Finite Group Databasebeta

← #41013853

Subgroups — #41013853

Order 27008.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269958C211yesmaximal, characteristic
2C2×#20269933C211yesmaximal, characteristic, complemented
3C2×#20269892C211yesmaximal, characteristic
4#27213007C211yesmaximal, characteristic, complemented
5#27212912C211yesmaximal, characteristic, complemented
6#27212912C211yesmaximal, characteristic, complemented
7C2×#99×C211C211yesmaximal, characteristic
8C23×Dic211C2211yescharacteristic
9#20269958—22no—
10#20269958—21no—
11#20269958—21no—
12#20269933—22no—
13#20269933—22no—
14C2×Dic211⋊C4—21no—
15C2×Dic211⋊C4C412yescomplemented
16C2×C22⋊Dic211C412yescomplemented
17C2×C22⋊Dic211C2211yescharacteristic
18C2×C22⋊Dic211C2211yescharacteristic
19C2×C22⋊Dic211C2211yescharacteristic
20C2×C4×Dic211—21no—
21#20269892—22no—
22C211⋊#95—22no—
23C22×D4×C211C2211yescharacteristic
24C2×C22⋊C4×C211C2211yescharacteristic
25C2×C22⋊C4×C211C2211yescharacteristic
26#99×C211—22no—
27C22×Dic211—21no—
28C22×Dic211—21no—
29C22×Dic211—21no—
30C22×Dic211—21no—
31C22×Dic211—21no—
32C22×Dic211—21no—
33C22×Dic211C2×C412yes—
34C22×Dic211C2×C412yes—
35C22×Dic211D412yes—
36C22×Dic211D411yescharacteristic
37C22×Dic211D411yescharacteristic
38Dic211⋊C4—24no—
39Dic211⋊C4—22no—
40Dic211⋊C4—22no—
41C22⋊Dic211—22no—
42C22⋊Dic211—22no—
43C22⋊Dic211—21no—
44C22⋊Dic211—21no—
45C22⋊Dic211—21no—
46C22⋊Dic211—21no—
47C22⋊Dic211—21no—
48C22⋊Dic211D412yes—
49C4×Dic211—24no—
50C24×C211C2×C411yescharacteristic
51C24×C211C2311yescharacteristic
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C22×C4×C211—21no—
59C22×C4×C211—21no—
60C22×C4×C211C2×C411yescharacteristic
61C22⋊C4×C211—21no—
62C22⋊C4×C211—21no—
63C22⋊C4×C211—21no—
64C22⋊C4×C211D412yes—
65C2×Dic211—22no—
66C2×Dic211—22no—
67C2×Dic211—21no—
68C2×Dic211—21no—
69C2×Dic211—22no—
70C2×Dic211—22no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—22no—
74C2×Dic211—22no—
75C2×Dic211—42no—
76C2×Dic211—42no—
77C2×Dic211—42no—
78C2×Dic211—41no—
79C2×Dic211C22⋊C412yes—
80C2×Dic211C22⋊C412yes—
81C23×C211—21no—
82C23×C211—21no—
83C23×C211—21no—
84C23×C211—42no—
85C23×C211—42no—
86C23×C211C22×C411yescharacteristic
87C23×C211C2×D411yescharacteristic
88C23×C211C2×D411yescharacteristic
89C23×C211C2×D411yescharacteristic
90C23×C211C2×D411yescharacteristic
91C23×C211#4111yescharacteristic
92C23×C211#4111yescharacteristic
93D4×C211—41no—
94D4×C211—41no—
95D4×C211—41no—
96D4×C211—41no—
97C2×C4×C211—21no—
98C2×C4×C211—21no—
99C2×C4×C211—21no—
100C2×C4×C211—22no—
101C2×C4×C211—42no—
102C2×C4×C211—42no—
103C2×C4×C211—41no—
104C22×C211—22no—
105C22×C211—22no—
106C22×C211—22no—
107C22×C211—41no—
108C22×C211—41no—
109C22×C211—41no—
110C22×C211—41no—
111C22×C211—41no—
112C22×C211—42no—
113C22×C211—81no—
114C22×C211—81no—
115C22×C211C2×C22⋊C411yescharacteristic
116C22×C211C4×D411yescharacteristic
117C22×C211C4×D411yescharacteristic
118C22×C211#12011yescharacteristic
119C22×C211#12111yescharacteristic
120C22×C211#12111yescharacteristic
121C22×C211#12311yescharacteristic
122C4×C211—42no—
123C4×C211—41no—
124C4×C211—81no—
125Dic211—22no—
126Dic211—22no—
127Dic211—42no—
128Dic211—42no—
129Dic211—81no—
