Finite Group Databasebeta

← #16829969

Subgroups — #16829969

Order 3712.

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

normal subgroups only

This group has 564 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×#11745313C211yesmaximal, characteristic, complemented
2C2×#11745311C211yesmaximal, characteristic, complemented
3C22×#90773C211yesmaximal, characteristic, complemented
4#14271734C211yesmaximal, characteristic, complemented
5#14271727C211yesmaximal, characteristic, complemented
6#14271556C212yesmaximal, complemented
7#14271551C212yesmaximal, complemented
8#14271544C212yesmaximal, complemented
9#14271534C212yesmaximal, complemented
10#14271118C211yesmaximal, characteristic, complemented
11C2×C29×#135C211yesmaximal, characteristic, complemented
12C2×#41×C29C2211yescharacteristic
13C2×#41×C29C2211yescharacteristic
14C22×D116C2211yescharacteristic
15C22×Dic58C2211yescharacteristic
16C29×#135—24no—
17C2×Q16×C29C2211yescharacteristic, complemented
18C2×SD16×C29C2211yescharacteristic, complemented
19C2×SD16×C29C2211yescharacteristic, complemented
20C2×D8×C29C2211yescharacteristic, complemented
21C22×C8×C29C2211yescharacteristic
22#11745313—24no—
23#11745311—24no—
24#11745309C2212yes—
25#11745306C2212yescomplemented
26C2×#90785C2212yescomplemented
27C2×#90784C2212yes—
28#11745302C2212yes—
29#11745299—22no—
30#11745298C2212yescomplemented
31#11745297—22no—
32C2×#90783C2212yescomplemented
33C2×#90782C2212yes—
34#11745292—22no—
35#11745288C2211yescharacteristic
36#11745286—22no—
37C2×#90777C2211yescharacteristic
38C2×#90773—24no—
39C2×#90773C2212yes—
40C2×#90773C2212yes—
41#11745163C2212yescomplemented
42C8×Dic29C2212yescomplemented
43C23×D29—21no—
44#41×C29—22no—
45#41×C29—22no—
46#41×C29—22no—
47#41×C29—22no—
48C2×Q8×C29C2311yescharacteristic
49C2×Q8×C29C2311yescharacteristic
50C2×D4×C29—21no—
51C2×D4×C29—21no—
52C2×D4×C29C2311yescharacteristic
53C2×D4×C29C2311yescharacteristic
54C22×C4×C29—21no—
55C22×C4×C29—21no—
56C22×C4×C29C2311yescharacteristic
57C2×C22⋊D29—24no—
58C22×Dic29—21no—
59C2×D116—22no—
60C2×D116—22no—
61C2×D116—21no—
62C2×D116C2312yes—
63C2×Dic58—22no—
64C2×Dic58—22no—
65C2×Dic58—21no—
66C2×Dic58C2312yes—
67Q16×C29—22no—
68SD16×C29—22no—
69SD16×C29—22no—
70D8×C29—22no—
71C2×C8×C29D414yescomplemented
72C2×C8×C29C2311yescharacteristic
73C2×C8×C29C2311yescharacteristic
74C22⋊Dic29—24no—
75C22⋊Dic29D412yes—
76#90785—24no—
77#90784—24no—
78#90783—24no—
79#90782—24no—
80#90781—24no—
81C4⋊Dic29D412yes—
82#90779—24no—
83C4×Dic29C2312yes—
84#90777D414yescomplemented
85C2×#6597C2312yes—
86#90773—24no—
87#90773—24no—
88C23×C29—21no—
89C23×C29—21no—
90C23×C29C2×D411yescharacteristic
91C22×D29—22no—
92C22×D29—42no—
93C22×D29—82no—
94Q8×C29—21no—
95Q8×C29—21no—
96Q8×C29—21no—
97Q8×C29—21no—
98D4×C29—21no—
99D4×C29—21no—
100D4×C29—21no—
101D4×C29—21no—
102D4×C29—42no—
103D4×C29—42no—
104C2×C4×C29—21no—
105C2×C4×C29—21no—
106C2×C4×C29—42no—
107C2×C4×C29—42no—
108C2×C4×C29C2×D411yescharacteristic
109C2×C4×C29C2×D412yes—
110C2×C4×C29C2×D412yes—
111C2×C4×C29C2411yescharacteristic
112C22⋊D29—84no—
113C2×Dic29—22no—
114C2×Dic29—42no—
115C2×Dic29C2×D414yes—
116D116—22no—
117D116—22no—
118D116—42no—
119Dic58—22no—
120Dic58—22no—
121Dic58—42no—
122C8×C29C2×D412yes—
123C8×C29C2×D412yes—
124#6597C2×D414yes—
125#3154—291nomaximal, Sylow
126C22×C29—22no—
127C22×C29—41no—
128C22×C29—41no—
129C22×C29—42no—
130C22×C29—42no—
131C22×C29#12712yes—
132C22×C29C22×D411yescharacteristic
133C2×D29—42no—
134C2×D29—44no—
135C2×D29—84no—
136C4×C29—41no—
137C4×C29—41no—
138C4×C29#12712yes—
139C4×C29C22×D411yescharacteristic
140C4×C29C22×D411yescharacteristic
141Dic29—24no—
142Dic29—42no—
143C2×#137—291no—
144C2×AGL(1,ℤ/8)—291no—
145C2×#135D2911yescharacteristic, complemented
146C22×SD16—291no—
147#536—291no—
148#535—291no—
149#497—292no—
