Finite Group Databasebeta

← #41014398

Subgroups — #41014398

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
1C2×#20269953C211yesmaximal, characteristic, complemented
2C2×#20269948C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic, complemented
5C2×(C2×D211)⋊C8C211yesmaximal, characteristic, complemented
6#27212909C211yesmaximal, characteristic, complemented
7C22×SD16×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211—21no—
9C22×C4×D211C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11#20269953—22no—
12#20269953—22no—
13#20269948—22no—
14#20269948—22no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×C4⋊Dic211C2211yescharacteristic
17C2×Dic211⋊C4—21no—
18C2×C22⋊Dic211—21no—
19C2×(C2×D211)⋊C4—21no—
20#20269901—22no—
21#20269898—22no—
22(C2×D211)⋊C8—22no—
23C22×Q8×C211C2211yescharacteristic
24C22×D4×C211C2211yescharacteristic
25C2×SD16×C211—22no—
26C2×SD16×C211—22no—
27C2×SD16×C211—22no—
28C22×C8×C211C2211yescharacteristic
29C23×D211D411yescharacteristic
30C2×C22⋊D211—24no—
31C2×C22⋊D211—42no—
32C2×C22⋊D211—42no—
33C22×Dic211—21no—
34C22×Dic211—21no—
35C22×Dic211D411yescharacteristic
36C2×C4×D211—21no—
37C2×C4×D211—21no—
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C4×D211—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C4⋊Dic211—21no—
46C4⋊Dic211—21no—
47C4⋊Dic211D412yes—
48C4⋊Dic211D412yescomplemented
49Dic211⋊C4—24no—
50C22⋊Dic211—24no—
51(C2×D211)⋊C4—24no—
52C24×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211—21no—
58C2×Q8×C211D412yescomplemented
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211—21no—
63C2×D4×C211—21no—
64C2×D4×C211D412yes—
65C22×C4×C211—21no—
66C22×C4×C211C2311yescharacteristic
67SD16×C211—28no—
68C2×C8×C211—21no—
69C2×C8×C211—21no—
70C2×C8×C211D412yescomplemented
71C22×D211—21no—
72C22×D211—21no—
73C22×D211—21no—
74C22×D211—21no—
75C22×D211—21no—
76C22×D211—21no—
77C22×D211SD1612yescomplemented
78C22⋊D211—48no—
79C2×Dic211—22no—
80C2×Dic211—21no—
81C2×Dic211—22no—
82C2×Dic211—21no—
83C2×Dic211—21no—
84C2×Dic211—41no—
85C2×Dic211—41no—
86C2×Dic211—41no—
87C2×Dic211—41no—
88C4×D211—41no—
89C4×D211—41no—
90C4×D211—41no—
91C4×D211—41no—
92C211⋊C8—41no—
93C23×C211—22no—
94C23×C211—41no—
95C23×C211—41no—
96C23×C211—41no—
97C23×C211—41no—
98C23×C211—41no—
99C23×C211—41no—
100C23×C211C2×D411yescharacteristic
101Q8×C211—22no—
102Q8×C211—22no—
103Q8×C211—41no—
104Q8×C211—41no—
105D4×C211—22no—
106D4×C211—22no—
107D4×C211—41no—
108D4×C211—41no—
109C2×C4×C211—22no—
110C2×C4×C211—41no—
111C2×C4×C211—41no—
112C2×C4×C211C2×D411yescharacteristic
113C2×C4×C211C2×D411yescharacteristic
114C2×C4×C211C2×D411yescharacteristic
115C2×C4×C211C2×D411yescharacteristic
116C2×C4×C211C2×D411yescharacteristic
117C2×C4×C211#4111yescharacteristic
118C8×C211—22no—
119C22×C211—42no—
120C22×C211—42no—
121C22×C211—42no—
122C22×C211—81no—
123C22×C211—81no—
124C22×C211—81no—
125C22×C211—81no—
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—22no—
134C2×D211—22no—
135C2×D211—22no—
136C2×D211—41no—
137C2×D211—41no—
138C2×D211—41no—
139C2×D211—41no—
140C4×C211—42no—
141C4×C211#12011yescharacteristic
142C4×C211#12111yescharacteristic
143C4×C211#12111yescharacteristic
144C4×C211#12711yescharacteristic
145Dic211—41no—
146Dic211—42no—
147Dic211—42no—
148C2×C211—82no—
149C2×C211#39211yescharacteristic
150C2×C211#44811yescharacteristic
151C2×C211#44911yescharacteristic
152C2×C211#45011yescharacteristic
153C2×C211#45211yescharacteristic
154C2×C211#46511yescharacteristic