130C2×C211—21no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211—21no—
134C2×C211—21no—
135C2×C211—21no—
136C2×C211—81no—
137C2×C211#38611yescharacteristic
138C2×C211#41012yes—
139C211#189911yescharacteristic, complemented, Sylow
140#1899—2111nomaximal, Sylow
141C2×#123—2111no—
142C2×#117—2111no—
143C2×#99—2111no—
144C2×#99D21111yescharacteristic
145#381—2111no—
146#342—2111no—
147#342—2111no—
148C22×D4C2×D21111yescharacteristic
149C23×C4—2111no—
150#123—4222no—
151#123—4221no—
152#123—4221no—
153#117—4222no—
154#117—4222no—
155C2×C4⋊C4—4221no—
156C2×C4⋊C4—2112no—
157C2×C22⋊C4—2111no—
158C2×C22⋊C4—2112no—
159C2×C22⋊C4—2111no—
160C2×C22⋊C4—2111no—
161C2×C22⋊C4C2×D21111yescharacteristic
162C2×C22⋊C4C2×D21111yescharacteristic
163C2×C42—4221no—
164#99—22no—
165#99—4222no—
166#95—4222no—
167C24C4×D21111yescharacteristic
168C24C22×D21111yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—4221no—
178C22×C4—4221no—
179C22×C4—4221no—
180C22×C4—4221no—
181C22×C4—4221no—
182C22×C4—4221no—
183C22×C4—2112no—
184C22×C4—2112no—
185C22×C4—2112no—
186C22×C4—2111no—
187C22×C4—2111no—
188C22×C4C4×D21111yescharacteristic
189C4⋊C4—4224no—
190C4⋊C4—4222no—
191C4⋊C4—4222no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—4221no—
196C22⋊C4—4221no—
197C22⋊C4—4221no—
198C22⋊C4—4221no—
199C22⋊C4—4221no—
200C22⋊C4—4222no—
201C22⋊C4—4222no—
202C22⋊C4—2112no—
203C22⋊C4C22⋊D21112yes—
204C42—4224no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—42no—
209C23—42no—
210C23C2×C4×D21111yescharacteristic
211C23D4×D21111yescharacteristic
212C23D4×D21111yescharacteristic
213C23D4×D21111yescharacteristic
214C23C2×C22⋊D21111yescharacteristic
215C23#1680996111yescharacteristic
216C23#1680996211yescharacteristic
217D4—41no—
218D4—41no—
219D4—41no—
220D4—41no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—8442no—
229C2×C4—8442no—
230C2×C4—8442no—
231C2×C4—8441no—
232C2×C4—4222no—
233C2×C4—4222no—
234C2×C4—4222no—
235C2×C4—4222no—
236C2×C4—4221no—
237C2×C4—4221no—
238C2×C4—4222no—
239C2×C4—4222no—
240C2×C4—4221no—
241C2×C4—4221no—
242C2×C4—2112no—
243C2×C4—2112no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—81no—
254C22—81no—
255C22C22⋊C4×D21111yescharacteristic
256C22(C22⋊D211)⋊C411yescharacteristic
257C22C4×C22⋊D21111yescharacteristic
258C22#2026994511yescharacteristic
259C22#2026994811yescharacteristic
260C22#2026995011yescharacteristic
261C22#2026995911yescharacteristic
262C4—42no—
263C4—41no—
264C4—81no—
265C4—8442no—
266C4—8442no—
267C4—16881no—
268C4—4222no—
269C4—4222no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—81no—
277C2#2721302211yescharacteristic
278C2#2721311812yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC24C22×D211
FittingC2×#99×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D4×C211C22
cosocleC24×C211C23
hypercenterC2×#99D211
hypocenterC211#1899
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#20269933C2semidirect
4. #27213007C2semidirect
5. #27212912C2semidirect
6. #27212912C2semidirect
15. C2×Dic211⋊C4C4semidirect
16. C2×C22⋊Dic211C4semidirect
139. C211#1899semidirect

Maximal subgroups

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

subgroupindex
C2×#202699582
C2×#202699332
C2×#202698922
#272130072
#272129122
C2×#99×C2112
#1899211