Finite Group Databasebeta

← #41013901

Subgroups — #41013901

Order 27008.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#41×Dic211C211yesmaximal, characteristic, complemented
2#27213163C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213120C211yesmaximal, characteristic, complemented
5#27212918C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7#575×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9Q8×Dic211—21no—
10D4×Dic211—22no—
11D4×Dic211—21no—
12#20269936—21no—
13#20269936C2211yescharacteristic
14#20269936C2211yescharacteristic
15C2×C4×Dic211—21no—
16C2×C4×Dic211C2211yescharacteristic
17#20269904—21no—
18#20269904—21no—
19#20269904—22no—
20#20269892—22no—
21C2×#41×C211C2211yescharacteristic
22C2×#41×C211C2211yescharacteristic
23#137×C211—21no—
24#137×C211C412yescomplemented
25AGL(1,ℤ/8)×C211—21no—
26AGL(1,ℤ/8)×C211C412yescomplemented
27#135×C211—22no—
28C2×M16×C211C2211yescharacteristic
29C22×Dic211—22no—
30C22×Dic211—21no—
31#16809946—21no—
32#16809946—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—22no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C22⋊Dic211—22no—
38C22⋊Dic211—21no—
39C22⋊Dic211D412yes—
40C4×Dic211—22no—
41C4×Dic211—21no—
42C4×Dic211—21no—
43C4×Dic211—21no—
44C4×Dic211D411yescharacteristic
45C4×Dic211D411yescharacteristic
46#41×C211—21no—
47#41×C211—21no—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52#41×C211C2×C412yes—
53#41×C211D412yes—
54C2×Q8×C211C2×C411yescharacteristic, complemented
55C2×Q8×C211D411yescharacteristic
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211C2×C411yescharacteristic, complemented
59C2×D4×C211D411yescharacteristic
60C22×C4×C211—21no—
61C22×C4×C211—21no—
62C22×C4×C211C2311yescharacteristic
63Q16×C211—22no—
64SD16×C211—22no—
65SD16×C211—22no—
66D8×C211—22no—
67M16×C211—21no—
68M16×C211C2×C412yescomplemented
69C2×C8×C211—21no—
70C2×Dic211—21no—
71C2×Dic211—22no—
72C2×Dic211—21no—
73C2×Dic211—42no—
74C2×Dic211—44no—
75C2×Dic211—41no—
76C211⋊C8—41no—
77C23×C211—21no—
78C23×C211—21no—
79C23×C211C2×D411yescharacteristic
80Q8×C211—21no—
81Q8×C211—21no—
82Q8×C211—21no—
83Q8×C211C22⋊C412yes—
84D4×C211—22no—
85D4×C211—22no—
86D4×C211—21no—
87D4×C211—21no—
88D4×C211—21no—
89D4×C211—41no—
90D4×C211—41no—
91D4×C211C22⋊C412yes—
92C2×C4×C211—21no—
93C2×C4×C211—22no—
94C2×C4×C211—22no—
95C2×C4×C211—21no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211C22×C411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C2×C4×C211#4111yescharacteristic
104C8×C211—22no—
105C22×C211—22no—
106C22×C211—41no—
107C22×C211—42no—
108C22×C211—41no—
109C22×C211—81no—
110C22×C211C4×D411yescharacteristic
111C22×C211#12011yescharacteristic
112C22×C211#12111yescharacteristic
113C4×C211—22no—
114C4×C211—41no—
115C4×C211C2×C22⋊C411yescharacteristic
116C4×C211C4×D411yescharacteristic
117C4×C211#12111yescharacteristic
118C4×C211#12311yescharacteristic
119Dic211—22no—
120Dic211—42no—
121Dic211—41no—
122Dic211—81no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211—42no—
127C2×C211—81no—
128C2×C211#38611yescharacteristic
129C211#190611yescharacteristic, complemented, Sylow
130#1906—2111nomaximal, Sylow
131#575D21111yescharacteristic
132C4×#41—2111no—
133#421—2111no—
134C2×#104—2111no—
