Finite Group Databasebeta

← #49170385

Subgroups — #49170385

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×#22627649C212yesmaximal, complemented
2D4×D1172C211yesmaximal, characteristic, complemented
3#34036503C211yesmaximal, characteristic, complemented
4#34036377C212yesmaximal, complemented
5#34036331C211yesmaximal, characteristic, complemented
6#34036324C211yesmaximal, characteristic, complemented
7#34036279C212yesmaximal, complemented
8#34036274C212yesmaximal, complemented
9C4×#17989787C211yesmaximal, characteristic, complemented
10#34036207C211yesmaximal, characteristic, complemented
11#547×C293C211yesmaximal, characteristic, complemented
12C22×D1172C2212yes—
13C2×D4×D293—21no—
14#22627649—28no—
15C2×#17989789C2212yes—
16C2×#17989789C2212yescomplemented
17C2×#17989789C2212yescomplemented
18C2×#17989787C2211yescharacteristic
19C2×#17989787C2212yescomplemented
20C2×#17989786C2212yescomplemented
21#22627621C2211yescharacteristic
22#22627599—21no—
23#22627598C2212yes—
24#22627594—22no—
25C4×D1172C2211yescharacteristic
26#22627557C2211yescharacteristic, complemented
27C4⋊#14556873C2211yescharacteristic, complemented
28#22627548C2212yescomplemented
29#22627545C2211yescharacteristic, complemented
30#22627544C2211yescharacteristic
31#22627544C2212yescomplemented
32C22⋊#14556873C2212yescomplemented
33C4×#14556873C2211yescharacteristic, complemented
34C2×#41×C293C2212yescomplemented
35C4⋊Q8×C293C2211yescharacteristic, complemented
36#123×C293—22no—
37C22⋊Q8×C293C2212yescomplemented
38#121×C293—21no—
39C4×D4×C293C2211yescharacteristic
40C4×D4×C293C2211yescharacteristic, complemented
41C2×C4⋊C4×C293—21no—
42C23×D293—22no—
43C2×D1172—24no—
44C2×D1172—22no—
45C2×D1172—21no—
46C2×D1172—21no—
47C2×D1172—42no—
48C2×D1172C2312yes—
49C2×D1172C2311yescharacteristic
50C2×C22⋊D293—22no—
51D4×D293—44no—
52C2×C4×D293—21no—
53#17989789—24no—
54#17989789—24no—
55#17989789—42no—
56#17989787—24no—
57#17989787D414yescomplemented
58#17989786—24no—
59C2×#14556873C2312yes—
60C2×#14556873C2311yescharacteristic
61C2×#14556873C2311yescharacteristic
62C4⋊Dic293C2311yescharacteristic
63#17989779—21no—
64#17989779—22no—
65#41×C293—24no—
66#41×C293—42no—
67C2×Q8×C293C2312yes—
68C2×D4×C293—22no—
69C2×D4×C293D411yescharacteristic
70C2×D4×C293C2311yescharacteristic
71C22×C4×C293—21no—
72C22×C4×C293—22no—
73C22×C4×C293C2312yes—
74C4⋊C4×C293—21no—
75C4⋊C4×C293—22no—
76C4⋊C4×C293—21no—
77C4⋊C4×C293D411yescharacteristic
78C4⋊C4×C293C2311yescharacteristic
79C22⋊C4×C293—22no—
80C22⋊C4×C293—21no—
81C22⋊C4×C293D412yes—
82C42×C293C2311yescharacteristic
83C22×D293—21no—
84C22×D293—22no—
85C22×D293—44no—
86C22×D293—44no—
87C22×D293—82no—
88D1172—22no—
89D1172—21no—
90D1172—22no—
91D1172—44no—
92D1172—42no—
93D1172—81no—
94D1172C2×D412yes—
95C22⋊D293—44no—
96C2×Dic293—21no—
97C4×D293—42no—
98#14556873—22no—
99#14556873—21no—
100#14556873C2×D412yescomplemented
101C23×C293—21no—
102C23×C293C2×D412yes—
103Q8×C293—22no—
104Q8×C293—22no—
105D4×C293—22no—
106D4×C293—21no—
107D4×C293—44no—
108D4×C293C2×D412yescomplemented
109C2×C4×C293—22no—
110C2×C4×C293—22no—
111C2×C4×C293—21no—
112C2×C4×C293—22no—
113C2×C4×C293—44no—
114C2×C4×C293—42no—
115C2×C4×C293C2×D411yescharacteristic
116C2×C4×C293C2×D412yes—
117C2×C4×C293C2×D411yescharacteristic
118C2×C4×C293C2411yescharacteristic
119C22×C293—22no—
120C22×C293—22no—
121C22×C293—22no—
122C22×C293—22no—
123C22×C293—41no—
124C22×C293—41no—
125C22×C293C22×D411yescharacteristic
126C2×D293—22no—
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×C293—42no—
136C4×C293—41no—
137C4×C293AGL(1,ℤ/8)12yescomplemented
138C4×C293C22×D411yescharacteristic
139C4×C293H(5,𝔽2)11yescharacteristic
140Dic293—41no—
141C2×C293—42no—
142C2×C293—42no—
143C2×C293D4211yescharacteristic
144C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C293#57711yescharacteristic
146D293—42no—
147D293—82no—
148C293#327211yescharacteristic, complemented, Sylow
149#3272—2931nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2932no—
