Finite Group Databasebeta

← #49168886

Subgroups — #49168886

Order 37504.

444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#34036684C211yesmaximal, characteristic, complemented
2C4×D4×D293C211yesmaximal, characteristic
3#34036466C212yesmaximal, complemented
4#34036406C211yesmaximal, characteristic, complemented
5#34036335C211yesmaximal, characteristic, complemented
6#34036334C211yesmaximal, characteristic, complemented
7#34036281C212yesmaximal
8#34036228C212yesmaximal, complemented
9#34036226C212yesmaximal, complemented
10#34036185C211yesmaximal, characteristic, complemented
11#474×C293C211yesmaximal, characteristic, complemented
12C2×D4×D293C2211yescharacteristic
13C22×C4×D293—21no—
14C2×#17989791C2212yes—
15C2×#17989790C2212yes—
16#22627624C2211yescharacteristic, complemented
17#22627615—21no—
18#22627604C2212yes—
19#22627600—21no—
20Q8×Dic293—21no—
21#22627589C2212yescomplemented
22C4×D1172C2211yescharacteristic
23D4×Dic293C2211yescharacteristic
24#22627586C2211yescharacteristic
25C4×C22⋊D293—21no—
26#22627584—21no—
27#22627582—21no—
28C4⋊C4×D293C2211yescharacteristic
29C22⋊C4×D293—21no—
30C42×D293C2211yescharacteristic
31#22627557C2212yescomplemented
32C4⋊#14556873C2211yescharacteristic, complemented
33#22627549C2212yescomplemented
34#22627547C2212yescomplemented
35#22627544C2212yes—
36#22627536C2212yes—
37#22627534C2212yescomplemented
38C4⋊Q8×C293C2211yescharacteristic, complemented
39C4×D4×C293C2211yescharacteristic
40#107×C293C2212yescomplemented
41C4⋊C8×C293C2211yescharacteristic, complemented
42#102×C293C2212yes—
43C23×D293—21no—
44C2×Dic586C2312yes—
45C2×D1172C2311yescharacteristic
46C2×C22⋊D293—21no—
47D4×D293—22no—
48D4×D293—22no—
49C22×Dic293—21no—
50C2×C4×D293—21no—
51C2×C4×D293—21no—
52C2×C4×D293—21no—
53C2×C4×D293—44no—
54C2×C4×D293D411yescharacteristic
55C2×C4×D293D411yescharacteristic
56C2×C4×D293C2311yescharacteristic
57#17989791—24no—
58#17989790—24no—
59C2×#14556873C2312yes—
60C4⋊Dic293—22no—
61C4⋊Dic293—21no—
62C4⋊Dic293—21no—
63C4⋊Dic293C2311yescharacteristic
64#17989781—21no—
65#17989781—22no—
66C22⋊Dic293—21no—
67#17989779—22no—
68#17989779—21no—
69#17989779—21no—
70C4×Dic293—22no—
71C4×Dic293D411yescharacteristic
72C4×Dic293D411yescharacteristic
73C4×Dic293C2311yescharacteristic
74C2×Q8×C293—21no—
75C2×D4×C293C2311yescharacteristic
76C22×C4×C293—21no—
77C2×C8×C293C2312yes—
78C4⋊C4×C293—21no—
79C4⋊C4×C293C2311yescharacteristic
80C4⋊C4×C293C2312yes—
81C22⋊C4×C293—21no—
82C42×C293C2311yescharacteristic
83C22×D293—24no—
84C22×D293—21no—
85C22×D293—42no—
86C22×D293—42no—
87C22×D293C2×D411yescharacteristic
88Dic586—22no—
89Dic586—22no—
90D1172—21no—
91D1172#4112yes—
92C22⋊D293—24no—
93C2×Dic293—22no—
94C2×Dic293—21no—
95C2×Dic293—42no—
96C2×Dic293C2×D411yescharacteristic
97C2×Dic293C2×D411yescharacteristic
98C2×Dic293C2×D411yescharacteristic
99C4×D293—22no—
100C4×D293—22no—
101C4×D293—21no—
102C4×D293—21no—
103C4×D293—42no—
104C4×D293—42no—
105C4×D293—42no—
106#14556873—22no—
107C23×C293—21no—
108Q8×C293—42no—
109D4×C293—21no—
110D4×C293#4112yes—
111C2×C4×C293—21no—
112C2×C4×C293—22no—
113C2×C4×C293—42no—
114C2×C4×C293C2×D411yescharacteristic
115C2×C4×C293C2×D411yescharacteristic
116C2×C4×C293C2411yescharacteristic
117C8×C293—22no—
118C22×C293—22no—
119C22×C293—41no—
120C22×C293—41no—
121C22×C293C22×D411yescharacteristic
122C2×D293—21no—
123C2×D293—21no—
124C2×D293—22no—
125C2×D293—48no—
126C2×D293—41no—
127C2×D293—41no—
128C2×D293#13712yes—
129C4×C293—21no—
130C4×C293—41no—
131C4×C293—42no—
132C4×C293AGL(1,ℤ/8)12yes—
133C4×C293C2×#4111yescharacteristic
134C4×C293C2×#4111yescharacteristic
135Dic293—21no—
136Dic293—21no—
137Dic293—21no—
138Dic293—42no—
139Dic293—41no—
140C2×C293—42no—
141C2×C293#52511yescharacteristic
142C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C293C2×#13711yescharacteristic
144D293—22no—
