Finite Group Databasebeta

← #41013407

Subgroups — #41013407

Order 27008.

356 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20269939C211yesmaximal, characteristic
2C2×Dic211⋊C8C211yesmaximal, characteristic, complemented
3C2×#20269899C211yesmaximal, characteristic
4C2×#20269897C211yesmaximal, characteristic
5#27213011C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C4×D211—21no—
10C22×C211⋊C8C2211yescharacteristic
11#20269939—22no—
12#20269939C414yescomplemented
13C2×C4⋊Dic211C2211yescharacteristic
14C2×(C2×D211)⋊C4—21no—
15C2×C4×Dic211C2211yescharacteristic
16Dic211⋊C8—22no—
17#20269899—22no—
18#20269897—22no—
19C211⋊#95—21no—
20C211⋊#95—21no—
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C23×D211—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844C2×C412yes—
29C2×D844D412yes—
30C2×D844Q812yes—
31C22×Dic211—21no—
32C22×Dic211D411yescharacteristic
33C22×Dic211D411yescharacteristic
34C2×C4×D211—24no—
35C2×C4×D211—42no—
36C2×C4×D211—42no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42(C2×D211)⋊C4—24no—
43C4×Dic211C2×C412yes—
44C4×Dic211D412yes—
45C22×C4×C211—21no—
46C22×C4×C211—21no—
47C22×C4×C211C2311yescharacteristic
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211C2×C412yes—
55C22×D211—22no—
56C22×D211—22no—
57C22×D211—22no—
58C22×D211—41no—
59C22×D211—41no—
60C22×D211—41no—
61C22×D211—41no—
62D844—22no—
63D844—22no—
64D844—22no—
65D844C4⋊C414yes—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211C22⋊C412yes—
74C2×Dic211C22⋊C412yes—
75C4×D211—48no—
76C211⋊C8—41no—
77C23×C211C2×D411yescharacteristic
78C2×C4×C211—22no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211—41no—
84C2×C4×C211C22×C411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×Q811yescharacteristic
88C2×C4×C211#4111yescharacteristic
89C2×C4×C211#4111yescharacteristic
90C8×C211—41no—
91C22×C211C2×C22⋊C411yescharacteristic
92C22×C211#12111yescharacteristic
93C22×C211C22⋊Q811yescharacteristic
94C22×C211#13511yescharacteristic
95C22×C211#13511yescharacteristic
96C22×C211AGL(1,ℤ/8)11yescharacteristic
97C22×C211AGL(1,ℤ/8)11yescharacteristic
98C2×D211—24no—
99C2×D211—42no—
100C2×D211—42no—
101C2×D211—42no—
102C2×D211—42no—
103C2×D211—42no—
104C2×D211—42no—
105C4×C211—42no—
106C4×C211—81no—
107C4×C211C2×C4⋊C411yescharacteristic
108C4×C211#11711yescharacteristic
109C4×C211#12111yescharacteristic
110C4×C211C22⋊Q811yescharacteristic
111Dic211—22no—
112Dic211—22no—
113Dic211—81no—
114C2×C211#38011yescharacteristic
115C2×C211#41611yescharacteristic
116C2×C211#41811yescharacteristic
117C2×C211#46311yescharacteristic
118C2×C211#46311yescharacteristic
119C2×C211#47811yescharacteristic
120C2×C211#47811yescharacteristic
121D211—44no—
122C211#180411yescharacteristic, complemented, Sylow
123#1804—2111nomaximal, Sylow
124C2×C4×D4—2111no—
125C2×C4⋊C8—2111no—
126C2×#102—2111no—
127C2×#102—2111no—
128#382—2111no—
129#340—2111no—
130#340D21111yescharacteristic, complemented
131C22×D4—2111no—
132C23×C4—4221no—
133C22×C8—2111no—
134C22×C8C2×D21111yescharacteristic
135C4×D4—4222no—
136C4×D4—2114no—
137C2×C4⋊C4—2111no—
138C2×C4⋊C4C2×D21111yescharacteristic
139C2×C4⋊C4C2×D21111yescharacteristic
140C2×C22⋊C4—4221no—
141C2×C42—2111no—
142C4⋊C8—4222no—
143#102—4222no—
144#102—4222no—
145#95—4221no—
146#95—4221no—
147C24—4221no—
148C2×D4—4221no—
149C2×D4—4221no—
150C2×D4—4221no—
151C2×D4—2112no—
152C2×D4—2112no—
153C2×D4—2112no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—8442no—
157C22×C4—8442no—
158C22×C4—4224no—
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—4221no—
173C4⋊C4—4221no—
174C4⋊C4C4×D21112yes—
175C22⋊C4—4224no—
176C42—2112no—
177C42—2112no—
178C23—8441no—
179C23—8441no—
180C23—8441no—
181C23—8441no—
182C23—4222no—
183C23—4222no—
184C23—4222no—
185C23D4×D21111yescharacteristic
186D4—4222no—
187D4—4222no—
188D4—4222no—
189D4—2114no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—8448no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—8441no—
200C2×C4—4221no—
201C2×C4—4221no—
202C2×C4—4221no—
203C2×C4—4221no—
204C2×C4—2112no—
205C2×C4—2112no—
206C2×C4C2×C4×D21111yescharacteristic
207C2×C4D4×D21111yescharacteristic
208C2×C4D4×D21111yescharacteristic
209C2×C4Q8×D21111yescharacteristic
210C2×C4#1680996211yescharacteristic
211C2×C4#1680996211yescharacteristic
212C8—41no—
213C8—8441no—
214C22—8442no—
215C22—8442no—
216C22—8442no—
217C22—8442no—
218C22—8442no—
219C22—8442no—
220C22—4224no—
221C22C22⋊C4×D21111yescharacteristic
222C22#2026995011yescharacteristic
223C22#2026995511yescharacteristic
224C22#2026999811yescharacteristic
225C22#2027000011yescharacteristic
226C22#2027000111yescharacteristic
227C22#2027000411yescharacteristic
228C4—42no—
229C4—81no—
230C4—16881no—
231C4—4222no—
232C4—4222no—
233C4C4⋊C4×D21111yescharacteristic
234C4#2026993411yescharacteristic
235C4#2026995011yescharacteristic
236C4#2026995511yescharacteristic
237C2—8444no—
238C2#2721300511yescharacteristic
239C2#2721314111yescharacteristic
240C2#2721314911yescharacteristic
241C2#2721336011yescharacteristic
242C2#2721336411yescharacteristic
243C2#2721344511yescharacteristic
244C2#2721344911yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
cosocleC22×C4×C211C23
hypercenter#340D211
hypocenterC211#1804
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×Dic211⋊C8C2semidirect
5. #27213011C2semidirect
6. #27212908C2semidirect
7. #340×C211C2semidirect
12. #20269939C4semidirect
122. C211#1804semidirect
130. #340D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699392
C2×Dic211⋊C82
C2×#202698992
C2×#202698972
#272130112
#272129082
#340×C2112
#1804211