Finite Group Databasebeta

← #41014401

Subgroups — #41014401

Order 27008.

444 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×#16809950C211yesmaximal, characteristic, complemented
2C2×#20269956C211yesmaximal, characteristic, complemented
3C2×#20269900C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic, complemented
5C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#121×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809950—22no—
10C2×#16809950—22no—
11C2×#16809950—22no—
12C22×C211⋊C8C2211yescharacteristic
13C22×C211⋊C8C2211yescharacteristic
14#20269956—22no—
15#20269956—22no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×Dic211⋊C4—21no—
18C2×C22⋊Dic211—21no—
19#20269900—22no—
20#20269898—22no—
21C22⋊(C211⋊C8)—22no—
22C22×D4×C211—21no—
23C22×D4×C211C2211yescharacteristic
24C23×C4×C211C2211yescharacteristic
25#121×C211—22no—
26#121×C211—22no—
27C2×C4⋊C4×C211C2211yescharacteristic
28C2×C22⋊C4×C211—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422D412yescomplemented
35C22×Dic211—21no—
36C22×Dic211—21no—
37#16809950—28no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8D412yescomplemented
44C4⋊Dic211—21no—
45C4⋊Dic211D412yescomplemented
46Dic211⋊C4—24no—
47C22⋊Dic211—24no—
48C24×C211—21no—
49C24×C211D411yescharacteristic
50C2×D4×C211—24no—
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—42no—
57C2×D4×C211—42no—
58C2×D4×C211D412yes—
59C22×C4×C211—21no—
60C22×C4×C211—21no—
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211—21no—
64C22×C4×C211—21no—
65C22×C4×C211—21no—
66C22×C4×C211D411yescharacteristic
67C22×C4×C211C2311yescharacteristic
68C4⋊C4×C211—21no—
69C4⋊C4×C211D412yes—
70C22⋊C4×C211—24no—
71Dic422—22no—
72Dic422—22no—
73Dic422—41no—
74Dic422—41no—
75C2×Dic211—22no—
76C2×Dic211—22no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C2×Dic211—41no—
81C211⋊C8—22no—
82C211⋊C8—41no—
83C23×C211—21no—
84C23×C211—21no—
85C23×C211—21no—
86C23×C211—21no—
87C23×C211—21no—
88C23×C211—21no—
89C23×C211—22no—
90C23×C211—41no—
91C23×C211—41no—
92C23×C211—41no—
93C23×C211—41no—
94C23×C211—41no—
95C23×C211—41no—
96C23×C211SD1612yescomplemented
97C23×C211C2×D411yescharacteristic
98D4×C211—22no—
99D4×C211—22no—
100D4×C211—48no—
101D4×C211—41no—
102D4×C211—41no—
103C2×C4×C211—21no—
104C2×C4×C211—22no—
105C2×C4×C211—21no—
106C2×C4×C211—21no—
107C2×C4×C211—41no—
108C2×C4×C211—41no—
109C2×C4×C211—41no—
110C2×C4×C211—41no—
111C2×C4×C211—41no—
112C2×C4×C211—41no—
113C2×C4×C211C2×D411yescharacteristic
114C2×C4×C211C2×D411yescharacteristic
115C2×C4×C211C2×D411yescharacteristic
116C2×C4×C211C2×D411yescharacteristic
117C2×C4×C211C2×D411yescharacteristic
118C2×C4×C211#4111yescharacteristic
119C22×C211—22no—
120C22×C211—22no—
121C22×C211—22no—
122C22×C211—41no—
123C22×C211—41no—
124C22×C211—41no—
125C22×C211—41no—
126C22×C211—42no—
127C22×C211—42no—
128C22×C211—42no—
129C22×C211—81no—
130C22×C211—81no—
131C22×C211—81no—
132C22×C211—81no—
133C22×C211#12011yescharacteristic
134C22×C211#12011yescharacteristic
135C22×C211#12111yescharacteristic
136C22×C211C2×SD1611yescharacteristic
137C22×C211#13511yescharacteristic
138C22×C211AGL(1,ℤ/8)11yescharacteristic
139C22×C211#13711yescharacteristic
140C4×C211—41no—
141C4×C211—42no—
142C4×C211#12011yescharacteristic
143C4×C211#12111yescharacteristic
144C4×C211#12111yescharacteristic
145C4×C211#12711yescharacteristic
146Dic211—42no—
147Dic211—42no—
148C2×C211—42no—
149C2×C211—82no—
150C2×C211#39211yescharacteristic
151C2×C211#44811yescharacteristic
