Finite Group Databasebeta

← #41013406

Subgroups — #41013406

Order 27008.

356 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 109 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×D4×Dic211C211yesmaximal, characteristic
2C2×Dic211⋊C8C211yesmaximal, characteristic, complemented
3C2×#20269898C211yesmaximal, characteristic
4#27213011C211yesmaximal, characteristic, complemented
5#27212909C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic
8C23×Dic211—21no—
9C22×C211⋊C8C2211yescharacteristic
10D4×Dic211—22no—
11D4×Dic211C414yescomplemented
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211—21no—
15C2×C4×Dic211C2211yescharacteristic
16Dic211⋊C8—22no—
17#20269898—22no—
18C211⋊#95—21no—
19C211⋊#95—21no—
20C22×D4×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#102×C211—22no—
24C22×Dic211—24no—
25C22×Dic211—21no—
26C22×Dic211—21no—
27C22×Dic211—42no—
28C22×Dic211—42no—
29C22×Dic211D411yescharacteristic
30C22×Dic211D411yescharacteristic
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211C2×C412yes—
38C22⋊Dic211—24no—
39C4×Dic211C2×C412yes—
40C4×Dic211D412yes—
41C24×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211C2×C412yes—
46C2×D4×C211D412yes—
47C2×D4×C211Q812yes—
48C22×C4×C211—21no—
49C22×C4×C211C2311yescharacteristic
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C2×C8×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—22no—
60C2×Dic211—48no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211C22⋊C412yes—
67C2×Dic211C22⋊C412yes—
68C211⋊C8—41no—
69C23×C211—22no—
70C23×C211—22no—
71C23×C211—22no—
72C23×C211—41no—
73C23×C211—41no—
74C23×C211—41no—
75C23×C211—41no—
76C23×C211C2×D411yescharacteristic
77D4×C211—22no—
78D4×C211—22no—
79D4×C211—22no—
80D4×C211C4⋊C414yes—
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×C211—24no—
92C22×C211—42no—
93C22×C211—42no—
94C22×C211—42no—
95C22×C211—42no—
96C22×C211—42no—
97C22×C211—42no—
98C22×C211C2×C22⋊C411yescharacteristic
99C22×C211#12111yescharacteristic
100C22×C211C22⋊Q811yescharacteristic
101C22×C211#13511yescharacteristic
102C22×C211#13511yescharacteristic
103C22×C211AGL(1,ℤ/8)11yescharacteristic
104C22×C211AGL(1,ℤ/8)11yescharacteristic
105C4×C211—81no—
106C4×C211C2×C4⋊C411yescharacteristic
107C4×C211#11711yescharacteristic
108C4×C211#12111yescharacteristic
109C4×C211C22⋊Q811yescharacteristic
110Dic211—22no—
111Dic211—22no—
112Dic211—42no—
113Dic211—81no—
114C2×C211—44no—
115C2×C211#38011yescharacteristic
116C2×C211#41611yescharacteristic
117C2×C211#41811yescharacteristic
118C2×C211#46311yescharacteristic
119C2×C211#46311yescharacteristic
120C2×C211#47811yescharacteristic
121C2×C211#47811yescharacteristic
122C211#180411yescharacteristic, complemented, Sylow
123#1804—2111nomaximal, Sylow
124C2×C4×D4—2111no—
125C2×C4⋊C8—2111no—
126C2×#102—2111no—
127C2×#102D21111yescharacteristic
128#382—2111no—
129#340—2111no—
130#340—2111no—
131C22×D4C2×D21111yescharacteristic
132C23×C4—4221no—
133C22×C8—2111no—
134C22×C8C2×D21111yescharacteristic
135C4×D4—4222no—
136C4×D4—2114no—
137C2×C4⋊C4—2111no—
138C2×C4⋊C4—2111no—
139C2×C4⋊C4C2×D21111yescharacteristic
140C2×C22⋊C4—4221no—
141C2×C42—2111no—
142C4⋊C8—4222no—
143#102—22no—
144#102—4222no—
145#95—4221no—
146#95—4221no—
147C24—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4C4×D21112yes—
152C2×D4C22⋊D21112yes—
153C2×D4Dic42212yes—
154C22×C4—21no—
155C22×C4—8442no—
156C22×C4—8442no—
157C22×C4—4221no—
158C22×C4—4221no—
159C22×C4—4224no—
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—4221no—
172C4⋊C4—4221no—
173C4⋊C4—4221no—
174C4⋊C4—2112no—
175C22⋊C4—4224no—
176C42—2112no—
177C42—2112no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D21111yescharacteristic
186D4—22no—
187D4—22no—
188D4—22no—
189D4Dic211⋊C414yes—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—8448no—
196C2×C4—8441no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—4222no—
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×C4C2×C22⋊D21111yescharacteristic
209C2×C4C2×Dic42211yescharacteristic
210C2×C4#1680996111yescharacteristic
211C2×C4#1680996111yescharacteristic
212C8—41no—
213C8—8441no—
214C22—24no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22C22⋊C4×D21111yescharacteristic
222C22#2026994911yescharacteristic
223C22#2026995411yescharacteristic
224C22#2026999611yescharacteristic
225C22#2026999811yescharacteristic
226C22#2027000111yescharacteristic
227C22#2027000411yescharacteristic
228C4—81no—
229C4—8442no—
230C4—16881no—
231C4—4222no—
232C4—4222no—
233C4C2×Dic211⋊C411yescharacteristic
234C4#2026993311yescharacteristic
235C4#2026994911yescharacteristic
236C4#2026995411yescharacteristic
237C2—44no—
238C2#2721300411yescharacteristic
239C2#2721313911yescharacteristic
240C2#2721314911yescharacteristic
241C2#2721335911yescharacteristic
242C2#2721336311yescharacteristic
243C2#2721344411yescharacteristic
244C2#2721344811yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#102D211
hypocenterC211#1804
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×Dic211⋊C8C2semidirect
4. #27213011C2semidirect
5. #27212909C2semidirect
6. #27212908C2semidirect
11. D4×Dic211C4semidirect
122. C211#1804semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic2112
C2×Dic211⋊C82
C2×#202698982
#272130112
#272129092
#272129082
C2×#102×C2112
#1804211