Finite Group Databasebeta

← #49170387

Subgroups — #49170387

Order 37504.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22627652C212yesmaximal, complemented
2#34036784C211yesmaximal, characteristic, complemented
3D4×D1172C211yesmaximal, characteristic, complemented
4#34036502C211yesmaximal, characteristic, complemented
5#34036375C212yesmaximal, complemented
6#34036335C211yesmaximal, characteristic, complemented
7#34036325C211yesmaximal, characteristic, complemented
8#34036282C212yesmaximal, complemented
9#34036276C212yesmaximal, complemented
10C4×D2344C211yesmaximal, characteristic, complemented
11#436×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2212yescomplemented
13C22×D1172C2212yes—
14C2×D4×D293—21no—
15#22627652—28no—
16C2×#17989791C2212yes—
17C2×#17989791C2212yescomplemented
18C2×#17989791C2212yescomplemented
19C2×D2344C2211yescharacteristic
20C2×D2344C2212yescomplemented
21#22627624C2211yescharacteristic, complemented
22#22627621C2211yescharacteristic
23#22627606—22no—
24#22627603C2212yescomplemented
25#22627599—21no—
26#22627598C2212yes—
27#22627595—21no—
28#22627594—22no—
29C4×D1172C2211yescharacteristic
30C4×D1172C2211yescharacteristic, complemented
31C4⋊C4×D293—21no—
32#22627558C2211yescharacteristic, complemented
33#22627549C2212yescomplemented
34#22627546C2211yescharacteristic
35#22627546C2211yescharacteristic, complemented
36#22627546C2212yescomplemented
37C2×M16×C293C2212yescomplemented
38C4×D4×C293C2211yescharacteristic
39C4⋊C8×C293C2211yescharacteristic, complemented
40C22⋊C8×C293C2212yescomplemented
41C4×C8×C293C2211yescharacteristic, complemented
42C23×D293—22no—
43#17989802—24no—
44#17989802—42no—
45C2×Dic586C2312yes—
46C2×D1172—24no—
47C2×D1172—22no—
48C2×D1172—21no—
49C2×D1172—21no—
50C2×D1172—42no—
51C2×D1172C2312yes—
52C2×D1172C2311yescharacteristic
53C2×D1172C2311yescharacteristic
54C2×C22⋊D293—22no—
55C2×C22⋊D293—22no—
56D4×D293—44no—
57C2×C4×D293—21no—
58C2×C4×D293—21no—
59C2×C4×D293—22no—
60#17989791—24no—
61#17989791—24no—
62#17989791—42no—
63D2344—24no—
64D2344D414yescomplemented
65C4⋊Dic293—21no—
66C4⋊Dic293—21no—
67C4⋊Dic293C2311yescharacteristic
68C4⋊Dic293C2311yescharacteristic
69#17989781—22no—
70C22⋊Dic293—22no—
71#17989779—21no—
72#17989779—21no—
73#17989779—22no—
74C2×D4×C293D411yescharacteristic
75C22×C4×C293C2312yes—
76M16×C293—24no—
77C2×C8×C293C2312yes—
78C2×C8×C293C2311yescharacteristic
79C2×C8×C293C2311yescharacteristic
80C4⋊C4×C293D411yescharacteristic
81C22⋊C4×C293D412yes—
82C42×C293C2311yescharacteristic
83C22×D293—21no—
84C22×D293—21no—
85C22×D293—22no—
86C22×D293—44no—
87C22×D293—44no—
88C22×D293—82no—
89Dic586—22no—
90Dic586—22no—
91D1172—22no—
92D1172—21no—
93D1172—21no—
94D1172—22no—
95D1172—44no—
96D1172—42no—
97D1172—81no—
98D1172C2×D412yes—
99D1172C2×D412yescomplemented
100C22⋊D293—44no—
101C22⋊D293—44no—
102C2×Dic293—21no—
103C2×Dic293—21no—
104C2×Dic293—22no—
105C4×D293—44no—
106C4×D293—42no—
107C4×D293—42no—
108C23×C293C2×D412yes—
109D4×C293—22no—
110C2×C4×C293—22no—
111C2×C4×C293—22no—
112C2×C4×C293C2×D411yescharacteristic
113C2×C4×C293C2×D412yes—
114C2×C4×C293C2×D411yescharacteristic
115C2×C4×C293C2411yescharacteristic
116C8×C293—22no—
117C8×C293—21no—
118C8×C293C2×D412yescomplemented
119C22×C293—22no—
120C22×C293—22no—
121C22×C293—22no—
122C22×C293C22×D411yescharacteristic
123C2×D293—22no—
124C2×D293—22no—
125C2×D293—41no—
126C2×D293—41no—
127C2×D293—41no—
128C2×D293—41no—
129C2×D293—42no—
130C2×D293—42no—
131C2×D293—42no—
132C2×D293—88no—
133C4×C293—22no—
134C4×C293—21no—
135C4×C293AGL(1,ℤ/8)12yescomplemented
136C4×C293C22×D411yescharacteristic
137C4×C293H(5,𝔽2)11yescharacteristic
138Dic293—42no—
139Dic293—41no—
140Dic293—41no—
141C2×C293—42no—
142C2×C293D4211yescharacteristic
143C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C293#57711yescharacteristic
145D293—42no—
146D293—42no—
147D293—82no—
148C293#327211yescharacteristic, complemented, Sylow
149#3272—2931nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2932no—
151#547—2931no—
152D42—2931no—
153#492—2931no—
154#468—2932no—
155#461—2931no—
156#459—2931no—
157#450—2932no—
158#449—2932no—
159C4×D8—2931no—
