Finite Group Databasebeta

← #41014533

Subgroups — #41014533

Order 27008.

326 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20269976C211yesmaximal, characteristic, complemented
2C2×#20269899C211yesmaximal, characteristic
3C2×#20269897C211yesmaximal, characteristic
4#27213047C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×Dic422—21no—
9C22×D844C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11#20269976—22no—
12#20269976—22no—
13C2×C4⋊Dic211C2211yescharacteristic
14C2×Dic211⋊C4—21no—
15C2×(C2×D211)⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17C2×C4×Dic211C2211yescharacteristic
18#20269899—22no—
19#20269897—22no—
20C22×C8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C23×D211—21no—
24C2×Dic422—24no—
25C2×Dic422—42no—
26C2×Dic422—42no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844Q812yescomplemented
33C22×Dic211—21no—
34C22×Dic211D411yescharacteristic
35C22×Dic211D411yescharacteristic
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211D412yes—
41Dic211⋊C4—24no—
42(C2×D211)⋊C4—42no—
43(C2×D211)⋊C4—42no—
44C4×Dic211—21no—
45C4×Dic211—21no—
46C22×C4×C211—21no—
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211—21no—
55C4⋊C4×C211D412yes—
56C22×D211—22no—
57C22×D211—41no—
58C22×D211—41no—
59C22×D211—41no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—41no—
63Dic422—48no—
64D844—22no—
65D844—22no—
66D844—41no—
67D844—41no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—21no—
71C2×Dic211—21no—
72C2×Dic211—22no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211D812yes—
76C2×Dic211SD1612yes—
77C211⋊C8—41no—
78C23×C211C2×D411yescharacteristic
79C2×C4×C211—22no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×Q811yescharacteristic
88C2×C4×C211#4111yescharacteristic
89C2×C4×C211#4111yescharacteristic
90C2×C4×C211#4111yescharacteristic
91C8×C211—41no—
92C22×C211#12011yescharacteristic
93C22×C211C22⋊Q811yescharacteristic
94C22×C211#12411yescharacteristic
95C22×C211C2×D811yescharacteristic
96C22×C211C2×SD1611yescharacteristic
97C22×C211AGL(1,ℤ/8)11yescharacteristic
98C22×C211#13711yescharacteristic
99C2×D211—42no—
100C2×D211—42no—
101C2×D211—42no—
102C2×D211—81no—
103C2×D211—81no—
104C2×D211—81no—
105C2×D211—81no—
106C4×C211—42no—
107C4×C211—81no—
108C4×C211C22⋊Q811yescharacteristic
109C4×C211C22⋊Q811yescharacteristic
110C4×C211#12411yescharacteristic
111C4×C211#12411yescharacteristic
112Dic211—22no—
113Dic211—22no—
114Dic211—42no—
115C2×C211#39311yescharacteristic
116C2×C211#44711yescharacteristic
117C2×C211#44811yescharacteristic
118C2×C211#47411yescharacteristic
119C2×C211#47611yescharacteristic
120C2×C211#48611yescharacteristic
121C2×C211#48911yescharacteristic
122D211—82no—
123C211#202311yescharacteristic, complemented, Sylow
124#2023—2111nomaximal, Sylow
125C2×C4⋊Q8—2111no—
126C2×#102—2111no—
127C2×#102—2111no—
128#390—2111no—
129#340—2111no—
130#340D21111yescharacteristic, complemented
131#336—2111no—
132C22×Q8—4221no—
133C22×D4—2111no—
134C22×C8—2111no—
135C22×C8C2×D21111yescharacteristic
136C4⋊Q8—4222no—
137C4⋊Q8—4222no—
138C2×C4⋊C4—4221no—
139C2×C4⋊C4—2111no—
140C2×C4⋊C4C2×D21111yescharacteristic
141C2×C4⋊C4C2×D21111yescharacteristic
142C2×C22⋊C4—4221no—
143C2×C22⋊C4—4221no—
144C2×C42—2111no—
145#102—4222no—
146#102—4222no—
147C24—4221no—
148C2×Q8—8442no—
149C2×Q8—8442no—
150C2×Q8—4224no—
151C2×D4—4221no—
152C2×D4—4221no—
153C2×D4—4221no—
154C2×D4—4221no—
155C2×D4—4221no—
156C2×D4—2112no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4—2111no—
162C22×C4C22×D21111yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—4221no—
167C2×C8—4221no—
168C2×C8—4221no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—4224no—
173C4⋊C4—4221no—
174C4⋊C4—2112no—
175C4⋊C4D84412yes—
176C22⋊C4—8442no—
177C22⋊C4—8442no—
178C42—4221no—
179C42—4221no—
180C23—8441no—
181C23—8441no—
182C23—8441no—
183C23—8441no—
184C23—8441no—
185C23—8441no—
186C23—4222no—
187C23D4×D21111yescharacteristic
188Q8—8448no—
189D4—8441no—
190D4—8441no—
191D4—4222no—
192D4—4222no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—8441no—
200C2×C4—8441no—
201C2×C4—4222no—
202C2×C4—4221no—
203C2×C4—4221no—
204C2×C4—4221no—
205C2×C4—4221no—
206C2×C4—2112no—
207C2×C4—2112no—
208C2×C4D4×D21111yescharacteristic
209C2×C4C2×D84411yescharacteristic
210C2×C4Q8×D21111yescharacteristic
211C2×C4#1680996111yescharacteristic
212C2×C4#1680996211yescharacteristic
213C2×C4#1680996211yescharacteristic
214C8—41no—
215C8—8441no—
216C22—8442no—
217C22—8442no—
218C22—8442no—
219C22—16881no—
220C22—16881no—
221C22—16881no—
222C22—16881no—
223C22#2026994711yescharacteristic
224C22#2026995511yescharacteristic
225C22#2026996311yescharacteristic
226C22D8×D21111yescharacteristic
227C22SD16×D21111yescharacteristic
228C22#2027000411yescharacteristic
229C22#2027000911yescharacteristic
230C4—42no—
231C4—81no—
232C4—8442no—
233C4—4222no—
234C4—4222no—
235C4#2026995511yescharacteristic
236C4#2026995711yescharacteristic
237C4#2026996311yescharacteristic
238C4#2026996511yescharacteristic
239C2—16882no—
240C2#2721305911yescharacteristic
241C2#2721328111yescharacteristic
242C2#2721328611yescharacteristic
243C2#2721342511yescharacteristic
244C2#2721343511yescharacteristic
245C2#2721349111yescharacteristic
246C2#2721350711yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
hypercenter#340D211
hypocenterC211#2023
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#20269976C2semidirect
5. #27212908C2semidirect
6. #27212900C2semidirect
7. #340×C211C2semidirect
32. C2×D844Q8semidirect
123. C211#2023semidirect
130. #340D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699762
C2×#202698992
C2×#202698972
#272130472
#272129082
#272129002
#340×C2112
#2023211