Finite Group Databasebeta

← #41014591

Subgroups — #41014591

Order 27008.

316 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 91 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×#20269901C211yesmaximal, characteristic, complemented
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
5#27213006C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic, complemented
7C2×C22⋊Q8×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×C22⋊Dic211—21no—
13#20269908—22no—
14#20269901—22no—
15#20269901—22no—
16C22⋊(C211⋊C8)—22no—
17C211⋊#95—21no—
18C22×Q8×C211C2211yescharacteristic
19C23×C4×C211C2211yescharacteristic
20C22⋊Q8×C211—22no—
21C22⋊Q8×C211—22no—
22C2×C4⋊C4×C211—21no—
23C2×C4⋊C4×C211C2211yescharacteristic
24C2×C22⋊C4×C211—21no—
25C22×Dic211—21no—
26C22×Dic211—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8D412yes—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C22⋊Dic211—42no—
38C24×C211D411yescharacteristic
39C2×Q8×C211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211D412yes—
45C22×C4×C211—21no—
46C22×C4×C211—21no—
47C22×C4×C211—21no—
48C22×C4×C211—21no—
49C22×C4×C211—21no—
50C22×C4×C211—21no—
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211D411yescharacteristic
54C22×C4×C211C2311yescharacteristic
55C4⋊C4×C211—24no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211D412yes—
58C22⋊C4×C211—24no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C211⋊C8—22no—
66C211⋊C8—41no—
67C23×C211—21no—
68C23×C211—21no—
69C23×C211—21no—
70C23×C211—21no—
71C23×C211—21no—
72C23×C211—21no—
73C23×C211SD1612yes—
74C23×C211C2×D411yescharacteristic
75Q8×C211—22no—
76Q8×C211—22no—
77Q8×C211—41no—
78Q8×C211—41no—
79C2×C4×C211—22no—
80C2×C4×C211—22no—
81C2×C4×C211—21no—
82C2×C4×C211—21no—
83C2×C4×C211—21no—
84C2×C4×C211—41no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211#4111yescharacteristic
96C2×C4×C211#4111yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C22×C211—22no—
99C22×C211—22no—
100C22×C211—22no—
101C22×C211—41no—
102C22×C211—41no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211#12011yescharacteristic
106C22×C211#12111yescharacteristic
107C22×C211#12311yescharacteristic
108C22×C211C2×SD1611yescharacteristic
109C22×C211#13511yescharacteristic
110C22×C211#13711yescharacteristic
111C22×C211#13711yescharacteristic
112C4×C211—41no—
113C4×C211—42no—
114C4×C211—42no—
115C4×C211#12111yescharacteristic
116C4×C211#12111yescharacteristic
117C4×C211#12311yescharacteristic
118C4×C211#12411yescharacteristic
119Dic211—81no—
120Dic211—81no—
121C2×C211—42no—
122C2×C211#39411yescharacteristic
123C2×C211#44811yescharacteristic
124C2×C211#45211yescharacteristic
125C2×C211#46511yescharacteristic
126C2×C211#47011yescharacteristic
127C2×C211#48311yescharacteristic
128C2×C211#48511yescharacteristic
129C211#203511yescharacteristic, complemented, Sylow
130#2035—2111nomaximal, Sylow
131C2×C22⋊Q8D21111yescharacteristic
132C2×#106—2111no—
133C2×#103—2111no—
134C2×#103—2111no—
135C2×C22⋊C8—2111no—
136#380—2111no—
137#340—2111no—
138C22×Q8C2×D21111yescharacteristic
139C23×C4C2×D21111yescharacteristic
140C22×C8—2111no—
141C22×C8—2111no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—2111no—
146C2×C4⋊C4—2111no—
147C2×C4⋊C4C2×D21111yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—4221no—
150#106—4222no—
151#103—4222no—
152#103—4222no—
153C22⋊C8—4222no—
154#95—4221no—
155C24C22⋊D21111yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D21112yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—4221no—
171C22×C4—4221no—
172C22×C4C22⋊D21111yescharacteristic
173C22×C4C22×D21111yescharacteristic
174C2×C8—4221no—
175C2×C8—4221no—
176C2×C8—4221no—
177C2×C8—4221no—
178C2×C8—4221no—
179C2×C8—2112no—
180C4⋊C4—24no—
181C4⋊C4—21no—
182C4⋊C4—4221no—
183C4⋊C4—4221no—
184C4⋊C4—4221no—
185C4⋊C4—4221no—
186C4⋊C4C22⋊D21112yes—
187C22⋊C4—24no—
188C22⋊C4—8442no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#1680994912yes—
196C23C2×C22⋊D21111yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—8441no—
215C2×C4—8441no—
216C2×C4—8441no—
217C2×C4—8441no—
218C2×C4—8441no—
219C2×C4—8441no—
220C2×C4D4×D21111yescharacteristic
221C2×C4C2×C22⋊D21111yescharacteristic
222C2×C4C2×C22⋊D21111yescharacteristic
223C2×C4#1680996111yescharacteristic
224C2×C4#1680996111yescharacteristic
225C2×C4#1680996111yescharacteristic
226C8—8441no—
227C8—4222no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#2026994611yescharacteristic
236C22#2026994911yescharacteristic
237C22#2026995811yescharacteristic
238C22C2×#1680994911yescharacteristic
239C22#2026999711yescharacteristic
240C22#2027000711yescharacteristic
241C22#2027000811yescharacteristic
242C4—41no—
243C4—42no—
244C4—42no—
245C4—16881no—
246C4—16881no—
247C4#2026994811yescharacteristic
248C4#2026994911yescharacteristic
249C4#2026995811yescharacteristic
250C4#2026996411yescharacteristic
251C2—42no—
252C2#2721306011yescharacteristic
253C2#2721328511yescharacteristic
254C2#2721330811yescharacteristic
255C2#2721337711yescharacteristic
256C2#2721340211yescharacteristic
257C2#2721347411yescharacteristic
258C2#2721348411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×C22⋊Q8×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
hypercenterC2×C22⋊Q8D211
hypocenterC211#2035
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20269908C2semidirect
2. C2×#20269901C2semidirect
3. C2×#20269901C2semidirect
6. #27212908C2semidirect
129. C211#2035semidirect

Maximal subgroups

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

subgroupindex
C2×#202699082
C2×#202699012
C2×C22⋊(C211⋊C8)2
#272130062
#272129082
C2×C22⋊Q8×C2112
#2035211