Finite Group Databasebeta

← #34079572

Subgroups — #34079572

Order 19328.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18012772C211yesmaximal, characteristic, complemented
2C2×#18012769C211yesmaximal, characteristic, complemented
3C2×#18012765C211yesmaximal, characteristic, complemented
4#23752758C211yesmaximal, characteristic, complemented
5C2×#18012726C211yesmaximal, characteristic, complemented
6#23752603C212yesmaximal, complemented
7#23752600C212yesmaximal, complemented
8#23752595C212yesmaximal, complemented
9#23752593C212yesmaximal, complemented
10M16×Dic151C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic, complemented
12C2×#14559004C2211yescharacteristic
13C22×Dic302C2211yescharacteristic
14#18012772—24no—
15#18012769—24no—
16#18012765—24no—
17C2×Dic604C2212yescomplemented
18C2×#14558994C2212yescomplemented
19C2×#14558993C2212yescomplemented
20C2×#14558992C2212yes—
21C2×#14558992C2212yescomplemented
22C2×#14558989C2212yes—
23C2×#14558988C2211yescharacteristic
24#18012737C2211yescharacteristic, complemented
25#18012735C2211yescharacteristic, complemented
26#18012728C2211yescharacteristic, complemented
27#18012726—22no—
28#18012726C2212yes—
29#18012726C2212yes—
30#18012726C2211yescharacteristic, complemented
31#18012711—22no—
32C4×C22⋊D151—22no—
33C2×C22⋊Dic151—22no—
34C2×C4×Dic151C2211yescharacteristic
35#18012649C2212yescomplemented
36C8×Dic151C2212yescomplemented
37C2×#41×C151C2211yescharacteristic
38C22×D4×C151C2211yescharacteristic
39AGL(1,ℤ/8)×C151—24no—
40C2×SD16×C151C2212yes—
41C2×D8×C151C2212yes—
42C2×M16×C151C2211yescharacteristic
43#14559004—22no—
44#14559004—22no—
45C2×Dic302—21no—
46C2×Dic302—22no—
47C2×Dic302—22no—
48C2×Dic302C2312yes—
49C2×Dic302C2311yescharacteristic
50C2×D604C2311yescharacteristic
51C2×C22⋊D151—22no—
52C2×C22⋊D151—21no—
53C22×Dic151—21no—
54C22×Dic151D412yes—
55C2×C4×D151—21no—
56Dic604—24no—
57#14558994—24no—
58#14558993—24no—
59#14558992—24no—
60#14558992—24no—
61#14558989—24no—
62#14558988D414yescomplemented
63C2×C151⋊C8C2312yes—
64Dic151⋊C4—22no—
65C22⋊Dic151—24no—
66C22⋊Dic151—24no—
67C22⋊Dic151—22no—
68(C2×D151)⋊C4—22no—
69C4×Dic151D412yes—
70C4×Dic151C2311yescharacteristic
71C4×Dic151C2311yescharacteristic
72C24×C151—21no—
73#41×C151—22no—
74#41×C151—22no—
75C2×Q8×C151C2311yescharacteristic
76C2×D4×C151—21no—
77C2×D4×C151—22no—
78C2×D4×C151—22no—
79C2×D4×C151—21no—
80C2×D4×C151C2312yes—
81C2×D4×C151C2311yescharacteristic
82C22×C4×C151—21no—
83C22×C4×C151C2311yescharacteristic
84SD16×C151—24no—
85D8×C151—24no—
86M16×C151D414yescomplemented
87C2×C8×C151C2312yes—
88C22×D151—21no—
89Dic302—21no—
90Dic302—22no—
91Dic302—22no—
92Dic302—21no—
93Dic302—42no—
94D604—21no—
95D604—21no—
96C22⋊D151—44no—
97C22⋊D151—42no—
98C2×Dic151—22no—
99C2×Dic151—21no—
100C2×Dic151—42no—
101C2×Dic151—42no—
102C2×Dic151C2×D412yes—
103C2×Dic151C2×D412yes—
104C4×D151—42no—
105C151⋊C8C2×D414yes—
106C23×C151—22no—
107C23×C151—21no—
108C23×C151—44no—
109C23×C151—42no—
110C23×C151C2×D411yescharacteristic
111Q8×C151—21no—
112Q8×C151—21no—
113D4×C151—21no—
114D4×C151—21no—
115D4×C151—22no—
116D4×C151—22no—
117D4×C151—42no—
118D4×C151—42no—
119C2×C4×C151—21no—
120C2×C4×C151—42no—
121C2×C4×C151C2×D411yescharacteristic
122C2×C4×C151C2×D412yes—
123C2×C4×C151C2×D412yes—
124C2×C4×C151C2411yescharacteristic
125C8×C151C2×D414yes—
126C22×C151—22no—
127C22×C151—41no—
128C22×C151—42no—
129C22×C151—44no—
130C22×C151—42no—
131C22×C151—84no—
132C22×C151#12712yes—
133C22×C151C22×D411yescharacteristic
134C2×D151—41no—
135C2×D151—42no—
136C4×C151—41no—
137C4×C151#12712yes—
138C4×C151C22×D411yescharacteristic
139C4×C151C22×D411yescharacteristic
140Dic151—22no—
141Dic151—22no—
142Dic151—41no—
143Dic151—42no—
144C2×C151—22no—
145C2×C151—81no—
146C2×C151—82no—
147C2×C151C2×#12711yescharacteristic
148C2×C151#57512yes—
149D151—81no—
