Finite Group Databasebeta

← #41013728

Subgroups — #41013728

Order 27008.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269997C211yesmaximal, characteristic
2C2×#20269983C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5#27212910C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7#408×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269997—22no—
10#20269997—22no—
11C2×#16809952C411yescharacteristic, complemented
12C2×#16809950C411yescharacteristic, complemented
13C2×#16809949C411yescharacteristic, complemented
14C2×#16809947C411yescharacteristic, complemented
15#20269983—22no—
16#20269983—22no—
17C2×#16809946C2211yescharacteristic
18C2×#16809945—21no—
19C22×C211⋊C8C2211yescharacteristic
20C2×C8×D211—21no—
21C2×C4×Dic211C2211yescharacteristic
22#20269904—22no—
23#20269903—22no—
24C2×#41×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26C22×C8×C211C2211yescharacteristic
27C22⋊C8×C211—21no—
28C22⋊C8×C211—21no—
29#16809962—21no—
30#16809962—21no—
31#16809962D412yes—
32#16809962D412yes—
33C2×Dic422C2×C411yescharacteristic
34C2×D844C2×C411yescharacteristic
35C2×C22⋊D211—21no—
36C22×Dic211—21no—
37C2×C4×D211—21no—
38#16809952—22no—
39#16809950—22no—
40#16809949—22no—
41#16809947—22no—
42#16809946—21no—
43#16809946D412yes—
44#16809945—24no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8—21no—
48C2×C211⋊C8C2×C411yescharacteristic
49C2×C211⋊C8C2×C411yescharacteristic
50C8×D211—24no—
51C4×Dic211—21no—
52C4×Dic211—21no—
53#41×C211—21no—
54#41×C211—21no—
55#41×C211—21no—
56#41×C211—21no—
57C2×Q8×C211C2×C411yescharacteristic
58C2×D4×C211—21no—
59C2×D4×C211C2×C411yescharacteristic
60C22×C4×C211—21no—
61C22×C4×C211C2311yescharacteristic
62M16×C211—21no—
63M16×C211—21no—
64C2×C8×C211—21no—
65C2×C8×C211—21no—
66C2×C8×C211—21no—
67C2×C8×C211D411yescharacteristic
68C2×C8×C211D411yescharacteristic
69C22×D211—21no—
70Dic422—21no—
71Dic422C22⋊C412yes—
72D844—21no—
73D844C22⋊C412yes—
74C22⋊D211—22no—
75C22⋊D211—22no—
76C2×Dic211—21no—
77C2×Dic211—22no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C4×D211—22no—
81C4×D211—22no—
82C211⋊C8—21no—
83C211⋊C8—21no—
84C211⋊C8—22no—
85C23×C211—21no—
86C23×C211#4111yescharacteristic
87Q8×C211—21no—
88Q8×C211—21no—
89D4×C211—21no—
90D4×C211—21no—
91D4×C211—41no—
92D4×C211—41no—
93C2×C4×C211—21no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211C22×C411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211#4111yescharacteristic
102C8×C211—21no—
103C8×C211—21no—
104C8×C211—41no—
105C22×C211—21no—
106C22×C211—21no—
107C22×C211—41no—
108C22×C211—41no—
109C22×C211—41no—
110C22×C211C4×D411yescharacteristic
111C22×C211#12111yescharacteristic
112C22×C211#12311yescharacteristic
113C2×D211—22no—
114C2×D211—41no—
115C2×D211—41no—
116C4×C211—41no—
117C4×C211C2×C22⋊C411yescharacteristic
118C4×C211C4×D411yescharacteristic
119C4×C211#12011yescharacteristic
120C4×C211#12111yescharacteristic
121Dic211—22no—
122Dic211—42no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—81no—
126C2×C211#38611yescharacteristic
127C2×C211#44311yescharacteristic
128C2×C211#44411yescharacteristic
129D211—42no—
130C211#187611yescharacteristic, complemented, Sylow