151#547D29311yescharacteristic, complemented
152D42—2931no—
153#492—2931no—
154#468—2932no—
155#461—2931no—
156#459—2931no—
157#450—2932no—
158#449—2932no—
159C4×D8—2931no—
160#436—2931no—
161C2×#41C2×D29312yescomplemented
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×M16—2932no—
171C4⋊Q8C2×D29311yescharacteristic, complemented
172#127—2931no—
173#123—22no—
174C22⋊Q8C2×D29312yescomplemented
175#121—21no—
176#121—2932no—
177#121—5861no—
178#120—5862no—
179C4×D4—2931no—
180C4×D4C2×D29311yescharacteristic
181C4×D4C2×D29311yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—2931no—
184C4⋊C8—2931no—
185#103—2932no—
186#102—2931no—
187#102—2931no—
188#102—2932no—
189C22⋊C8—2932no—
190C4×C8—2931no—
191C24—5862no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D29312yes—
195C2×D4—22no—
196C2×D4—2931no—
197C2×D4—2932no—
198C2×D4—5864no—
199C2×D4—5862no—
200C2×D4—5861no—
201C2×D4—5862no—
202C2×D4—5861no—
203C2×D4—11722no—
204C2×D4—11724no—
205C2×D4C22⋊D29311yescharacteristic
206C2×D4C22×D29311yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—5861no—
210C22×C4C22×D29312yes—
211SD16—5864no—
212SD16—5864no—
213SD16—11722no—
214D8—2934no—
215D8—5864no—
216M16—5864no—
217C2×C8—2931no—
218C2×C8—2931no—
219C2×C8—2932no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—2931no—
224C4⋊C4C22⋊D29311yescharacteristic
225C4⋊C4C22×D29311yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—5861no—
229C22⋊C4—5862no—
230C22⋊C4C22⋊D29312yes—
231C42C22×D29311yescharacteristic
232C23—21no—
233C23—23442no—
234C23—5861no—
235C23—5862no—
236C23—11724no—
237C23—11724no—
238C23C2×C22⋊D29312yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—2932no—
245D4—23441no—
246D4—5862no—
247D4—5861no—
248D4—5862no—
249D4—11724no—
250D4—11724no—
251D4—11722no—
252D4D4×D29312yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—5861no—
260C2×C4—11722no—
261C2×C4C2×C22⋊D29311yescharacteristic
262C2×C4C2×C22⋊D29312yes—
263C2×C4C2×C22⋊D29311yescharacteristic
264C2×C4C23×D29311yescharacteristic
265C8—2932no—
266C8—5862no—
267C8—5861no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—23448no—
275C22—5862no—
276C22—11722no—
277C22—11722no—
278C22—11722no—
279C22—11721no—
280C22—11721no—
281C22C22×C22⋊D29311yescharacteristic
282C4—22no—
283C4—21no—
284C4—42no—
285C4—41no—
286C4—11721no—
287C4#2262764912yescomplemented
288C4C2×D4×D29311yescharacteristic
289C4#2262766911yescharacteristic
290C2—42no—
291C2—42no—
292C2—23442no—
293C2—11722no—
294C2D4×C22⋊D29311yescharacteristic
295C2C2×#2262764911yescharacteristic
296C2#3403693711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#547×C293C2
socleC22×C293C22×D4
hypercenter#547D293
hypocenterC293#3272
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#22627649C2semidirect
2. D4×D1172C2semidirect
3. #34036503C2semidirect
4. #34036377C2semidirect
5. #34036331C2semidirect
6. #34036324C2semidirect
7. #34036279C2semidirect
8. #34036274C2semidirect
9. C4×#17989787C2semidirect
10. #34036207C2semidirect
11. #547×C293C2semidirect
16. C2×#17989789C22semidirect
17. C2×#17989789C22semidirect
19. C2×#17989787C22semidirect
20. C2×#17989786C22semidirect
26. #22627557C22semidirect
27. C4⋊#14556873C22semidirect
28. #22627548C22semidirect
29. #22627545C22semidirect
31. #22627544C22semidirect
32. C22⋊#14556873C22semidirect
33. C4×#14556873C22semidirect
34. C2×#41×C293C22semidirect
35. C4⋊Q8×C293C22semidirect
37. C22⋊Q8×C293C22semidirect
40. C4×D4×C293C22semidirect
57. #17989787D4semidirect
100. #14556873C2×D4semidirect
108. D4×C293C2×D4semidirect
137. C4×C293AGL(1,ℤ/8)semidirect
148. C293#3272semidirect
151. #547D293semidirect
161. C2×#41C2×D293semidirect
171. C4⋊Q8C2×D293semidirect
174. C22⋊Q8C2×D293semidirect
181. C4×D4C2×D293semidirect
252. D4D4×D293semidirect
287. C4#22627649semidirect

Maximal subgroups

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

subgroupindex
C2×#226276492
D4×D11722
#340365032
#340363772
#340363312
#340363242
#340362792
#340362742
C4×#179897872
#340362072
#547×C2932
#3272293