150C151#315311yescharacteristic, complemented, Sylow
151#3153—1511nomaximal, Sylow
152C2×#137—1511no—
153C2×#137—1511no—
154C2×AGL(1,ℤ/8)—1511no—
155C2×AGL(1,ℤ/8)D15111yescharacteristic, complemented
156#532—1511no—
157C2×#124—1511no—
158#497—1512no—
159#496—1512no—
160#495—1512no—
161#495—1512no—
162C4×M16—1511no—
163C2×#41—1511no—
164C2×#41C2×D15111yescharacteristic
165C22×Q8—1511no—
166C22×D4C2×D15111yescharacteristic
167#137—3024no—
168#137—3024no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—3024no—
171C2×Q16—1512no—
172C2×Q16—1512no—
173C2×SD16—1512no—
174C2×SD16—1512no—
175C2×SD16—1512no—
176C2×SD16C2×D15112yes—
177C2×D8—1512no—
178C2×D8C2×D15112yes—
179C2×M16—1511no—
180C2×M16C2×D15111yescharacteristic
181C4⋊Q8—1511no—
182#127—1511no—
183#124—3022no—
184#124—1511no—
185#124—1512no—
186#124—1512no—
187#124—1511no—
188#121—3022no—
189C4×D4—3022no—
190C2×C22⋊C4—3022no—
191C2×C42—1511no—
192#97—1512no—
193C4×C8—1512no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—3022no—
198#41—3022no—
199C2×Q8—3021no—
200C2×Q8—3022no—
201C2×Q8—3022no—
202C2×Q8—1512no—
203C2×Q8—1511no—
204C2×Q8C22×D15111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3021no—
210C2×D4—3022no—
211C2×D4—1511no—
212C2×D4C22×D15112yes—
213C2×D4C22×D15111yescharacteristic
214C22×C4—21no—
215C22×C4—3021no—
216C22×C4—3021no—
217C22×C4—1512no—
218C22×C4C22×D15111yescharacteristic
219Q16—3024no—
220Q16—3024no—
221SD16—24no—
222SD16—3024no—
223SD16—3024no—
224SD16—3024no—
225D8—24no—
226D8—3024no—
227M16—1514no—
228M16C22⋊D15114yescomplemented
229C2×C8—1512no—
230C2×C8C22×D15112yes—
231C4⋊C4—3022no—
232C22⋊C4—3022no—
233C22⋊C4—3024no—
234C22⋊C4—3024no—
235C22⋊C4—3022no—
236C42—1512no—
237C42—1511no—
238C42—1511no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—3021no—
244C23D4×D15111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—3021no—
248Q8—3022no—
249Q8—3022no—
250Q8—3021no—
251Q8—6042no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3021no—
259D4—3021no—
260D4—6044no—
261D4—6042no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—3022no—
265C2×C4—3021no—
266C2×C4—6042no—
267C2×C4—6042no—
268C2×C4—6042no—
269C2×C4—1512no—
270C2×C4—1512no—
271C2×C4D4×D15111yescharacteristic
272C2×C4D4×D15112yes—
273C2×C4C2×C22⋊D15112yes—
274C2×C4C23×D15111yescharacteristic
275C8—1514no—
276C8C2×C22⋊D15114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—6041no—
284C22—6042no—
285C22#1801273512yes—
286C22C2×D4×D15111yescharacteristic
287C4—41no—
288C4—3022no—
289C4—3022no—
290C4—6041no—
291C4—6042no—
292C4#1801273512yes—
293C4C2×D4×D15111yescharacteristic
294C4C22×C22⋊D15111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—12081no—
299C2C2×#1801273511yescharacteristic
300C2#2375300812yes—
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×AGL(1,ℤ/8)×C151C2
socleC22×C151C22×D4
hypercenterC2×AGL(1,ℤ/8)D151
hypocenterC151#3153
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#18012772C2semidirect
2. C2×#18012769C2semidirect
3. C2×#18012765C2semidirect
4. #23752758C2semidirect
5. C2×#18012726C2semidirect
6. #23752603C2semidirect
7. #23752600C2semidirect
8. #23752595C2semidirect
9. #23752593C2semidirect
10. M16×Dic151C2semidirect
11. C2×AGL(1,ℤ/8)×C151C2semidirect
17. C2×Dic604C22semidirect
18. C2×#14558994C22semidirect
19. C2×#14558993C22semidirect
21. C2×#14558992C22semidirect
24. #18012737C22semidirect
25. #18012735C22semidirect
26. #18012728C22semidirect
30. #18012726C22semidirect
35. #18012649C22semidirect
36. C8×Dic151C22semidirect
62. #14558988D4semidirect
86. M16×C151D4semidirect
150. C151#3153semidirect
155. C2×AGL(1,ℤ/8)D151semidirect
228. M16C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127722
C2×#180127692
C2×#180127652
#237527582
C2×#180127262
#237526032
#237526002
#237525952
#237525932
M16×Dic1512
C2×AGL(1,ℤ/8)×C1512
#3153151