Finite Group Databasebeta

← #41013914

Subgroups — #41013914

Order 27008.

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

normal subgroups only

This group has 312 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×#20270010C211yesmaximal, characteristic
2C2×#20269985C211yesmaximal, characteristic
3#27213145C211yesmaximal, characteristic
4C2×#20269900C211yesmaximal, characteristic, complemented
5C2×#20269895C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7#343×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10#20270010—22no—
11#20270010C414yescomplemented
12C2×Dic844—21no—
13C2×#16809951—21no—
14#20269985—22no—
15#20269985—22no—
16C2×#16809946C2211yescharacteristic
17C2×#16809945—21no—
18C22×C211⋊C8C2211yescharacteristic
19C2×C8×D211—21no—
20#20269900—21no—
21#20269900—21no—
22#20269900—21no—
23#20269897—21no—
24#20269895—22no—
25C2×M16×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28#16809962—21no—
29#16809962—21no—
30#16809962C2×C412yes—
31#16809962D412yes—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422—21no—
36C2×Dic422—21no—
37C2×Dic422C2×C412yes—
38C2×Dic422D411yescharacteristic
39C2×D844D411yescharacteristic
40C2×C22⋊D211—21no—
41C22×Dic211—21no—
42C2×C4×D211—21no—
43Dic844—24no—
44#16809951—24no—
45#16809946—21no—
46#16809946—21no—
47#16809945—24no—
48C2×C211⋊C8—21no—
49C2×C211⋊C8—21no—
50C2×C211⋊C8—21no—
51C2×C211⋊C8D411yescharacteristic
52C2×C211⋊C8D411yescharacteristic
53C8×D211—24no—
54C22×C4×C211—21no—
55C22×C4×C211C2311yescharacteristic
56M16×C211—21no—
57M16×C211—21no—
58M16×C211C2×C412yes—
59M16×C211D412yes—
60C2×C8×C211—21no—
61C2×C8×C211—21no—
62C4⋊C4×C211—21no—
63C4⋊C4×C211—21no—
64C22×D211—21no—
65Dic422—21no—
66Dic422—22no—
67Dic422—22no—
68Dic422—41no—
69Dic422—41no—
70Dic422C22⋊C412yes—
71D844—21no—
72D844C22⋊C412yes—
73C22⋊D211—22no—
74C22⋊D211—22no—
75C2×Dic211—21no—
76C2×Dic211—22no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C4×D211—22no—
80C4×D211—22no—
81C211⋊C8—21no—
82C211⋊C8—21no—
83C211⋊C8—41no—
84C23×C211#4111yescharacteristic
85C2×C4×C211—41no—
86C2×C4×C211—81no—
87C2×C4×C211C22×C411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211#4111yescharacteristic
93C8×C211—22no—
94C8×C211—22no—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211C4×D411yescharacteristic
98C22×C211#12111yescharacteristic
99C22×C211#12311yescharacteristic
100C2×D211—22no—
101C2×D211—41no—
102C2×D211—41no—
103C4×C211—81no—
104C4×C211C2×C22⋊C411yescharacteristic
105C4×C211C4×D411yescharacteristic
106C4×C211#12011yescharacteristic
107C4×C211#12111yescharacteristic
108Dic211—22no—
109Dic211—42no—
110C2×C211—21no—
111C2×C211—21no—
112C2×C211#38611yescharacteristic
113C2×C211#47111yescharacteristic
114C2×C211#47311yescharacteristic
115D211—42no—
116C211#190811yescharacteristic, complemented, Sylow
117#1908—2111nomaximal, Sylow
118C2×#137—2111no—
119C2×#131—2111no—
120#417—2111no—
121C2×#103—2111no—
122C2×#101—2111no—
123#343D21111yescharacteristic, complemented
124#341—2111no—
125C2×#41—2111no—
126C22×Q8—2111no—
127#137—4222no—
128#137—2114no—
129C2×Q16—4221no—
130C2×SD16—4221no—
131#131—4222no—
132#131—4222no—
133C2×M16—4221no—
134C2×M16—2111no—
135C2×M16C2×D21111yescharacteristic
136C2×M16C2×D21111yescharacteristic
137C22×C8—4221no—
138C22×C8—2111no—
139C2×C4⋊C4C2×D21111yescharacteristic
140#103—4221no—
141#103—4221no—
142#103—4221no—
143#102—4221no—
144#101—4222no—
145#41—4221no—
146#41—4221no—
147#41—2112no—
148#41—2112no—
149C2×Q8—4221no—
150C2×Q8—4221no—
151C2×Q8—4221no—
152C2×Q8—4221no—
153C2×Q8—4221no—
154C2×Q8—2112no—
155C2×Q8—2111no—
156C2×D4—4221no—
157C2×D4—2111no—
158C22×C4—21no—
159C22×C4—4221no—
160C22×C4—4221no—
161C22×C4C22×D21111yescharacteristic
162Q16—4224no—
163SD16—4224no—
164M16—21no—
165M16—21no—
166M16—4224no—
167M16—4221no—
168M16—4221no—
169M16C4×D21112yes—
170M16D84412yes—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—4224no—
174C2×C8—4221no—
175C2×C8—4221no—
176C2×C8—4221no—
177C2×C8—2111no—
178C2×C8—2111no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C23—4221no—
182C23#1680996211yescharacteristic
183Q8—8441no—
184Q8—8441no—
185Q8—4222no—
186Q8—4222no—
187Q8—4221no—
188Q8—2112no—
189D4—4222no—
190D4—4222no—
191D4—4221no—
192D4—2112no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—8441no—
196C2×C4—8441no—
197C2×C4—4222no—
198C2×C4—4221no—
199C2×C4—4222no—
200C2×C4—4222no—
201C2×C4C2×C4×D21111yescharacteristic
202C2×C4D4×D21111yescharacteristic
203C2×C4D4×D21111yescharacteristic
204C2×C4D4×D21111yescharacteristic
205C2×C4C2×D84411yescharacteristic
206C2×C4#1680996311yescharacteristic
207C8—22no—
208C8—22no—
209C8—8441no—
210C8—4221no—
211C8—4221no—
212C22—21no—
213C22—21no—
214C22—8441no—
215C22—8441no—
216C22—4222no—
217C22C4×D84411yescharacteristic
218C22#2026995011yescharacteristic
219C22#2026996211yescharacteristic
220C4—81no—
221C4—8442no—
222C4—4222no—
223C4C22⋊C4×D21111yescharacteristic
224C4#2026993911yescharacteristic
225C4#2026994711yescharacteristic
226C4#2026995211yescharacteristic
227C2—21no—
228C2—21no—
229C2—8442no—
230C2#2721302411yescharacteristic
231C2#2721340811yescharacteristic
232C2#2721342011yescharacteristic
subgroupG/H
centerC22#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#121
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#343D211
hypocenterC211#1908
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
4. C2×#20269900C2semidirect
5. C2×#20269895C2semidirect
6. #27212910C2semidirect
7. #343×C211C2semidirect
11. #20270010C4semidirect
116. C211#1908semidirect
123. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700102
C2×#202699852
#272131452
C2×#202699002
C2×#202698952
#272129102
#343×C2112
#1908211