Finite Group Databasebeta

← #34079915

Subgroups — #34079915

Order 19328.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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⋊Q8×D151C211yesmaximal, characteristic, complemented
2C2×#18012714C211yesmaximal, characteristic
3#23752547C211yesmaximal, characteristic, complemented
4#23752527C212yesmaximal, complemented
5#23752461C212yesmaximal, complemented
6#23752459C212yesmaximal, complemented
7#23752365C211yesmaximal, characteristic, complemented
8#23752362C211yesmaximal, characteristic, complemented
9#23752352C212yesmaximal, complemented
10#23752174C211yesmaximal, characteristic, complemented
11#484×C151C211yesmaximal, characteristic, complemented
12C2×Q8×D151C2211yescharacteristic, complemented
13C22×D604—21no—
14C22×D604C2211yescharacteristic
15C22×C4×D151C2211yescharacteristic
16C2×#14558993C2212yescomplemented
17C2×#14558991C2212yescomplemented
18C2×#14558987C2212yescomplemented
19#18012719C2211yescharacteristic, complemented
20#18012718C2211yescharacteristic
21#18012717—21no—
22#18012716—21no—
23#18012715C2211yescharacteristic, complemented
24#18012714—21no—
25#18012714—21no—
26#18012714C2212yes—
27#18012714C2212yes—
28C4⋊C4×D151—21no—
29C4⋊C4×D151C2211yescharacteristic, complemented
30C2×(C2×D151)⋊C4—21no—
31C22⋊C4×D151—21no—
32#18012672C2212yescomplemented
33#18012662C2212yescomplemented
34#18012661C2212yescomplemented
35#18012659C2212yescomplemented
36C22⋊(C151⋊C8)C2211yescharacteristic
37(C2×D151)⋊C8C2212yescomplemented
38C22⋊Q8×C151C2211yescharacteristic
39C2×C4⋊C4×C151C2211yescharacteristic
40#107×C151C2212yescomplemented
41#103×C151C2212yescomplemented
42C22⋊C8×C151C2211yescharacteristic
43C23×D151—21no—
44C23×D151D411yescharacteristic
45C2×Dic302—21no—
46C2×Dic302C2311yescharacteristic
47Q8×D151—42no—
48C2×D604—22no—
49C2×D604—22no—
50C2×D604—21no—
51C2×D604—42no—
52C2×D604—44no—
53C2×D604C2312yes—
54C22×Dic151D411yescharacteristic
55C2×C4×D151—21no—
56C2×C4×D151—21no—
57C2×C4×D151—42no—
58C2×C4×D151D411yescharacteristic
59C2×C4×D151D411yescharacteristic
60C2×C4×D151C2311yescharacteristic
61C2×C4×D151C2311yescharacteristic
62#14558993—42no—
63#14558991—42no—
64#14558987—42no—
65C2×C151⋊C8C2312yes—
66C4⋊Dic151—21no—
67C4⋊Dic151C2311yescharacteristic
68Dic151⋊C4—21no—
69Dic151⋊C4—21no—
70Dic151⋊C4—21no—
71C22⋊Dic151—21no—
72(C2×D151)⋊C4—21no—
73(C2×D151)⋊C4—21no—
74(C2×D151)⋊C4—22no—
75(C2×D151)⋊C4—22no—
76C2×Q8×C151C2311yescharacteristic
77C22×C4×C151—21no—
78C22×C4×C151C2311yescharacteristic
79C2×C8×C151C2312yes—
80C4⋊C4×C151—21no—
81C4⋊C4×C151—21no—
82C4⋊C4×C151C2312yes—
83C4⋊C4×C151C2311yescharacteristic
84C22⋊C4×C151—21no—
85C22×D151—22no—
86C22×D151—22no—
87C22×D151—22no—
88C22×D151—41no—
89C22×D151—42no—
90C22×D151—42no—
91C22×D151—42no—
92C22×D151C2×D411yescharacteristic
93C22×D151C2×D411yescharacteristic
94Dic302—22no—
95Dic302—42no—
96D604—22no—
97D604—22no—
98D604—42no—
99D604—44no—
100D604—44no—
101C2×Dic151—21no—
102C2×Dic151—21no—
103C2×Dic151—41no—
104C2×Dic151C2×D411yescharacteristic
105C2×Dic151C2×D411yescharacteristic
106C4×D151—22no—
107C4×D151—22no—
108C4×D151—41no—
109C4×D151—41no—
110C4×D151—42no—
111C4×D151—42no—
112C151⋊C8—22no—
113C23×C151C2×D411yescharacteristic
114Q8×C151—22no—
115C2×C4×C151—22no—
116C2×C4×C151—21no—
117C2×C4×C151—22no—
118C2×C4×C151—21no—
119C2×C4×C151—42no—
120C2×C4×C151C2×D411yescharacteristic
121C2×C4×C151C2411yescharacteristic
122C8×C151—22no—
123C22×C151—21no—
124C22×C151—21no—
125C22×C151#13712yes—
126C22×C151C22×D411yescharacteristic
127C2×D151—21no—
128C2×D151—21no—
129C2×D151—21no—
130C2×D151—21no—
131C2×D151—21no—
132C2×D151—44no—
133C2×D151—42no—
134C2×D151—42no—
135C2×D151—42no—
136C2×D151—84no—
137C2×D151AGL(1,ℤ/8)12yes—
138C4×C151—21no—
139C4×C151—41no—
140C4×C151—42no—
141C4×C151—41no—
142C4×C151H(5,𝔽2)12yes—
143Dic151—21no—
144Dic151—21no—
145Dic151—41no—
146Dic151—41no—
147C2×C151—22no—
148C2×C151#53411yescharacteristic
149C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C151C2×#13711yescharacteristic
