Finite Group Databasebeta

← #41013897

Subgroups — #41013897

Order 27008.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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#27213944C211yesmaximal, characteristic
2#41×Dic211C211yesmaximal, characteristic, complemented
3C2×#20269903C211yesmaximal, characteristic, complemented
4#27213119C211yesmaximal, characteristic, complemented
5#27212917C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#421×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007C412yescomplemented
11#20270003—21no—
12#20270002C412yescomplemented
13#20269997—22no—
14C2×#16809946C2211yescharacteristic
15Q8×Dic211—21no—
16D4×Dic211—22no—
17D4×Dic211—21no—
18#20269936—21no—
19#20269936C2211yescharacteristic
20C2×C4×Dic211—21no—
21C2×C4×Dic211C2211yescharacteristic
22#20269903—21no—
23#20269903—21no—
24#20269891—22no—
25C2×#41×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27#117×C211C2211yescharacteristic
28#104×C211—22no—
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33C2×Dic422C2×C411yescharacteristic, complemented
34C2×D844C2×C411yescharacteristic, complemented
35C2×C22⋊D211—21no—
36C22×Dic211—22no—
37C22×Dic211—21no—
38C2×C4×D211—21no—
39#16809952—22no—
40#16809950—22no—
41#16809949—22no—
42#16809947—22no—
43#16809946—21no—
44#16809946C2×C412yescomplemented
45C2×C211⋊C8—21no—
46C4⋊Dic211—22no—
47C4⋊Dic211—21no—
48C22⋊Dic211—22no—
49C22⋊Dic211D412yes—
50C4×Dic211—22no—
51C4×Dic211—21no—
52C4×Dic211—21no—
53C4×Dic211D411yescharacteristic
54C4×Dic211D411yescharacteristic
55#41×C211—21no—
56#41×C211—21no—
57#41×C211C2×C412yes—
58#41×C211D412yes—
59C2×Q8×C211D411yescharacteristic
60C2×D4×C211—21no—
61C2×D4×C211D411yescharacteristic
62C22×C4×C211—21no—
63C22×C4×C211C2311yescharacteristic
64M16×C211—21no—
65M16×C211—21no—
66C2×C8×C211—21no—
67C4⋊C4×C211—21no—
68C22⋊C4×C211—21no—
69C42×C211—21no—
70C22×D211—21no—
71Dic422—21no—
72Dic422—21no—
73D844—21no—
74D844—21no—
75C22⋊D211—41no—
76C22⋊D211—41no—
77C2×Dic211—21no—
78C2×Dic211—21no—
79C2×Dic211—22no—
80C2×Dic211—21no—
81C2×Dic211—42no—
82C2×Dic211—44no—
83C4×D211—41no—
84C4×D211—41no—
85C211⋊C8—22no—
86C23×C211—21no—
87C23×C211C2×D411yescharacteristic
88Q8×C211—21no—
89Q8×C211C22⋊C412yes—
90D4×C211—22no—
91D4×C211—22no—
92D4×C211—21no—
93D4×C211C22⋊C412yes—
94C2×C4×C211—21no—
95C2×C4×C211—22no—
96C2×C4×C211—22no—
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—41no—
105C22×C211—22no—
106C22×C211—41no—
107C22×C211—42no—
108C22×C211C4×D411yescharacteristic
109C22×C211#12011yescharacteristic
110C22×C211#12111yescharacteristic
111C2×D211—41no—
112C2×D211—81no—
113C4×C211—22no—
114C4×C211—81no—
115C4×C211C2×C22⋊C411yescharacteristic
116C4×C211C4×D411yescharacteristic
117C4×C211#12111yescharacteristic
118C4×C211#12311yescharacteristic
119Dic211—22no—
120Dic211—42no—
121Dic211—41no—
122Dic211—41no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211—42no—
127C2×C211#38611yescharacteristic
128D211—81no—
129C211#190611yescharacteristic, complemented, Sylow
130#1906—2111nomaximal, Sylow
131#575—2111no—
132C4×#41—2111no—