150#496—292no—
151#495—292no—
152#492—292no—
153#405—291no—
154C2×C29—22no—
155C2×C29—81no—
156C2×C29—81no—
157C2×C29C2×#12711yescharacteristic
158C2×C29#57712yes—
159D29—82no—
160C2×#41C2×D2911yescharacteristic
161C2×#41C2×D2911yescharacteristic
162C22×Q8—291no—
163C22×D4—291no—
164#137—584no—
165AGL(1,ℤ/8)—584no—
166#135—24no—
167C2×Q16—292no—
168C2×Q16C2×D2911yescharacteristic, complemented
169C2×SD16—292no—
170C2×SD16—292no—
171C2×SD16—292no—
172C2×SD16—292no—
173C2×SD16—584no—
174C2×SD16C2×D2911yescharacteristic, complemented
175C2×SD16C2×D2911yescharacteristic, complemented
176C2×D8—292no—
177C2×D8C2×D2911yescharacteristic, complemented
178C2×M16—291no—
179C22×C8C2×D2911yescharacteristic
180C4⋊Q8—292no—
181#127—292no—
182#124—292no—
183#124—292no—
184#123—582no—
185C22⋊Q8—582no—
186#121—582no—
187#120—582no—
188#117—291no—
189#97—292no—
190C4×C8—292no—
191C29#315411yescharacteristic, complemented, Sylow
192C24—581no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—292no—
198C2×Q8—581no—
199C2×Q8—582no—
200C2×Q8—582no—
201C2×Q8C22×D2911yescharacteristic
202C2×Q8C22×D2911yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—292no—
206C2×D4—584no—
207C2×D4—582no—
208C2×D4—581no—
209C2×D4—582no—
210C2×D4C22×D2911yescharacteristic
211C2×D4C22×D2911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—581no—
215C22×C4C22×D2911yescharacteristic
216Q16—22no—
217Q16—584no—
218SD16—22no—
219SD16—22no—
220SD16—584no—
221SD16—584no—
222SD16—584no—
223SD16—584no—
224D8—22no—
225D8—584no—
226M16—294no—
227C2×C8—292no—
228C2×C8C22⋊D2914yescomplemented
229C2×C8C22×D2911yescharacteristic
230C2×C8C22×D2911yescharacteristic
231C4⋊C4—292no—
232C4⋊C4—584no—
233C22⋊C4—292no—
234C22⋊C4—584no—
235C22⋊C4—584no—
236C42—292no—
237C23—21no—
238C23—21no—
239C23—582no—
240C23—1162no—
241C23—2322no—
242C23D4×D2911yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—582no—
248Q8—582no—
249Q8—1162no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—582no—
257D4—582no—
258D4—1162no—
259D4—2324no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—294no—
265C2×C4—582no—
266C2×C4—1162no—
267C2×C4D4×D2911yescharacteristic
268C2×C4D4×D2912yes—
269C2×C4C2×C22⋊D2912yes—
270C2×C4C23×D2911yescharacteristic
271C8—294no—
272C8C2×C22⋊D2912yes—
273C8C2×C22⋊D2912yes—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—1162no—
280C22—1164no—
281C22—2324no—
282C22#1174530212yes—
283C22C2×D4×D2911yescharacteristic
284C4—41no—
285C4—41no—
286C4—584no—
287C4—1162no—
288C4#1174530212yes—
289C4C2×D4×D2911yescharacteristic
290C4C22×C22⋊D2911yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—2322no—
295C2C2×#1174530211yescharacteristic
296C2#1427197012yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×#135C2
socleC22×C29C22×D4
hypercenterC2×#135D29
hypocenterC29#3154
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#11745313C2semidirect
2. C2×#11745311C2semidirect
3. C22×#90773C2semidirect
4. #14271734C2semidirect
5. #14271727C2semidirect
6. #14271556C2semidirect
7. #14271551C2semidirect
8. #14271544C2semidirect
9. #14271534C2semidirect
10. #14271118C2semidirect
11. C2×C29×#135C2semidirect
17. C2×Q16×C29C22semidirect
18. C2×SD16×C29C22semidirect
19. C2×SD16×C29C22semidirect
20. C2×D8×C29C22semidirect
25. #11745306C22semidirect
26. C2×#90785C22semidirect
30. #11745298C22semidirect
32. C2×#90783C22semidirect
41. #11745163C22semidirect
42. C8×Dic29C22semidirect
71. C2×C8×C29D4semidirect
84. #90777D4semidirect
145. C2×#135D29semidirect
168. C2×Q16C2×D29semidirect
174. C2×SD16C2×D29semidirect
175. C2×SD16C2×D29semidirect
177. C2×D8C2×D29semidirect
191. C29#3154semidirect
228. C2×C8C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117453132
C2×#117453112
C22×#907732
#142717342
#142717272
#142715562
#142715512
#142715442
#142715342
#142711182
C2×C29×#1352
#315429