Finite Group Databasebeta

← #49169261

Subgroups — #49169261

Order 37504.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22627657C211yesmaximal, characteristic, complemented
2C2×#22627652C211yesmaximal, characteristic, complemented
3C2×#22627646C211yesmaximal, characteristic, complemented
4C22×#17989791C211yesmaximal, characteristic, complemented
5#34036705C211yesmaximal, characteristic, complemented
6#34036698C211yesmaximal, characteristic, complemented
7#34036523C212yesmaximal, complemented
8#34036518C212yesmaximal, complemented
9#34036513C212yesmaximal, complemented
10#34036502C212yesmaximal, complemented
11#405×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C2×#17989802C2211yescharacteristic
14C22×Dic586C2211yescharacteristic
15C22×D1172C2211yescharacteristic
16#22627657—24no—
17#22627652—24no—
18#22627646—24no—
19C2×Dic1172C2212yescomplemented
20C2×Dic1172C2211yescharacteristic, complemented
21C2×#17989791—24no—
22C2×#17989791C2212yes—
23C2×#17989791C2212yes—
24C2×#17989791C2212yes—
25C2×#17989791C2211yescharacteristic, complemented
26C2×#17989791C2212yescomplemented
27C2×#17989791C2211yescharacteristic, complemented
28C2×D2344C2212yes—
29C2×D2344C2211yescharacteristic, complemented
30#22627624C2212yescomplemented
31#22627621C2212yes—
32#22627612C2212yes—
33#22627612C2212yescomplemented
34#22627608—22no—
35#22627604—22no—
36#22627599—22no—
37#22627594—22no—
38C2×M16×C293C2211yescharacteristic
39C22×C8×C293C2211yescharacteristic
40#117×C293C2211yescharacteristic
41#97×C293C2212yescomplemented
42C4×C8×C293C2212yescomplemented
43C23×D293—21no—
44#17989802—22no—
45#17989802—22no—
46#17989802—22no—
47#17989802—22no—
48C2×Dic586—21no—
49C2×Dic586—22no—
50C2×Dic586—22no—
51C2×Dic586C2311yescharacteristic
52C2×Dic586C2312yes—
53C2×Dic586C2311yescharacteristic
54C2×D1172—22no—
55C2×D1172—22no—
56C2×D1172—21no—
57C2×D1172—24no—
58C2×D1172C2312yes—
59C2×D1172C2311yescharacteristic
60C2×D1172C2311yescharacteristic
61C2×C22⋊D293—21no—
62C2×C22⋊D293—21no—
63C22×Dic293—21no—
64C2×C4×D293—21no—
65C2×C4×D293—21no—
66Dic1172—22no—
67Dic1172—24no—
68#17989791—24no—
69#17989791—24no—
70#17989791—24no—
71#17989791—22no—
72#17989791—22no—
73#17989791—24no—
74D2344—24no—
75D2344—22no—
76C4⋊Dic293—24no—
77#17989779—24no—
78#17989779—24no—
79C22×C4×C293C2311yescharacteristic
80M16×C293D414yescomplemented
81C2×C8×C293D414yescomplemented
82C2×C8×C293C2312yes—
83C2×C8×C293C2311yescharacteristic
84C2×C8×C293C2311yescharacteristic
85C4⋊C4×C293D412yes—
86C22⋊C4×C293D412yes—
87C42×C293C2312yes—
88C22×D293—21no—
89C22×D293—22no—
90C22×D293—21no—
91C22×D293—42no—
92C22×D293—82no—
93Dic586—22no—
94Dic586—21no—
95Dic586—22no—
96Dic586—21no—
97Dic586—21no—
98Dic586—21no—
99Dic586—42no—
100D1172—21no—
101D1172—22no—
102D1172—21no—
103D1172—21no—
104D1172—22no—
105D1172—21no—
106D1172—42no—
107D1172—84no—
108C22⋊D293—42no—
109C22⋊D293—42no—
110C2×Dic293—21no—
111C2×Dic293—22no—
112C2×Dic293—21no—
113C2×Dic293—42no—
114C4×D293—42no—
115C4×D293—42no—
116C23×C293C2×D411yescharacteristic
117C2×C4×C293C2×D414yes—
118C2×C4×C293C2×D411yescharacteristic
119C2×C4×C293C2×D412yes—
120C2×C4×C293C2×D412yes—
121C2×C4×C293C2411yescharacteristic
122C8×C293C2×D412yes—
123C8×C293C2×D412yes—
124C8×C293C2×D414yes—
125C22×C293—22no—
126C22×C293#12712yes—
127C22×C293C22×D411yescharacteristic
128C2×D293—41no—
129C2×D293—41no—
130C2×D293—42no—
131C2×D293—42no—
132C2×D293—44no—
133C2×D293—42no—
134C2×D293—84no—
135C4×C293—24no—
136C4×C293#12712yes—
137C4×C293C22×D411yescharacteristic
138C4×C293C22×D411yescharacteristic
139Dic293—41no—
140Dic293—41no—
141Dic293—42no—
142C2×C293—22no—
143C2×C293C2×#12711yescharacteristic
144C2×C293#57712yes—
145D293—81no—
146D293—81no—
147D293—82no—
148C293#315411yescharacteristic, complemented, Sylow
149#3154—2931nomaximal, Sylow
150C2×#137—2931no—
151C2×AGL(1,ℤ/8)—2931no—
152C2×#135—2931no—
153C22×SD16—2931no—
154#536—2931no—
155#535—2931no—