151D151—22no—
152D151—41no—
153D151—82no—
154C151#319311yescharacteristic, complemented, Sylow
155#3193—1511nomaximal, Sylow
156C2×C22⋊Q8—1511no—
157C2×#121—1511no—
158#484—1511no—
159#484D15111yescharacteristic, complemented
160#480—1512no—
161#468—1512no—
162#468—1512no—
163#450—1511no—
164#450—1511no—
165#448—1512no—
166#407—1511no—
167C22×Q8—1511no—
168C22×D4—3021no—
169C22×D4—1511no—
170C23×C4—1511no—
171C2×SD16—1512no—
172C2×SD16—1512no—
173C2×M16—1512no—
174C22⋊Q8—3021no—
175C22⋊Q8—3021no—
176C22⋊Q8—1511no—
177C22⋊Q8—1511no—
178C22⋊Q8—1511no—
179C22⋊Q8C2×D15111yescharacteristic
180#121—3021no—
181#121—3021no—
182#121—1512no—
183#121—1512no—
184C2×C4⋊C4—3021no—
185C2×C4⋊C4—1511no—
186C2×C4⋊C4C2×D15111yescharacteristic
187C2×C22⋊C4—3021no—
188C2×C22⋊C4—3021no—
189#107—1512no—
190#107C2×D15112yescomplemented
191#103—1512no—
192#103C2×D15112yescomplemented
193#102—1512no—
194#102—1512no—
195C22⋊C8—1511no—
196C22⋊C8—1512no—
197C22⋊C8C2×D15111yescharacteristic
198C24—3021no—
199C24—1511no—
200C2×Q8—3021no—
201C2×Q8—6042no—
202C2×Q8—1511no—
203C2×Q8C22×D15111yescharacteristic
204C2×D4—3022no—
205C2×D4—3022no—
206C2×D4—3021no—
207C2×D4—6042no—
208C2×D4—6044no—
209C2×D4—1512no—
210C22×C4—21no—
211C22×C4—3021no—
212C22×C4—3021no—
213C22×C4—6042no—
214C22×C4—1511no—
215C22×C4—1511no—
216C22×C4—1511no—
217C22×C4—1511no—
218C22×C4—1511no—
219C22×C4C22×D15111yescharacteristic
220SD16—6042no—
221SD16—6042no—
222M16—6042no—
223C2×C8—1512no—
224C2×C8C22×D15112yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—3021no—
228C4⋊C4—3021no—
229C4⋊C4—3021no—
230C4⋊C4—3021no—
231C4⋊C4—1511no—
232C4⋊C4C22×D15111yescharacteristic
233C4⋊C4C22×D15112yes—
234C22⋊C4—21no—
235C22⋊C4—3021no—
236C22⋊C4—3021no—
237C22⋊C4—3022no—
238C22⋊C4—3022no—
239C22⋊C4—3021no—
240C23—3022no—
241C23—3022no—
242C23—3022no—
243C23—6041no—
244C23—6042no—
245C23—6042no—
246C23—6042no—
247C23—1511no—
248C23—1511no—
249C23D4×D15111yescharacteristic
250Q8—22no—
251Q8—3022no—
252Q8—6042no—
253D4—3022no—
254D4—3022no—
255D4—6042no—
256D4—6044no—
257D4—6044no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—3022no—
264C2×C4—3022no—
265C2×C4—3021no—
266C2×C4—3021no—
267C2×C4—6041no—
268C2×C4—6041no—
269C2×C4—6042no—
270C2×C4—6042no—
271C2×C4—6041no—
272C2×C4—1511no—
273C2×C4—1511no—
274C2×C4D4×D15111yescharacteristic
275C2×C4C23×D15111yescharacteristic
276C8—22no—
277C8—3022no—
278C22—21no—
279C22—21no—
280C22—3021no—
281C22—3021no—
282C22—3021no—
283C22—3021no—
284C22—3021no—
285C22—6044no—
286C22—6042no—
287C22—6042no—
288C22—6042no—
289C22—1512no—
290C22—12084no—
291C22#1801277112yes—
292C22C2×D4×D15111yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3021no—
298C4—3021no—
299C4—6041no—
300C4—6041no—
301C4#1801278512yes—
302C2—22no—
303C2—3022no—
304C2—6041no—
305C2—12082no—
306C2#2375277311yescharacteristic
307C2AGL(1,ℤ/8)×D15111yescharacteristic
308C2C2×#1801277111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#484×C151C2
socleC22×C151C22×D4
prime radicalC2×#18012714C2
hypercenter#484D151
hypocenterC151#3193
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D151C2semidirect
3. #23752547C2semidirect
4. #23752527C2semidirect
5. #23752461C2semidirect
6. #23752459C2semidirect
7. #23752365C2semidirect
8. #23752362C2semidirect
9. #23752352C2semidirect
10. #23752174C2semidirect
11. #484×C151C2semidirect
12. C2×Q8×D151C22semidirect
16. C2×#14558993C22semidirect
17. C2×#14558991C22semidirect
18. C2×#14558987C22semidirect
19. #18012719C22semidirect
23. #18012715C22semidirect
29. C4⋊C4×D151C22semidirect
32. #18012672C22semidirect
33. #18012662C22semidirect
34. #18012661C22semidirect
35. #18012659C22semidirect
37. (C2×D151)⋊C8C22semidirect
40. #107×C151C22semidirect
41. #103×C151C22semidirect
154. C151#3193semidirect
159. #484D151semidirect
190. #107C2×D151semidirect
192. #103C2×D151semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1512
C2×#180127142
#237525472
#237525272
#237524612
#237524592
#237523652
#237523622
#237523522
#237521742
#484×C1512
#3193151