Finite Group Databasebeta

← #41013659

Subgroups — #41013659

Order 27008.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20269908C211yesmaximal, characteristic, complemented
2C2×#20269898C211yesmaximal, characteristic
3C2×#20269898C211yesmaximal, characteristic
4#27213044C211yesmaximal, characteristic, complemented
5#27212908C211yesmaximal, characteristic, complemented
6#27212901C211yesmaximal, characteristic, complemented
7C2×C4×D4×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×Dic211⋊C4—21no—
13#20269908—22no—
14#20269898—22no—
15#20269898C414yescomplemented
16C211⋊#95—21no—
17C22×D4×C211C2211yescharacteristic
18C23×C4×C211—21no—
19C4×D4×C211—22no—
20C4×D4×C211—22no—
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C22⋊C4×C211—21no—
23C2×C42×C211C2211yescharacteristic
24C22×Dic211—21no—
25C22×Dic211—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8C2×C412yescomplemented
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211C2×C412yescomplemented
36Dic211⋊C4—42no—
37C24×C211—21no—
38C2×D4×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211C2×C412yes—
42C2×D4×C211D412yes—
43C2×D4×C211Q812yes—
44C22×C4×C211—24no—
45C22×C4×C211—21no—
46C22×C4×C211—42no—
47C22×C4×C211—42no—
48C22×C4×C211D411yescharacteristic
49C22×C4×C211D411yescharacteristic
50C22×C4×C211C2311yescharacteristic
51C4⋊C4×C211—21no—
52C4⋊C4×C211D412yes—
53C22⋊C4×C211—24no—
54C42×C211—21no—
55C42×C211—21no—
56C2×Dic211—22no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C211⋊C8—22no—
63C211⋊C8—41no—
64C23×C211—22no—
65C23×C211—22no—
66C23×C211—22no—
67C23×C211—41no—
68C23×C211—41no—
69C23×C211—41no—
70C23×C211—41no—
71C23×C211C2×D411yescharacteristic
72D4×C211—22no—
73D4×C211—22no—
74D4×C211—22no—
75D4×C211C4⋊C414yescomplemented
76C2×C4×C211—21no—
77C2×C4×C211—21no—
78C2×C4×C211—21no—
79C2×C4×C211—22no—
80C2×C4×C211—21no—
81C2×C4×C211—21no—
82C2×C4×C211—48no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211SD1612yes—
86C2×C4×C211C22×C411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×Q811yescharacteristic
90C2×C4×C211#4111yescharacteristic
91C2×C4×C211#4111yescharacteristic
92C22×C211—24no—
93C22×C211—42no—
94C22×C211—42no—
95C22×C211—42no—
96C22×C211—42no—
97C22×C211—42no—
98C22×C211—42no—
99C22×C211C4×D411yescharacteristic
100C22×C211#12011yescharacteristic
101C22×C211C22⋊Q811yescharacteristic
102C22×C211C2×SD1611yescharacteristic
103C22×C211#13511yescharacteristic
104C22×C211AGL(1,ℤ/8)11yescharacteristic
105C22×C211AGL(1,ℤ/8)11yescharacteristic
106C4×C211—22no—
107C4×C211—41no—
108C4×C211—42no—
109C4×C211C2×C4⋊C411yescharacteristic
110C4×C211C4×D411yescharacteristic
111C4×C211C22⋊Q811yescharacteristic
112C4×C211#12311yescharacteristic
113Dic211—42no—
114Dic211—81no—
115C2×C211—44no—
116C2×C211#38511yescharacteristic
117C2×C211C4×SD1611yescharacteristic
118C2×C211#44211yescharacteristic
119C2×C211#44911yescharacteristic
120C2×C211#45011yescharacteristic
121C2×C211#47611yescharacteristic
122C2×C211#47811yescharacteristic
123C211#186411yescharacteristic, complemented, Sylow
124#1864—2111nomaximal, Sylow
125C2×C4×D4D21111yescharacteristic
126C2×#106—2111no—
127C2×#102—2111no—
128C2×#102—2111no—
129#389—2111no—
130#340—2111no—
131#336—2111no—
132C22×D4C2×D21111yescharacteristic
133C23×C4—21no—
134C22×C8—2111no—
135C22×C8—2111no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C4⋊C4—4221no—
139C2×C4⋊C4—2111no—
140C2×C4⋊C4—2111no—
141C2×C4⋊C4C2×D21111yescharacteristic
142C2×C22⋊C4—21no—
143C2×C42C2×D21111yescharacteristic
144#106—4222no—
145#102—4222no—
146#102—2114no—
147#95—4221no—
148C24—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4C4×D21112yes—
153C2×D4C22⋊D21112yes—
154C2×D4Dic42212yes—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—4221no—
160C22×C4—4221no—
161C22×C4C22⋊D21111yescharacteristic
162C22×C4C22⋊D21111yescharacteristic
163C22×C4C22×D21111yescharacteristic
164C2×C8—4221no—
165C2×C8—4221no—
166C2×C8—4221no—
167C2×C8—4221no—
168C2×C8—4221no—
169C2×C8—2112no—
170C4⋊C4—21no—
171C4⋊C4—8442no—
172C4⋊C4—4221no—
173C4⋊C4—4221no—
174C4⋊C4—4221no—
175C4⋊C4—2112no—
176C4⋊C4C22⋊D21112yes—
177C22⋊C4—24no—
178C42—21no—
179C42—21no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23C2×C22⋊D21111yescharacteristic
188D4—22no—
189D4—22no—
190D4—22no—
191D4Dic211⋊C414yescomplemented
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—48no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—8441no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—8441no—
206C2×C4—4222no—
207C2×C4#1680995012yes—
208C2×C4C2×C4×D21111yescharacteristic
209C2×C4C2×C22⋊D21111yescharacteristic
210C2×C4C2×C22⋊D21111yescharacteristic
211C2×C4C2×Dic42211yescharacteristic
212C2×C4#1680996211yescharacteristic
213C2×C4#1680996211yescharacteristic
214C8—8441no—
215C8—4222no—
216C22—24no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22C4×C22⋊D21111yescharacteristic
224C22#2026994611yescharacteristic
225C22#2026995611yescharacteristic
226C22C2×#1680995011yescharacteristic
227C22#2026999711yescharacteristic
228C22#2027000211yescharacteristic
229C22#2027000311yescharacteristic
230C4—22no—
231C4—41no—
232C4—42no—
233C4—8442no—
234C4—16881no—
235C4C2×Dic211⋊C411yescharacteristic
236C4C4×C22⋊D21111yescharacteristic
237C4#2026995611yescharacteristic
238C4#2026996011yescharacteristic
239C2—44no—
240C2#2721301911yescharacteristic
241C2C4×#1680995011yescharacteristic
242C2#2721325611yescharacteristic
243C2#2721329111yescharacteristic
244C2#2721329711yescharacteristic
245C2#2721343611yescharacteristic
246C2#2721344611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×C4×D4×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×C4×D4D211
hypocenterC211#1864
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20269908C2semidirect
4. #27213044C2semidirect
5. #27212908C2semidirect
6. #27212901C2semidirect
15. #20269898C4semidirect
31. C2×C211⋊C8C2×C4semidirect
35. C4⋊Dic211C2×C4semidirect
75. D4×C211C4⋊C4semidirect
123. C211#1864semidirect
191. D4Dic211⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#202699082
C2×#202698982
#272130442
#272129082
#272129012
C2×C4×D4×C2112
#1864211