Finite Group Databasebeta

← #41014403

Subgroups — #41014403

Order 27008.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16809951C211yesmaximal, characteristic, complemented
2C2×#20269956C211yesmaximal, characteristic, complemented
3C2×#20269951C211yesmaximal, characteristic
4C2×#20269902C211yesmaximal, characteristic, complemented
5C2×#20269899C211yesmaximal, characteristic, complemented
6#27212909C211yesmaximal, characteristic, complemented
7C2×C22⋊C8×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C22×C22⋊D211—21no—
11C2×#16809951—22no—
12C2×#16809951—22no—
13C2×#16809951—22no—
14#20269956—22no—
15#20269956—22no—
16#20269951—22no—
17#20269951—22no—
18C2×C4⋊Dic211C2211yescharacteristic
19C2×C4⋊Dic211C2211yescharacteristic
20C2×Dic211⋊C4—21no—
21C2×C22⋊Dic211—21no—
22C2×(C2×D211)⋊C4—21no—
23#20269902—22no—
24#20269899—22no—
25C23×C4×C211C2211yescharacteristic
26C22×C8×C211C2211yescharacteristic
27C22×C8×C211C2211yescharacteristic
28C22⋊C8×C211—22no—
29C23×D211—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422D412yescomplemented
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844—21no—
40C2×D844—21no—
41C2×D844D412yes—
42C2×C22⋊D211—24no—
43C2×C22⋊D211—42no—
44C2×C22⋊D211—42no—
45C22×Dic211—21no—
46C22×Dic211—21no—
47C22×Dic211—21no—
48#16809951—28no—
49C4⋊Dic211—21no—
50C4⋊Dic211—21no—
51C4⋊Dic211D412yes—
52C4⋊Dic211D412yescomplemented
53Dic211⋊C4—24no—
54C22⋊Dic211—24no—
55(C2×D211)⋊C4—24no—
56C24×C211D411yescharacteristic
57C22×C4×C211—21no—
58C22×C4×C211—21no—
59C22×C4×C211—21no—
60C22×C4×C211—21no—
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211D411yescharacteristic
64C22×C4×C211C2311yescharacteristic
65C2×C8×C211—21no—
66C2×C8×C211—21no—
67C2×C8×C211—21no—
68C2×C8×C211—21no—
69C2×C8×C211—21no—
70C2×C8×C211D412yescomplemented
71C22×D211—22no—
72C22×D211—41no—
73C22×D211—41no—
74C22×D211—41no—
75C22×D211—41no—
76C22×D211—41no—
77C22×D211—41no—
78Dic422—22no—
79Dic422—22no—
80Dic422—41no—
81Dic422—41no—
82D844—22no—
83D844—22no—
84D844—41no—
85D844—41no—
86C22⋊D211—48no—
87C2×Dic211—22no—
88C2×Dic211—22no—
89C2×Dic211—22no—
90C2×Dic211—41no—
91C2×Dic211—41no—
92C2×Dic211—41no—
93C2×Dic211—41no—
94C2×Dic211—41no—
95C2×Dic211—41no—
96C23×C211—21no—
97C23×C211—21no—
98C23×C211—21no—
99C23×C211—21no—
100C23×C211—21no—
101C23×C211—21no—
102C23×C211SD1612yescomplemented
103C23×C211C2×D411yescharacteristic
104C2×C4×C211—21no—
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×C211C2×D411yescharacteristic
112C2×C4×C211C2×D411yescharacteristic
113C2×C4×C211C2×D411yescharacteristic
114C2×C4×C211C2×D411yescharacteristic
115C2×C4×C211C2×D411yescharacteristic
116C2×C4×C211#4111yescharacteristic
117C8×C211—22no—
118C8×C211—41no—
119C22×C211—22no—
120C22×C211—22no—
121C22×C211—22no—
122C22×C211—41no—
123C22×C211—41no—
124C22×C211—41no—
125C22×C211—41no—
126C22×C211#12011yescharacteristic
127C22×C211#12011yescharacteristic
128C22×C211#12111yescharacteristic
129C22×C211C2×SD1611yescharacteristic
130C22×C211#13511yescharacteristic
131C22×C211AGL(1,ℤ/8)11yescharacteristic
132C22×C211#13711yescharacteristic
133C2×D211—42no—
134C2×D211—42no—
135C2×D211—42no—
136C2×D211—81no—
137C2×D211—81no—
138C2×D211—81no—
139C2×D211—81no—
140C4×C211—41no—
141C4×C211#12011yescharacteristic
142C4×C211#12111yescharacteristic
143C4×C211#12111yescharacteristic
144C4×C211#12711yescharacteristic
145Dic211—42no—
146Dic211—42no—
147Dic211—42no—
148C2×C211—42no—
149C2×C211#39211yescharacteristic
150C2×C211#44811yescharacteristic
151C2×C211#44911yescharacteristic
152C2×C211#45011yescharacteristic
153C2×C211#45211yescharacteristic
154C2×C211#46511yescharacteristic
155C2×C211#46811yescharacteristic
156D211—82no—
157C211#200011yescharacteristic, complemented, Sylow
158#2000—2111nomaximal, Sylow
