Finite Group Databasebeta

← #41014463

Subgroups — #41014463

Order 27008.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#27213944C211yesmaximal, characteristic, complemented
2#27213657C211yesmaximal, characteristic, complemented
3#27213164C211yesmaximal, characteristic, complemented
4#27213163C211yesmaximal, characteristic, complemented
5#27213110C211yesmaximal, characteristic, complemented
6#27212902C211yesmaximal, characteristic, complemented
7#525×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007—21no—
11#20270003—21no—
12#20270002—21no—
13#20269997—22no—
14C2×#16809946C2211yescharacteristic
15C2×#16809946C2211yescharacteristic
16#20269960—21no—
17#20269956—21no—
18#20269951—21no—
19#20269946—21no—
20C4×C22⋊D211—22no—
21#20269936C2211yescharacteristic
22#20269905—22no—
23#20269904—22no—
24C22⋊(C211⋊C8)—22no—
25C2×#41×C211C2211yescharacteristic
26C23×C4×C211C2211yescharacteristic
27#123×C211—21no—
28C22⋊Q8×C211—21no—
29#121×C211—21no—
30#120×C211—21no—
31C4×D4×C211—22no—
32#117×C211C2211yescharacteristic
33#16809962—21no—
34#16809962—21no—
35#16809962—21no—
36#16809962—21no—
37C2×Dic422D411yescharacteristic, complemented
38C2×D844D411yescharacteristic, complemented
39C2×C22⋊D211—22no—
40C2×C22⋊D211—21no—
41C2×C4×D211—21no—
42#16809952—22no—
43#16809950—22no—
44#16809949—22no—
45#16809947—22no—
46#16809946—21no—
47#16809946—21no—
48#16809946—21no—
49#16809946—21no—
50C2×C211⋊C8—21no—
51C2×C211⋊C8D412yescomplemented
52C4⋊Dic211—21no—
53Dic211⋊C4—22no—
54C22⋊Dic211—22no—
55C22⋊Dic211—21no—
56(C2×D211)⋊C4—22no—
57C4×Dic211D412yescomplemented
58C24×C211D411yescharacteristic
59#41×C211—21no—
60#41×C211—21no—
61#41×C211—21no—
62#41×C211—21no—
63C2×Q8×C211D411yescharacteristic, complemented
64C2×D4×C211—22no—
65C2×D4×C211—21no—
66C2×D4×C211D411yescharacteristic, complemented
67C22×C4×C211—21no—
68C22×C4×C211—21no—
69C22×C4×C211—21no—
70C22×C4×C211—21no—
71C22×C4×C211—21no—
72C22×C4×C211—21no—
73C22×C4×C211—21no—
74C22×C4×C211D411yescharacteristic
75C22×C4×C211C2311yescharacteristic
76C4⋊C4×C211—22no—
77C4⋊C4×C211—21no—
78C22⋊C4×C211—22no—
79C22⋊C4×C211—22no—
80C22⋊C4×C211—21no—
81C42×C211D412yescomplemented
82C22×D211—21no—
83Dic422—21no—
84Dic422—21no—
85D844—21no—
86D844—21no—
87C22⋊D211—44no—
88C22⋊D211—41no—
89C22⋊D211—41no—
90C2×Dic211—22no—
91C2×Dic211—21no—
92C4×D211—41no—
93C4×D211—41no—
94C211⋊C8—22no—
95C211⋊C8—41no—
96C23×C211—21no—
97C23×C211—21no—
98C23×C211—21no—
99C23×C211—21no—
100C23×C211—42no—
101C23×C211C2×D411yescharacteristic
102Q8×C211—21no—
103Q8×C211—21no—
104D4×C211—21no—
105D4×C211—21no—
106D4×C211—44no—
107D4×C211—41no—
108D4×C211—41no—
109C2×C4×C211—22no—
110C2×C4×C211—21no—
111C2×C4×C211—21no—
112C2×C4×C211—21no—
113C2×C4×C211—21no—
114C2×C4×C211—41no—
115C2×C4×C211—41no—
116C2×C4×C211—41no—
117C2×C4×C211—41no—
118C2×C4×C211—41no—
119C2×C4×C211—41no—
120C2×C4×C211C2×D411yescharacteristic
121C2×C4×C211C2×D411yescharacteristic
122C2×C4×C211C2×D411yescharacteristic
123C2×C4×C211C2×D411yescharacteristic
124C2×C4×C211C2×D411yescharacteristic
125C2×C4×C211#4111yescharacteristic
126C22×C211—22no—
127C22×C211—22no—
128C22×C211—22no—
129C22×C211—41no—
130C22×C211—41no—
131C22×C211—41no—
132C22×C211—41no—
133C22×C211—42no—
134C22×C211—41no—
135C22×C211—41no—
136C22×C211#12011yescharacteristic
137C22×C211#12011yescharacteristic
138C22×C211#12111yescharacteristic
139C2×D211—42no—
140C2×D211—41no—
141C4×C211—42no—
142C4×C211—41no—
143C4×C211—41no—
144C4×C211#12011yescharacteristic
145C4×C211#12111yescharacteristic
146C4×C211#12111yescharacteristic
147C4×C211#12711yescharacteristic
148Dic211—42no—
149Dic211—41no—
150C2×C211—21no—
151C2×C211—21no—
