Finite Group Databasebeta

← #41014410

Subgroups — #41014410

Order 27008.

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

normal subgroups only

This group has 478 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#27213710C211yesmaximal, characteristic, complemented
2C2×#20269973C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213151C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C22⋊D211—22no—
10C2×#16809946C2211yescharacteristic
11#20269973—22no—
12#20269973—22no—
13#20269973—21no—
14#20269966—21no—
15#20269951—22no—
16#20269945—22no—
17#20269936C2211yescharacteristic
18C2×C4×Dic211C2211yescharacteristic
19#20269904—22no—
20#20269898—22no—
21#20269893—22no—
22C2×#41×C211C2211yescharacteristic
23C22×D4×C211C2211yescharacteristic
24AGL(1,ℤ/8)×C211—22no—
25AGL(1,ℤ/8)×C211—22no—
26C2×SD16×C211—21no—
27C2×D8×C211—21no—
28C2×M16×C211C2211yescharacteristic
29C23×D211D412yes—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×C22⋊D211—28no—
37C2×C22⋊D211—24no—
38C2×C22⋊D211—48no—
39C22×Dic211—21no—
40#16809946—21no—
41#16809946—21no—
42C2×C211⋊C8—21no—
43C4⋊Dic211D412yescomplemented
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—24no—
46C4×Dic211—21no—
47C4×Dic211—21no—
48C4×Dic211D412yescomplemented
49C24×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52#41×C211—21no—
53#41×C211—21no—
54C2×Q8×C211D411yescharacteristic, complemented
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211D411yescharacteristic, complemented
62C2×D4×C211D412yescomplemented
63C22×C4×C211—21no—
64C22×C4×C211C2311yescharacteristic
65SD16×C211—24no—
66D8×C211—24no—
67M16×C211—21no—
68M16×C211D412yescomplemented
69C2×C8×C211—21no—
70C22×D211—22no—
71C22×D211—22no—
72C22×D211—22no—
73C22×D211—44no—
74D844—41no—
75D844—41no—
76D844—41no—
77D844—41no—
78C22⋊D211—416no—
79C22⋊D211—84no—
80C2×Dic211—24no—
81C2×Dic211—22no—
82C2×Dic211—22no—
83C211⋊C8—41no—
84C23×C211—21no—
85C23×C211—22no—
86C23×C211—44no—
87C23×C211—41no—
88C23×C211—41no—
89C23×C211C2×D411yescharacteristic
90Q8×C211—21no—
91Q8×C211—21no—
92D4×C211—21no—
93D4×C211—22no—
94D4×C211—22no—
95D4×C211—21no—
96D4×C211—41no—
97D4×C211—41no—
98D4×C211—41no—
99D4×C211—41no—
100C2×C4×C211—21no—
101C2×C4×C211—41no—
102C2×C4×C211—41no—
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211#4111yescharacteristic
109C8×C211—22no—
110C22×C211—22no—
111C22×C211—44no—
112C22×C211—41no—
113C22×C211—42no—
114C22×C211—42no—
115C22×C211—82no—
116C22×C211—82no—
117C22×C211#12011yescharacteristic
118C22×C211#12011yescharacteristic
119C22×C211#12111yescharacteristic
120C2×D211—44no—
121C2×D211—42no—
122C2×D211—44no—
123C2×D211—82no—
124C4×C211—41no—
125C4×C211#12011yescharacteristic
126C4×C211#12111yescharacteristic
127C4×C211#12111yescharacteristic
128C4×C211#12711yescharacteristic
129Dic211—24no—
130Dic211—42no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211—81no—
134C2×C211—82no—
135C2×C211#39211yescharacteristic
136C2×C211#45312yes—
137D211—82no—
138C211#200211yescharacteristic, complemented, Sylow
139#2002—2111nomaximal, Sylow
140C2×AGL(1,ℤ/8)D21111yescharacteristic, complemented
141#535—2111no—
142C2×#127—2111no—
143C2×#104—2111no—
144#418—2111no—
145C2×#100—2111no—
146#339—2111no—