155C2×C211#46811yescharacteristic
156D211—42no—
157C211#200011yescharacteristic, complemented, Sylow
158#2000—2111nomaximal, Sylow
159C22×SD16D21111yescharacteristic, complemented
160C2×C22⋊Q8—2111no—
161C2×#121—2111no—
162C2×#103—2111no—
163C2×#102—2111no—
164C2×C22⋊C8—2111no—
165#340—2111no—
166C22×Q8C2×D21111yescharacteristic
167C22×D4—4221no—
168C22×D4C2×D21111yescharacteristic
169C23×C4—2111no—
170C2×SD16—22no—
171C2×SD16—22no—
172C2×SD16—22no—
173C22×C8—2111no—
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—4222no—
187C24—21no—
188C24—2111no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8C22⋊D21112yescomplemented
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—8442no—
201C2×D4—8442no—
202C2×D4—4224no—
203C2×D4C22⋊D21112yes—
204C22×C4—21no—
205C22×C4—4221no—
206C22×C4—4221no—
207C22×C4—4221no—
208C22×C4—4221no—
209C22×C4—4221no—
210C22×C4—4221no—
211C22×C4—4221no—
212C22×C4—4221no—
213C22×C4—2111no—
214C22×C4C22×D21111yescharacteristic
215SD16—28no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—4221no—
219C2×C8—4221no—
220C2×C8—4221no—
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—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—4221no—
237C23—4221no—
238C23—4221no—
239C23—4221no—
240C23—4221no—
241C23—4221no—
242C23—2112no—
243C23D4×D21111yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—8448no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—8441no—
257C2×C4—8441no—
258C2×C4—8441no—
259C2×C4—8441no—
260C2×C4—8441no—
261C2×C4—8441no—
262C2×C4—8441no—
263C2×C4—8441no—
264C2×C4—4221no—
265C2×C4—4222no—
266C2×C4—4222no—
267C2×C4—4221no—
268C2×C4—4221no—
269C2×C4D4×D21111yescharacteristic
270C2×C4D4×D21111yescharacteristic
271C2×C4C2×C22⋊D21111yescharacteristic
272C2×C4C2×C22⋊D21111yescharacteristic
273C2×C4C2×C22⋊D21111yescharacteristic
274C2×C4#1680996111yescharacteristic
275C8—22no—
276C8—8441no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—8441no—
285C22—8441no—
286C22—8441no—
287C22—8441no—
288C22—4222no—
289C22—4222no—
290C22—4222no—
291C22#2026994511yescharacteristic
292C22#2026994511yescharacteristic
293C22#2026994811yescharacteristic
294C22SD16×D21111yescharacteristic
295C22#2026999811yescharacteristic
296C22#2027000411yescharacteristic
297C22#2027000611yescharacteristic
298C4—42no—
299C4—8442no—
300C4—8442no—
301C4—8441no—
302C4#2026994611yescharacteristic
303C4#2026994911yescharacteristic
304C4#2026994911yescharacteristic
305C4#2026997311yescharacteristic
306C2—82no—
307C2—8442no—
308C2#2721305411yescharacteristic
309C2#2721328311yescharacteristic
310C2#2721329011yescharacteristic
311C2#2721329511yescharacteristic
312C2#2721330511yescharacteristic
313C2#2721337211yescharacteristic
314C2#2721338911yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×SD16×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269948C2
hypercenterC22×SD16D211
hypocenterC211#2000
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#20269953C2semidirect
3. C2×#20269901C2semidirect
4. C2×#20269898C2semidirect
5. C2×(C2×D211)⋊C8C2semidirect
6. #27212909C2semidirect
7. C22×SD16×C211C2semidirect
48. C4⋊Dic211D4semidirect
58. C2×Q8×C211D4semidirect
70. C2×C8×C211D4semidirect
77. C22×D211SD16semidirect
157. C211#2000semidirect
159. C22×SD16D211semidirect
194. C2×Q8C22⋊D211semidirect
221. C2×C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699532
C2×#202699482
C2×#202699012
C2×#202698982
C2×(C2×D211)⋊C82
#272129092
C22×SD16×C2112
#2000211