yes—
59C2×D4×C211—21no—
60C2×D4×C211D411yescharacteristic
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211C2311yescharacteristic
64Q16×C211—24no—
65SD16×C211—24no—
66M16×C211—21no—
67M16×C211D412yes—
68C2×C8×C211—21no—
69Dic422—41no—
70Dic422—41no—
71Dic422—41no—
72Dic422—41no—
73C2×Dic211—24no—
74C2×Dic211—22no—
75C2×Dic211—22no—
76C2×Dic211—22no—
77C2×Dic211—22no—
78C2×Dic211—22no—
79C211⋊C8—41no—
80C23×C211—21no—
81C23×C211C2×D411yescharacteristic
82Q8×C211—22no—
83Q8×C211—21no—
84Q8×C211—22no—
85Q8×C211—21no—
86Q8×C211—41no—
87Q8×C211—41no—
88D4×C211—21no—
89D4×C211—21no—
90D4×C211—41no—
91D4×C211—41no—
92C2×C4×C211—21no—
93C2×C4×C211—22no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211#4111yescharacteristic
104C8×C211—22no—
105C22×C211—22no—
106C22×C211—41no—
107C22×C211—42no—
108C22×C211#12011yescharacteristic
109C22×C211#12011yescharacteristic
110C22×C211#12111yescharacteristic
111C4×C211—41no—
112C4×C211—42no—
113C4×C211#12011yescharacteristic
114C4×C211#12111yescharacteristic
115C4×C211#12111yescharacteristic
116C4×C211#12711yescharacteristic
117Dic211—24no—
118Dic211—42no—
119Dic211—42no—
120C2×C211—21no—
121C2×C211—21no—
122C2×C211—81no—
123C2×C211#39211yescharacteristic
124C2×C211#45612yes—
125C211#200511yescharacteristic, complemented, Sylow
126#2005—2111nomaximal, Sylow
127C2×#137D21111yescharacteristic
128#536—2111no—
129C2×C4⋊Q8—2111no—
130C2×#104—2111no—
131#419—2111no—
132C2×#101—2111no—
133#339—2111no—
134C2×#41C2×D21111yescharacteristic
135C22×Q8—2111no—
136C22×Q8C2×D21111yescharacteristic
137#137—22no—
138#137—22no—
139C2×Q16—21no—
140C2×SD16—21no—
141C2×M16—2111no—
142C2×M16C2×D21111yescharacteristic
143C4⋊Q8—4222no—
144C4⋊Q8—4221no—
145C4⋊Q8—4222no—
146#124—4221no—
147#123—4222no—
148C22⋊Q8—4222no—
149#117—2111no—
150C2×C4⋊C4—4222no—
151C2×C42—2111no—
152#104—4222no—
153#103—4222no—
154#101—4222no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—4221no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—4221no—
169C2×Q8—4221no—
170C2×Q8C22⋊D21111yescharacteristic
171C2×Q8C22⋊D21112yes—
172C2×D4—21no—
173C2×D4C22⋊D21111yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—4221no—
177C22×C4—2112no—
178C22×C4C22×D21111yescharacteristic
179Q16—24no—
180SD16—24no—
181M16—21no—
182M16—4221no—
183M16—4221no—
184M16C22⋊D21112yes—
185C2×C8—21no—
186C2×C8—4221no—
187C4⋊C4—4228no—
188C4⋊C4—4224no—
189C4⋊C4—2112no—
190C22⋊C4—4224no—
191C22⋊C4—4221no—
192C42—4221no—
193C42—4221no—
194C42—2112no—
195C23—21no—
196C23D4×D21111yescharacteristic
197Q8—22no—
198Q8—21no—
199Q8—22no—
200Q8—21no—
201Q8—41no—
202Q8—41no—
203Q8—8441no—
204Q8—8441no—
205Q8—8441no—
206Q8—8441no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—4222no—
218C2×C4—4222no—
219C2×C4—4222no—
220C2×C4—4222no—
221C2×C4—4224no—
222C2×C4—4222no—
223C2×C4D4×D21111yescharacteristic
224C2×C4D4×D21111yescharacteristic
225C2×C4C2×C22⋊D21111yescharacteristic
226C2×C4C2×C22⋊D21111yescharacteristic
227C2×C4C2×C22⋊D21111yescharacteristic
228C2×C4#1680996111yescharacteristic
229C8—22no—
230C8—8441no—
231C22—22no—
232C22—42no—
233C22—41no—
234C22#2026994511yescharacteristic
235C22#2026994511yescharacteristic
236C22#2026994811yescharacteristic
237C4—41no—
238C4—42no—
239C4—8442no—
240C4—8442no—
241C4—4224no—
242C4#2026994611yescharacteristic
243C4#2026994911yescharacteristic
244C4#2026994911yescharacteristic
245C4#2026997311yescharacteristic
246C2—21no—
247C2—21no—
248C2—81no—
249C2#2721305411yescharacteristic
250C2#2721332712yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#137×C211C2
socleC22×C211#120
prime radicalC2×#41×C211C22
hypercenterC2×#137D211
hypocenterC211#2005
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#20269975C2semidirect
4. #27213155C2semidirect
5. C2×#20269895C2semidirect
6. #27212907C2semidirect
125. C211#2005semidirect

Maximal subgroups

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

subgroupindex
#272137172
C2×#202699752
C2×#202699042
#272131552
C2×#202698952
#272129072
C2×#137×C2112
#2005211