152C2×C211#44911yescharacteristic
153C2×C211#45011yescharacteristic
154C2×C211#45211yescharacteristic
155C2×C211#46511yescharacteristic
156C2×C211#46811yescharacteristic
157C211#200011yescharacteristic, complemented, Sylow
158#2000—2111nomaximal, Sylow
159C22×SD16—2111no—
160C2×C22⋊Q8—2111no—
161C2×#121D21111yescharacteristic
162C2×#103—2111no—
163C2×#102—2111no—
164C2×C22⋊C8—2111no—
165#340—2111no—
166C22×Q8—2111no—
167C22×D4—21no—
168C22×D4C2×D21111yescharacteristic
169C23×C4C2×D21111yescharacteristic
170C2×SD16—4222no—
171C2×SD16—4222no—
172C2×SD16—4222no—
173C22×C8—2111no—
174C22×C8—2111no—
175C22⋊Q8—4222no—
176C22⋊Q8—4222no—
177#121—22no—
178#121—22no—
179C2×C4⋊C4—4221no—
180C2×C4⋊C4—2111no—
181C2×C4⋊C4C2×D21111yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—4221no—
184#103—4222no—
185#102—4222no—
186C22⋊C8—4222no—
187C24—21no—
188C24C22⋊D21111yescharacteristic
189C2×Q8—4221no—
190C2×Q8—4221no—
191C2×Q8—4221no—
192C2×Q8—4221no—
193C2×Q8—4221no—
194C2×Q8—2112no—
195C2×D4—24no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4C22⋊D21112yes—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—4221no—
212C22×C4—4221no—
213C22×C4C22⋊D21111yescharacteristic
214C22×C4C22×D21111yescharacteristic
215SD16—4228no—
216C2×C8—4221no—
217C2×C8—4221no—
218C2×C8—4221no—
219C2×C8—4221no—
220C2×C8—4221no—
221C2×C8—2112no—
222C4⋊C4—21no—
223C4⋊C4—4224no—
224C4⋊C4—4221no—
225C4⋊C4—2112no—
226C4⋊C4C22⋊D21112yes—
227C22⋊C4—24no—
228C22⋊C4—4224no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23#1680995012yescomplemented
243C23C2×C22⋊D21111yescharacteristic
244Q8—8441no—
245Q8—8441no—
246Q8—4222no—
247Q8—4222no—
248D4—22no—
249D4—22no—
250D4—48no—
251D4—41no—
252D4—41no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—8441no—
264C2×C4—8441no—
265C2×C4—8441no—
266C2×C4—8441no—
267C2×C4—4222no—
268C2×C4—4222no—
269C2×C4D4×D21111yescharacteristic
270C2×C4D4×D21111yescharacteristic
271C2×C4D4×D21111yescharacteristic
272C2×C4C2×C22⋊D21111yescharacteristic
273C2×C4C2×C22⋊D21111yescharacteristic
274C2×C4#1680996111yescharacteristic
275C8—8441no—
276C8—4222no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22#2026994511yescharacteristic
292C22#2026994611yescharacteristic
293C22#2026994911yescharacteristic
294C22C2×#1680995011yescharacteristic
295C22#2026999711yescharacteristic
296C22#2027000311yescharacteristic
297C22#2027000711yescharacteristic
298C4—42no—
299C4—41no—
300C4—8442no—
301C4—8442no—
302C4#2026994511yescharacteristic
303C4#2026994811yescharacteristic
304C4#2026994911yescharacteristic
305C4#2026997311yescharacteristic
306C2—42no—
307C2—82no—
308C2#2721305411yescharacteristic
309C2#2721328411yescharacteristic
310C2#2721328911yescharacteristic
311C2#2721329711yescharacteristic
312C2#2721330811yescharacteristic
313C2#2721337311yescharacteristic
314C2#2721339011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×#121×C211C2
socleC23×C211C2×D4
hypercenterC2×#121D211
hypocenterC211#2000
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#16809950C2semidirect
2. C2×#20269956C2semidirect
3. C2×#20269900C2semidirect
4. C2×#20269898C2semidirect
5. C2×C22⋊(C211⋊C8)C2semidirect
6. #27212908C2semidirect
34. C2×Dic422D4semidirect
43. C2×C211⋊C8D4semidirect
45. C4⋊Dic211D4semidirect
96. C23×C211SD16semidirect
157. C211#2000semidirect
242. C23#16809950semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
C2×#202699562
C2×#202699002
C2×#202698982
C2×C22⋊(C211⋊C8)2
#272129082
C2×#121×C2112
#2000211