159C22×SD16—2111no—
160C2×C22⋊Q8—2111no—
161C2×#121—2111no—
162C2×#103—2111no—
163C2×#102—2111no—
164C2×C22⋊C8D21111yescharacteristic, complemented
165#340—2111no—
166C22×Q8—2111no—
167C22×D4—4221no—
168C22×D4—2111no—
169C23×C4C2×D21111yescharacteristic
170C2×SD16—4222no—
171C2×SD16—4222no—
172C2×SD16—4222no—
173C22×C8C2×D21111yescharacteristic
174C22×C8C2×D21111yescharacteristic
175C22⋊Q8—4222no—
176C22⋊Q8—4222no—
177#121—4222no—
178#121—4222no—
179C2×C4⋊C4—4221no—
180C2×C4⋊C4—2111no—
181C2×C4⋊C4—2111no—
182C2×C22⋊C4—4221no—
183C2×C22⋊C4—4221no—
184#103—4222no—
185#102—4222no—
186C22⋊C8—22no—
187C24—4221no—
188C24D84411yescharacteristic
189C2×Q8—4221no—
190C2×Q8—4221no—
191C2×Q8—4221no—
192C2×Q8—4221no—
193C2×Q8—4221no—
194C2×Q8—2112no—
195C2×D4—8442no—
196C2×D4—8442no—
197C2×D4—4224no—
198C2×D4—4221no—
199C2×D4—4221no—
200C2×D4—4221no—
201C2×D4—4221no—
202C2×D4—4221no—
203C2×D4—2112no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—4221no—
211C22×C4—4221no—
212C22×C4—4221no—
213C22×C4D84411yescharacteristic
214C22×C4C22×D21111yescharacteristic
215SD16—4228no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8C22⋊D21112yescomplemented
222C4⋊C4—4224no—
223C4⋊C4—4221no—
224C4⋊C4—4221no—
225C4⋊C4—2112no—
226C4⋊C4—2112no—
227C22⋊C4—4224no—
228C22⋊C4—4224no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—8441no—
236C23—8441no—
237C23—8441no—
238C23—8441no—
239C23—8441no—
240C23—8441no—
241C23—4222no—
242C23#1680995112yescomplemented
243C23C2×D84411yescharacteristic
244Q8—8441no—
245Q8—8441no—
246Q8—4222no—
247Q8—4222no—
248D4—8448no—
249D4—8441no—
250D4—8441no—
251D4—4222no—
252D4—4222no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—8441no—
261C2×C4—8441no—
262C2×C4—8441no—
263C2×C4—8441no—
264C2×C4—8441no—
265C2×C4—8441no—
266C2×C4—4222no—
267C2×C4—4222no—
268C2×C4—4222no—
269C2×C4D4×D21111yescharacteristic
270C2×C4D4×D21111yescharacteristic
271C2×C4D4×D21111yescharacteristic
272C2×C4D4×D21111yescharacteristic
273C2×C4C2×C22⋊D21111yescharacteristic
274C2×C4#1680996211yescharacteristic
275C8—22no—
276C8—41no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—8442no—
285C22—8442no—
286C22—8442no—
287C22—16881no—
288C22—16881no—
289C22—16881no—
290C22—16881no—
291C22#2026994711yescharacteristic
292C22#2026994711yescharacteristic
293C22#2026995111yescharacteristic
294C22C2×#1680995111yescharacteristic
295C22#2026999911yescharacteristic
296C22#2027000511yescharacteristic
297C22#2027001011yescharacteristic
298C4—41no—
299C4—8442no—
300C4—8442no—
301C4—8442no—
302C4#2026994511yescharacteristic
303C4#2026995011yescharacteristic
304C4#2026995011yescharacteristic
305C4#2026997311yescharacteristic
306C2—42no—
307C2—16882no—
308C2#2721305511yescharacteristic
309C2#2721328711yescharacteristic
310C2#2721329311yescharacteristic
311C2#2721329911yescharacteristic
312C2#2721331011yescharacteristic
313C2#2721337511yescharacteristic
314C2#2721339211yescharacteristic
subgroupG/H
centerC23C2×D844
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×C22⋊C8×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269951C2
hypercenterC2×C22⋊C8D211
hypocenterC211#2000
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#16809951C2semidirect
2. C2×#20269956C2semidirect
4. C2×#20269902C2semidirect
5. C2×#20269899C2semidirect
6. #27212909C2semidirect
7. C2×C22⋊C8×C211C2semidirect
35. C2×Dic422D4semidirect
52. C4⋊Dic211D4semidirect
70. C2×C8×C211D4semidirect
102. C23×C211SD16semidirect
157. C211#2000semidirect
164. C2×C22⋊C8D211semidirect
221. C2×C8C22⋊D211semidirect
242. C23#16809951semidirect

Maximal subgroups

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

subgroupindex
C22×#168099512
C2×#202699562
C2×#202699512
C2×#202699022
C2×#202698992
#272129092
C2×C22⋊C8×C2112
#2000211