156#497—2932no—
157#496—2932no—
158#495—2932no—
159#492—2932no—
160#405D29311yescharacteristic, complemented
161C2×#41—2931no—
162C2×#41—2931no—
163C22×Q8—2931no—
164C22×D4—2931no—
165#137—5864no—
166AGL(1,ℤ/8)—5864no—
167#135—5864no—
168C2×Q16—2931no—
169C2×Q16—2932no—
170C2×SD16—2932no—
171C2×SD16—2932no—
172C2×SD16—2932no—
173C2×SD16—2931no—
174C2×SD16—2931no—
175C2×SD16—2932no—
176C2×SD16—5864no—
177C2×D8—2932no—
178C2×D8—2931no—
179C2×M16C2×D29311yescharacteristic
180C22×C8C2×D29311yescharacteristic
181C4⋊Q8—2932no—
182#127—2932no—
183#124—2932no—
184#124—2932no—
185#123—5862no—
186C22⋊Q8—5862no—
187#121—5862no—
188#120—5862no—
189#117C2×D29311yescharacteristic
190#97C2×D29312yescomplemented
191C4×C8C2×D29312yescomplemented
192C24—5861no—
193#41—5862no—
194#41—5862no—
195#41—5862no—
196#41—5862no—
197C2×Q8—2931no—
198C2×Q8—2932no—
199C2×Q8—2931no—
200C2×Q8—5861no—
201C2×Q8—5862no—
202C2×Q8—5862no—
203C2×D4—2931no—
204C2×D4—2932no—
205C2×D4—2931no—
206C2×D4—5861no—
207C2×D4—5861no—
208C2×D4—5864no—
209C2×D4—5862no—
210C2×D4—5862no—
211C2×D4—5861no—
212C22×C4—5861no—
213C22×C4—5861no—
214C22×C4—5861no—
215C22×C4C22×D29311yescharacteristic
216Q16—5862no—
217Q16—5864no—
218SD16—5864no—
219SD16—5864no—
220SD16—5862no—
221SD16—5862no—
222SD16—5864no—
223SD16—5864no—
224D8—5862no—
225D8—5864no—
226M16D117214yescomplemented
227C2×C8D117214yescomplemented
228C2×C8C22×D29311yescharacteristic
229C2×C8C22×D29312yes—
230C2×C8C22×D29311yescharacteristic
231C4⋊C4—5864no—
232C4⋊C4D117212yes—
233C22⋊C4—5864no—
234C22⋊C4—5864no—
235C22⋊C4D117212yes—
236C42C22×D29312yes—
237C23—23442no—
238C23—5861no—
239C23—5862no—
240C23—5861no—
241C23—11722no—
242C23C2×D117211yescharacteristic
243Q8—5862no—
244Q8—5861no—
245Q8—5862no—
246Q8—5861no—
247Q8—5861no—
248Q8—5861no—
249Q8—11722no—
250D4—23444no—
251D4—5861no—
252D4—5862no—
253D4—5861no—
254D4—5861no—
255D4—5862no—
256D4—5861no—
257D4—11722no—
258D4—11722no—
259D4—11722no—
260C2×C4—5861no—
261C2×C4—5862no—
262C2×C4—5861no—
263C2×C4—11722no—
264C2×C4—11722no—
265C2×C4—11722no—
266C2×C4C2×D117214yes—
267C2×C4C2×D117211yescharacteristic
268C2×C4C2×D117212yes—
269C2×C4C2×D117212yes—
270C2×C4C23×D29311yescharacteristic
271C8C2×D117212yes—
272C8C2×D117212yes—
273C8C2×D117214yes—
274C22—22no—
275C22—23444no—
276C22—11721no—
277C22—11721no—
278C22—11722no—
279C22—11722no—
280C22—11724no—
281C22—11722no—
282C22#2262762112yes—
283C22C22×D117211yescharacteristic
284C4—24no—
285C4—11721no—
286C4—11721no—
287C4—11722no—
288C4#2262762112yes—
289C4C22×D117211yescharacteristic
290C4C22×D117211yescharacteristic
291C2—22no—
292C2—23442no—
293C2—23441no—
294C2—23441no—
295C2C2×#2262762111yescharacteristic
296C2#3403694112yes—
subgroupG/H
centerC22C22×D1172
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#405×C293C2
socleC22×C293C22×D4
hypercenter#405D293
hypocenterC293#3154
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#22627657C2semidirect
2. C2×#22627652C2semidirect
3. C2×#22627646C2semidirect
4. C22×#17989791C2semidirect
5. #34036705C2semidirect
6. #34036698C2semidirect
7. #34036523C2semidirect
8. #34036518C2semidirect
9. #34036513C2semidirect
10. #34036502C2semidirect
11. #405×C293C2semidirect
19. C2×Dic1172C22semidirect
20. C2×Dic1172C22semidirect
25. C2×#17989791C22semidirect
26. C2×#17989791C22semidirect
27. C2×#17989791C22semidirect
29. C2×D2344C22semidirect
30. #22627624C22semidirect
33. #22627612C22semidirect
41. #97×C293C22semidirect
42. C4×C8×C293C22semidirect
80. M16×C293D4semidirect
81. C2×C8×C293D4semidirect
148. C293#3154semidirect
160. #405D293semidirect
190. #97C2×D293semidirect
191. C4×C8C2×D293semidirect
226. M16D1172semidirect
227. C2×C8D1172semidirect

Maximal subgroups

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

subgroupindex
C2×#226276572
C2×#226276522
C2×#226276462
C22×#179897912
#340367052
#340366982
#340365232
#340365182
#340365132
#340365022
#405×C2932
#3154293