131#1876—2111nomaximal, Sylow
132C2×#135—2111no—
133C2×#131—2111no—
134C2×#104—2111no—
135C2×#104—2111no—
136#408D21111yescharacteristic, complemented
137#341—2111no—
138#336—2111no—
139C2×#41—2111no—
140C2×#41C2×D21111yescharacteristic
141#135—4222no—
142#135—4222no—
143C2×Q16—2111no—
144C2×SD16—2111no—
145C2×SD16—2111no—
146C2×D8—2111no—
147#131—4222no—
148#131—4222no—
149C2×M16—4221no—
150C2×M16—2111no—
151C2×M16C2×D21111yescharacteristic
152C22×C8—4221no—
153C22×C8—2111no—
154C22×C8C2×D21111yescharacteristic
155C2×C42—2111no—
156#104—4222no—
157#104—4222no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—4221no—
165#41—4221no—
166#41—2112no—
167#41—2112no—
168C2×Q8—2111no—
169C2×Q8C4×D21111yescharacteristic
170C2×D4—21no—
171C2×D4—4221no—
172C2×D4—2111no—
173C2×D4C4×D21111yescharacteristic
174C22×C4—21no—
175C22×C4—4221no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178Q16—4222no—
179SD16—4222no—
180SD16—4222no—
181D8—4222no—
182M16—21no—
183M16—21no—
184M16—4224no—
185M16—4221no—
186M16—2112no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—4224no—
191C2×C8—4221no—
192C2×C8—4221no—
193C2×C8—4221no—
194C2×C8—2111no—
195C2×C8—2111no—
196C2×C8C22⋊D21111yescharacteristic
197C2×C8C22⋊D21111yescharacteristic
198C42—4221no—
199C42—4221no—
200C23—21no—
201C23—4221no—
202C23#1680996211yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—4221no—
206Q8—2112no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—4222no—
212D4—4222no—
213D4—4221no—
214D4—2112no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—8441no—
219C2×C4—8441no—
220C2×C4—4222no—
221C2×C4—4221no—
222C2×C4—4222no—
223C2×C4—4222no—
224C2×C4C2×C4×D21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C2×C4D4×D21111yescharacteristic
227C2×C4D4×D21111yescharacteristic
228C2×C4C2×C22⋊D21111yescharacteristic
229C2×C4#1680996111yescharacteristic
230C8—21no—
231C8—21no—
232C8—41no—
233C8—4221no—
234C8—4221no—
235C8—4222no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—8441no—
242C22—8441no—
243C22—4222no—
244C22C4×C22⋊D21111yescharacteristic
245C22#2026995011yescharacteristic
246C22#2026995911yescharacteristic
247C4—41no—
248C4—8442no—
249C4—4222no—
250C4C22⋊C4×D21111yescharacteristic
251C4(C22⋊D211)⋊C411yescharacteristic
252C4#2026994511yescharacteristic
253C4#2026994811yescharacteristic
254C2—21no—
255C2—21no—
256C2—81no—
257C2—8442no—
258C2#2721302211yescharacteristic
259C2#2721326111yescharacteristic
260C2#2721326611yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#408×C211C2
socleC22×C211C4×D4
prime radicalC2×#20269997C2
cosocleC22×C4×C211C23
hypercenter#408D211
hypocenterC211#1876
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#20269983C2semidirect
3. C2×#20269904C2semidirect
4. C2×#20269903C2semidirect
5. #27212910C2semidirect
6. #27212900C2semidirect
7. #408×C211C2semidirect
11. C2×#16809952C4semidirect
12. C2×#16809950C4semidirect
13. C2×#16809949C4semidirect
14. C2×#16809947C4semidirect
130. C211#1876semidirect
136. #408D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699972
C2×#202699832
C2×#202699042
C2×#202699032
#272129102
#272129002
#408×C2112
#1876211