152C2×C211—21no—
153C2×C211—42no—
154C2×C211—81no—
155C2×C211#39211yescharacteristic
156D211—81no—
157C211#201011yescharacteristic, complemented, Sylow
158#2010—2111nomaximal, Sylow
159#575—2111no—
160#525—2111no—
161#525D21111yescharacteristic, complemented
162#421—2111no—
163#421—2111no—
164#407—2111no—
165H(3,ℤ/4)—2111no—
166C2×#41—2111no—
167C2×#41C2×D21111yescharacteristic
168C23×C4C2×D21111yescharacteristic
169#137—4221no—
170#137—4221no—
171AGL(1,ℤ/8)—4221no—
172AGL(1,ℤ/8)—4221no—
173#135—4222no—
174C2×M16—2111no—
175C2×M16—2111no—
176#123—21no—
177#123—4221no—
178C22⋊Q8—21no—
179C22⋊Q8—4221no—
180#121—21no—
181#121—4221no—
182#120—21no—
183#120—4221no—
184C4×D4—22no—
185C4×D4—4222no—
186#117—2111no—
187#117C2×D21111yescharacteristic
188#104—4222no—
189#104—4222no—
190C22⋊C8—4222no—
191C24C22⋊D21111yescharacteristic
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—4221no—
197#41—4221no—
198#41—4221no—
199#41—4221no—
200C2×Q8—2111no—
201C2×Q8C22⋊D21111yescharacteristic, complemented
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—4222no—
205C2×D4—4221no—
206C2×D4—2111no—
207C2×D4C22⋊D21111yescharacteristic, complemented
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—4221no—
216C22×C4C22⋊D21111yescharacteristic
217C22×C4C22×D21111yescharacteristic
218Q16—4222no—
219SD16—4222no—
220SD16—4222no—
221D8—4222no—
222M16—4221no—
223M16—4221no—
224M16—4221no—
225M16—4221no—
226C2×C8—4221no—
227C2×C8—2112no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—4222no—
231C4⋊C4—4221no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—21no—
235C22⋊C4—4222no—
236C22⋊C4—4222no—
237C22⋊C4—4221no—
238C42—2112no—
239C42C22⋊D21112yescomplemented
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—42no—
245C23—4221no—
246C23C2×C22⋊D21111yescharacteristic
247Q8—21no—
248Q8—21no—
249Q8—4221no—
250Q8—4221no—
251D4—21no—
252D4—21no—
253D4—44no—
254D4—41no—
255D4—41no—
256D4—8444no—
257D4—8441no—
258D4—8441no—
259D4—4221no—
260D4—4221no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—8441no—
273C2×C4—8441no—
274C2×C4—4222no—
275C2×C4—4221no—
276C2×C4D4×D21111yescharacteristic
277C2×C4D4×D21111yescharacteristic
278C2×C4D4×D21111yescharacteristic
279C2×C4C2×C22⋊D21111yescharacteristic
280C2×C4C2×C22⋊D21111yescharacteristic
281C2×C4#1680996111yescharacteristic
282C8—8441no—
283C8—4222no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—8442no—
295C22—8441no—
296C22#2026994511yescharacteristic
297C22#2026994611yescharacteristic
298C22#2026994911yescharacteristic
299C4—42no—
300C4—41no—
301C4—41no—
302C4—8442no—
303C4—8441no—
304C4#2026994511yescharacteristic
305C4#2026994811yescharacteristic
306C4#2026994911yescharacteristic
307C4#2026997311yescharacteristic
308C2—21no—
309C2—21no—
310C2—21no—
311C2—42no—
312C2—81no—
313C2—16881no—
314C2#2721305411yescharacteristic
subgroupG/H
centerC4#20269973
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#525×C211C2
socleC2×C211#392
hypercenter#525D211
hypocenterC211#2010
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #27213944C2semidirect
2. #27213657C2semidirect
3. #27213164C2semidirect
4. #27213163C2semidirect
5. #27213110C2semidirect
6. #27212902C2semidirect
7. #525×C211C2semidirect
37. C2×Dic422D4semidirect
38. C2×D844D4semidirect
51. C2×C211⋊C8D4semidirect
57. C4×Dic211D4semidirect
63. C2×Q8×C211D4semidirect
66. C2×D4×C211D4semidirect
81. C42×C211D4semidirect
157. C211#2010semidirect
161. #525D211semidirect
201. C2×Q8C22⋊D211semidirect
207. C2×D4C22⋊D211semidirect
239. C42C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#272136572
#272131642
#272131632
#272131102
#272129022
#525×C2112
#2010211