145D293—42no—
146C293#311011yescharacteristic, complemented, Sylow
147#3110—2931nomaximal, Sylow
148#533—2931no—
149C2×C4×D4—2931no—
150#484—2932no—
151#474—2931no—
152#474D29311yescharacteristic, complemented
153#461—2931no—
154#461—2931no—
155#450—2932no—
156#440—2932no—
157#440—2932no—
158#432—2931no—
159C22×D4—2931no—
160C23×C4—5861no—
161C2×SD16—2932no—
162C2×SD16—2932no—
163C4⋊Q8—2931no—
164C4⋊Q8C2×D29311yescharacteristic, complemented
165#125—5861no—
166C22⋊Q8—2932no—
167C22⋊Q8—5861no—
168C4×Q8—2932no—
169C4×Q8—5861no—
170C4×D4—2931no—
171C4×D4—2931no—
172C4×D4—2931no—
173C4×D4—5861no—
174C4×D4—5861no—
175C4×D4C2×D29311yescharacteristic
176#117—5861no—
177C2×C4⋊C4—2931no—
178C2×C22⋊C4—5861no—
179C2×C42—2931no—
180#107—2932no—
181#107C2×D29312yescomplemented
182C4⋊C8—2931no—
183C4⋊C8C2×D29311yescharacteristic, complemented
184#103—2932no—
185#103—2932no—
186#102—2932no—
187#102C2×D29312yes—
188C22⋊C8—2932no—
189#97—2932no—
190C24—5861no—
191C2×Q8—21no—
192C2×Q8—2932no—
193C2×D4—2931no—
194C2×D4—5861no—
195C2×D4—5862no—
196C2×D4—5862no—
197C2×D4C22×D29311yescharacteristic
198C22×C4—21no—
199C22×C4—2931no—
200C22×C4—2931no—
201C22×C4—2931no—
202C22×C4—5861no—
203C22×C4—5861no—
204C22×C4—5861no—
205C22×C4—5861no—
206C22×C4—11724no—
207SD16—5864no—
208SD16—5864no—
209C2×C8—2932no—
210C2×C8C22×D29312yes—
211C4⋊C4—21no—
212C4⋊C4—2931no—
213C4⋊C4—5861no—
214C4⋊C4—5862no—
215C4⋊C4—5862no—
216C4⋊C4—5861no—
217C4⋊C4—5861no—
218C4⋊C4C22×D29312yes—
219C4⋊C4C22×D29311yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—5861no—
222C22⋊C4—5862no—
223C22⋊C4—5861no—
224C22⋊C4—5861no—
225C42—2931no—
226C42—2931no—
227C42—2931no—
228C42—5862no—
229C42C22×D29311yescharacteristic
230C23—21no—
231C23—2931no—
232C23—5864no—
233C23—5861no—
234C23—11722no—
235C23—11722no—
236Q8—42no—
237Q8—5862no—
238Q8—5862no—
239D4—21no—
240D4—2932no—
241D4—5864no—
242D4—5861no—
243D4#1798980312yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—2931no—
248C2×C4—2931no—
249C2×C4—2931no—
250C2×C4—5862no—
251C2×C4—5862no—
252C2×C4—5861no—
253C2×C4—5861no—
254C2×C4—5862no—
255C2×C4—5861no—
256C2×C4—11722no—
257C2×C4—11722no—
258C2×C4—11722no—
259C2×C4—11722no—
260C2×C4D4×D29311yescharacteristic
261C2×C4D4×D29311yescharacteristic
262C2×C4C23×D29311yescharacteristic
263C8—22no—
264C8—5862no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—2932no—
269C22—5861no—
270C22—5861no—
271C22—5862no—
272C22—11728no—
273C22—11721no—
274C22—11721no—
275C22C2×D4×D29311yescharacteristic
276C4—21no—
277C4—41no—
278C4—42no—
279C4—5861no—
280C4—5861no—
281C4—5861no—
282C4—11722no—
283C4—11721no—
284C4#2262764812yes—
285C4#41×D29311yescharacteristic
286C4C2×#1798980311yescharacteristic
287C2—42no—
288C2—5862no—
289C2—11722no—
290C2#3403664211yescharacteristic
291C2C2×#2262764811yescharacteristic
292C2#137×D29311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#474×C293C2
socleC22×C293C22×D4
prime radicalC2×D4×D293C22
hypercenter#474D293
hypocenterC293#3110
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #34036684C2semidirect
3. #34036466C2semidirect
4. #34036406C2semidirect
5. #34036335C2semidirect
6. #34036334C2semidirect
8. #34036228C2semidirect
9. #34036226C2semidirect
10. #34036185C2semidirect
11. #474×C293C2semidirect
16. #22627624C22semidirect
21. #22627589C22semidirect
31. #22627557C22semidirect
32. C4⋊#14556873C22semidirect
33. #22627549C22semidirect
34. #22627547C22semidirect
37. #22627534C22semidirect
38. C4⋊Q8×C293C22semidirect
40. #107×C293C22semidirect
41. C4⋊C8×C293C22semidirect
146. C293#3110semidirect
152. #474D293semidirect
164. C4⋊Q8C2×D293semidirect
181. #107C2×D293semidirect
183. C4⋊C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366842
C4×D4×D2932
#340364662
#340364062
#340363352
#340363342
#340362812
#340362282
#340362262
#340361852
#474×C2932
#3110293