135#410—2111no—
136#344—2111no—
137#339—2111no—
138C2×#41C2×D21111yescharacteristic
139C2×#41C2×D21111yescharacteristic
140#137—21no—
141#137Dic21112yescomplemented
142AGL(1,ℤ/8)—21no—
143AGL(1,ℤ/8)Dic21112yescomplemented
144#135—22no—
145C2×M16—2111no—
146C2×M16C2×D21111yescharacteristic
147C4×Q8—4221no—
148C4×D4—4221no—
149C4×D4—4222no—
150#117—4221no—
151#117—2111no—
152#117—2111no—
153C2×C42—4221no—
154C2×C42—2111no—
155#104—4222no—
156#104—4221no—
157#104—4221no—
158#99—4222no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41C2×Dic21112yes—
166#41C22⋊D21112yes—
167C2×Q8C2×Dic21111yescharacteristic, complemented
168C2×Q8C22⋊D21111yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C2×Dic21111yescharacteristic, complemented
172C2×D4C22⋊D21111yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—4222no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178Q16—22no—
179SD16—22no—
180SD16—22no—
181D8—22no—
182M16—21no—
183M16—4221no—
184M16—4221no—
185M16C2×Dic21112yescomplemented
186C2×C8—21no—
187C2×C8—4221no—
188C4⋊C4—4222no—
189C4⋊C4—4221no—
190C4⋊C4—4221no—
191C22⋊C4—4221no—
192C22⋊C4—4222no—
193C22⋊C4—2112no—
194C42—4222no—
195C42—4221no—
196C42—4221no—
197C42—4221no—
198C42—2111no—
199C42—2111no—
200C23—21no—
201C23—21no—
202C23D4×D21111yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—21no—
206Q8C22⋊Dic21112yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—21no—
211D4—21no—
212D4—41no—
213D4—41no—
214D4C22⋊Dic21112yes—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—8442no—
222C2×C4—8444no—
223C2×C4—8441no—
224C2×C4—4222no—
225C2×C4—4221no—
226C2×C4—4221no—
227C2×C4C22×Dic21111yescharacteristic
228C2×C4D4×D21111yescharacteristic
229C2×C4C2×C22⋊D21111yescharacteristic
230C2×C4C2×C22⋊D21111yescharacteristic
231C2×C4#1680996111yescharacteristic
232C2×C4#1680996111yescharacteristic
233C8—22no—
234C8—8441no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—81no—
240C22D4×Dic21111yescharacteristic
241C22#2026994511yescharacteristic
242C22#2026994911yescharacteristic
243C4—22no—
244C4—41no—
245C4—8442no—
246C4—8441no—
247C4—16881no—
248C4—4222no—
249C4C2×C22⋊Dic21111yescharacteristic
250C4D4×Dic21111yescharacteristic
251C4#2026994911yescharacteristic
252C4#2026995811yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—81no—
258C2#2721302611yescharacteristic
subgroupG/H
centerC4#20269949
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radical#575×C211C2
socleC2×C211#386
cosocleC22×C4×C211C23
hypercenter#575D211
hypocenterC211#1906
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #41×Dic211C2semidirect
2. #27213163C2semidirect
3. C2×#20269904C2semidirect
4. #27213120C2semidirect
5. #27212918C2semidirect
6. #27212907C2semidirect
24. #137×C211C4semidirect
26. AGL(1,ℤ/8)×C211C4semidirect
54. C2×Q8×C211C2×C4semidirect
58. C2×D4×C211C2×C4semidirect
68. M16×C211C2×C4semidirect
129. C211#1906semidirect
141. #137Dic211semidirect
143. AGL(1,ℤ/8)Dic211semidirect
167. C2×Q8C2×Dic211semidirect
171. C2×D4C2×Dic211semidirect
185. M16C2×Dic211semidirect

Maximal subgroups

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

subgroupindex
#41×Dic2112
#272131632
C2×#202699042
#272131202
#272129182
#272129072
#575×C2112
#1906211