147C2×#41C2×D21111yescharacteristic
148C22×D4—4222no—
149C22×D4—2111no—
150C22×D4C2×D21111yescharacteristic
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)—22no—
153C2×SD16—21no—
154C2×D8—21no—
155C2×M16—2111no—
156C2×M16C2×D21111yescharacteristic
157#127—4222no—
158#127—4221no—
159#127—4222no—
160#124—4221no—
161#121—4222no—
162#120—4222no—
163#117—2111no—
164C2×C42—2111no—
165#104—4222no—
166#102—4222no—
167#100—4222no—
168C24—21no—
169C24—2112no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8C22⋊D21111yescharacteristic, complemented
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—8448no—
182C2×D4—4228no—
183C2×D4—4224no—
184C2×D4—4221no—
185C2×D4—4221no—
186C2×D4—4221no—
187C2×D4—4221no—
188C2×D4—4221no—
189C2×D4—4221no—
190C2×D4C22⋊D21111yescharacteristic, complemented
191C2×D4C22⋊D21112yescomplemented
192C22×C4—21no—
193C22×C4—4221no—
194C22×C4C22×D21111yescharacteristic
195SD16—24no—
196D8—24no—
197M16—21no—
198M16—4221no—
199M16—4221no—
200M16C22⋊D21112yescomplemented
201C2×C8—21no—
202C2×C8—4221no—
203C4⋊C4—2112no—
204C22⋊C4—4224no—
205C22⋊C4—4221no—
206C42—4221no—
207C42—4221no—
208C42—2112no—
209C23—22no—
210C23—21no—
211C23—44no—
212C23—41no—
213C23—41no—
214C23—8444no—
215C23—4222no—
216C23—4222no—
217C23—4222no—
218C23D4×D21111yescharacteristic
219Q8—21no—
220Q8—21no—
221D4—21no—
222D4—22no—
223D4—22no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—84416no—
230D4—8441no—
231D4—8441no—
232D4—8441no—
233D4—8441no—
234D4—16884no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—4222no—
239C2×C4—4224no—
240C2×C4—4222no—
241C2×C4D4×D21111yescharacteristic
242C2×C4D4×D21111yescharacteristic
243C2×C4C2×C22⋊D21111yescharacteristic
244C2×C4C2×C22⋊D21111yescharacteristic
245C2×C4C2×C22⋊D21111yescharacteristic
246C2×C4#1680996111yescharacteristic
247C8—22no—
248C8—8441no—
249C22—22no—
250C22—44no—
251C22—42no—
252C22—41no—
253C22—42no—
254C22—82no—
255C22—82no—
256C22—8444no—
257C22—8444no—
258C22—8442no—
259C22—16882no—
260C22#2026994511yescharacteristic
261C22#2026994511yescharacteristic
262C22#2026994811yescharacteristic
263C4—41no—
264C4—8442no—
265C4—4224no—
266C4#2026994611yescharacteristic
267C4#2026994911yescharacteristic
268C4#2026994911yescharacteristic
269C4#2026997311yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—81no—
274C2—16882no—
275C2#2721305411yescharacteristic
276C2#2721331212yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×AGL(1,ℤ/8)×C211C2
socleC22×C211#120
hypercenterC2×AGL(1,ℤ/8)D211
hypocenterC211#2002
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27213710C2semidirect
2. C2×#20269973C2semidirect
3. C2×#20269904C2semidirect
4. #27213151C2semidirect
5. C2×#20269893C2semidirect
6. #27212907C2semidirect
7. C2×AGL(1,ℤ/8)×C211C2semidirect
43. C4⋊Dic211D4semidirect
48. C4×Dic211D4semidirect
54. C2×Q8×C211D4semidirect
61. C2×D4×C211D4semidirect
62. C2×D4×C211D4semidirect
68. M16×C211D4semidirect
138. C211#2002semidirect
140. C2×AGL(1,ℤ/8)D211semidirect
174. C2×Q8C22⋊D211semidirect
190. C2×D4C22⋊D211semidirect
191. C2×D4C22⋊D211semidirect
200. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272137102
C2×#202699732
C2×#202699042
#272131512
C2×#202698932
#272129072
C2×AGL(1,ℤ/8)×C2112
#2002211