133#421D21111yescharacteristic, complemented
134C2×#104—2111no—
135#410—2111no—
136#344—2111no—
137#339—2111no—
138C2×#41—2111no—
139C2×#41C2×D21111yescharacteristic
140#137—4221no—
141#137—2112no—
142AGL(1,ℤ/8)—4221no—
143AGL(1,ℤ/8)—2112no—
144#135—4222no—
145C2×M16—2111no—
146C2×M16C2×D21111yescharacteristic
147C4×Q8—4221no—
148C4×D4—4221no—
149C4×D4—4222no—
150#117—4221no—
151#117—2111no—
152#117C2×D21111yescharacteristic
153C2×C42—4221no—
154C2×C42—2111no—
155#104—22no—
156#104—4221no—
157#104—4221no—
158#99—4222no—
159#41—21no—
160#41—21no—
161#41—4221no—
162#41—4221no—
163#41—4221no—
164#41—4221no—
165#41C4×D21112yes—
166#41C22⋊D21112yes—
167C2×Q8—2111no—
168C2×Q8D84411yescharacteristic
169C2×D4—21no—
170C2×D4—4221no—
171C2×D4—2111no—
172C2×D4D84411yescharacteristic
173C22×C4—21no—
174C22×C4—4222no—
175C22×C4—4221no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178Q16—4222no—
179SD16—4222no—
180SD16—4222no—
181D8—4222no—
182M16—21no—
183M16—21no—
184M16—4221no—
185M16—2112no—
186C2×C8—21no—
187C2×C8—4221no—
188C4⋊C4—21no—
189C4⋊C4—4222no—
190C4⋊C4—4221no—
191C22⋊C4—21no—
192C22⋊C4—4222no—
193C22⋊C4—2112no—
194C42—21no—
195C42—4222no—
196C42—4221no—
197C42—4221no—
198C42—2111no—
199C42—2111no—
200C23—21no—
201C23—4221no—
202C23D4×D21111yescharacteristic
203Q8—21no—
204Q8—4221no—
205Q8—4221no—
206Q8(C2×D211)⋊C412yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—8441no—
211D4—8441no—
212D4—4221no—
213D4—4221no—
214D4(C2×D211)⋊C412yes—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—8442no—
220C2×C4—8444no—
221C2×C4—8441no—
222C2×C4—8441no—
223C2×C4—4221no—
224C2×C4—4222no—
225C2×C4—4221no—
226C2×C4—4221no—
227C2×C4C2×C4×D21111yescharacteristic
228C2×C4D4×D21111yescharacteristic
229C2×C4C2×C22⋊D21111yescharacteristic
230C2×C4C2×D84411yescharacteristic
231C2×C4#1680996111yescharacteristic
232C2×C4#1680996111yescharacteristic
233C8—41no—
234C8—4222no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—8441no—
239C22—16881no—
240C22(C22⋊D211)⋊C411yescharacteristic
241C22#2026994711yescharacteristic
242C22#2026994911yescharacteristic
243C4—22no—
244C4—81no—
245C4—8442no—
246C4—8441no—
247C4—8441no—
248C4—4222no—
249C4C2×(C2×D211)⋊C411yescharacteristic
250C4(C22⋊D211)⋊C411yescharacteristic
251C4#2026994911yescharacteristic
252C4#2026996111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—16881no—
258C2#2721302311yescharacteristic
subgroupG/H
centerC4#20269949
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#421×C211C2
socleC2×C211#386
prime radical#27213944C2
cosocleC22×C4×C211C23
hypercenter#421D211
hypocenterC211#1906
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #41×Dic211C2semidirect
3. C2×#20269903C2semidirect
4. #27213119C2semidirect
5. #27212917C2semidirect
6. #27212905C2semidirect
7. #421×C211C2semidirect
10. #20270007C4semidirect
12. #20270002C4semidirect
33. C2×Dic422C2×C4semidirect
34. C2×D844C2×C4semidirect
44. #16809946C2×C4semidirect
129. C211#1906semidirect
133. #421D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#41×Dic2112
C2×#202699032
#272131192
#272129172
#272129052
#421×C2112
#1906211