160#436D29311yescharacteristic, complemented
161C2×#41—2932no—
162C22×D4—2932no—
163C22×D4—5861no—
164AGL(1,ℤ/8)—5868no—
165C2×SD16—2932no—
166C2×SD16—2932no—
167C2×SD16—2932no—
168C2×D8—2931no—
169C2×D8—2932no—
170C2×M16C2×D29312yescomplemented
171C4⋊Q8—2931no—
172#127—2931no—
173#123—5862no—
174C22⋊Q8—2932no—
175#121—2932no—
176#121—5861no—
177#121—5861no—
178#120—5862no—
179C4×D4—2931no—
180C4×D4—2931no—
181C4×D4C2×D29311yescharacteristic
182C2×C4⋊C4—5861no—
183#107—2931no—
184C4⋊C8C2×D29311yescharacteristic, complemented
185#103—2932no—
186#102—2931no—
187#102—2931no—
188#102—2932no—
189C22⋊C8C2×D29312yescomplemented
190C4×C8C2×D29311yescharacteristic, complemented
191C24—5862no—
192#41—5864no—
193#41—11722no—
194C2×Q8—2932no—
195C2×D4—2931no—
196C2×D4—2931no—
197C2×D4—2932no—
198C2×D4—5864no—
199C2×D4—5862no—
200C2×D4—5862no—
201C2×D4—5861no—
202C2×D4—5862no—
203C2×D4—5861no—
204C2×D4—11722no—
205C2×D4—11724no—
206C2×D4D117211yescharacteristic
207C22×C4—5861no—
208C22×C4—5862no—
209C22×C4—5861no—
210C22×C4C22×D29312yes—
211SD16—5864no—
212SD16—5864no—
213SD16—11722no—
214D8—2934no—
215D8—5864no—
216M16—24no—
217C2×C8C22×D29311yescharacteristic
218C2×C8C22×D29312yes—
219C2×C8C22×D29311yescharacteristic
220C4⋊C4—2931no—
221C4⋊C4—2931no—
222C4⋊C4—5861no—
223C4⋊C4—5862no—
224C4⋊C4—5861no—
225C4⋊C4D117211yescharacteristic
226C22⋊C4—5862no—
227C22⋊C4—5861no—
228C22⋊C4—5861no—
229C22⋊C4—5862no—
230C22⋊C4D117212yes—
231C42C22×D29311yescharacteristic
232C23—23442no—
233C23—5861no—
234C23—5861no—
235C23—5862no—
236C23—11724no—
237C23—11724no—
238C23C2×D117212yes—
239Q8—5862no—
240Q8—5862no—
241D4—22no—
242D4—2932no—
243D4—2932no—
244D4—23441no—
245D4—5862no—
246D4—5861no—
247D4—5861no—
248D4—5862no—
249D4—11724no—
250D4—11724no—
251D4—11724no—
252D4—11722no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—5861no—
256C2×C4—5861no—
257C2×C4—5862no—
258C2×C4—11724no—
259C2×C4—11722no—
260C2×C4—11722no—
261C2×C4C2×D117211yescharacteristic
262C2×C4C2×D117212yes—
263C2×C4C2×D117211yescharacteristic
264C2×C4C23×D29311yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D29312yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—23448no—
272C22—5862no—
273C22—5862no—
274C22—11722no—
275C22—11722no—
276C22—11722no—
277C22—11721no—
278C22—11721no—
279C22—11721no—
280C22—11721no—
281C22C22×D117211yescharacteristic
282C4—22no—
283C4—21no—
284C4—11722no—
285C4—11721no—
286C4—11721no—
287C4#2262765212yescomplemented
288C4C2×D4×D29311yescharacteristic
289C4#2262767011yescharacteristic
290C2—42no—
291C2—23442no—
292C2—11722no—
293C2—11722no—
294C2D4×D117211yescharacteristic
295C2C2×#2262765211yescharacteristic
296C2#3403694111yescharacteristic
subgroupG/H
centerC22C22×D1172
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#436×C293C2
socleC22×C293C22×D4
hypercenter#436D293
hypocenterC293#3272
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#22627652C2semidirect
2. #34036784C2semidirect
3. D4×D1172C2semidirect
4. #34036502C2semidirect
5. #34036375C2semidirect
6. #34036335C2semidirect
7. #34036325C2semidirect
8. #34036282C2semidirect
9. #34036276C2semidirect
10. C4×D2344C2semidirect
11. #436×C293C2semidirect
12. C2×#17989802C22semidirect
17. C2×#17989791C22semidirect
18. C2×#17989791C22semidirect
20. C2×D2344C22semidirect
21. #22627624C22semidirect
24. #22627603C22semidirect
30. C4×D1172C22semidirect
32. #22627558C22semidirect
33. #22627549C22semidirect
35. #22627546C22semidirect
36. #22627546C22semidirect
37. C2×M16×C293C22semidirect
39. C4⋊C8×C293C22semidirect
40. C22⋊C8×C293C22semidirect
41. C4×C8×C293C22semidirect
64. D2344D4semidirect
99. D1172C2×D4semidirect
118. C8×C293C2×D4semidirect
135. C4×C293AGL(1,ℤ/8)semidirect
148. C293#3272semidirect
160. #436D293semidirect
170. C2×M16C2×D293semidirect
184. C4⋊C8C2×D293semidirect
189. C22⋊C8C2×D293semidirect
190. C4×C8C2×D293semidirect
267. C8D4×D293semidirect
287. C4#22627652semidirect

Maximal subgroups

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

subgroupindex
C2×#226276522
#340367842
D4×D11722
#340365022
#340363752
#340363352
#340363252
#340362822
#340362762
C4×D23442
#436×C2932
#3272293