Finite Group Databasebeta

← #49169234

Subgroups — #49169234

Order 37504.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22627651C212yesmaximal, complemented
2C2×#22627648C212yesmaximal, complemented
3#34036678C211yesmaximal, characteristic, complemented
4#127×D293C211yesmaximal, characteristic, complemented
5#34036519C211yesmaximal, characteristic, complemented
6#34036518C211yesmaximal, characteristic, complemented
7#34036516C211yesmaximal, characteristic, complemented
8#34036504C212yesmaximal, complemented
9#34036500C212yesmaximal, complemented
10#34036081C211yesmaximal, characteristic, complemented
11#496×C293C211yesmaximal, characteristic, complemented
12C2×#17989803C2211yescharacteristic
13C2×#17989801C2211yescharacteristic
14C2×D4×D293—22no—
15C2×D4×D293C2212yes—
16#22627651—28no—
17#22627648—28no—
18C2×#17989791C2212yescomplemented
19C2×#17989790C2212yescomplemented
20C2×#17989789C2212yescomplemented
21C2×D2344C2212yescomplemented
22C2×#17989787C2212yescomplemented
23C2×#17989787C2212yescomplemented
24C2×#17989785C2212yes—
25#22627622C2211yescharacteristic, complemented
26#22627621C2211yescharacteristic
27#22627620—22no—
28#22627620C2212yes—
29#22627620C2211yescharacteristic, complemented
30#22627612C2211yescharacteristic, complemented
31#22627597—22no—
32#22627584—22no—
33C42×D293C2211yescharacteristic
34#22627535C2211yescharacteristic, complemented
35C8×Dic293C2212yescomplemented
36C2×SD16×C293C2212yescomplemented
37C2×D8×C293C2212yescomplemented
38#127×C293C2211yescharacteristic
39#124×C293C2211yescharacteristic, complemented
40#97×C293C2211yescharacteristic, complemented
41C23×D293—22no—
42#17989803—24no—
43#17989801—24no—
44C2×Dic586C2311yescharacteristic
45C2×D1172—21no—
46C2×D1172—22no—
47C2×D1172C2311yescharacteristic
48C2×D1172C2312yes—
49C2×C22⋊D293—21no—
50C2×C22⋊D293—22no—
51C2×C22⋊D293—22no—
52C2×C22⋊D293—24no—
53D4×D293—24no—
54D4×D293—24no—
55D4×D293—48no—
56C22×Dic293—21no—
57C2×C4×D293—21no—
58C2×C4×D293D412yes—
59C2×C4×D293C2311yescharacteristic
60#17989791—24no—
61#17989790—24no—
62#17989789—24no—
63D2344—24no—
64#17989787—24no—
65#17989787—24no—
66#17989785D418yescomplemented
67C2×#14556873C2312yes—
68#17989781—22no—
69#17989779—22no—
70C4×Dic293D412yes—
71C4×Dic293C2311yescharacteristic
72C2×Q8×C293C2311yescharacteristic
73C2×D4×C293—22no—
74C2×D4×C293C2312yes—
75C2×D4×C293C2311yescharacteristic
76SD16×C293—24no—
77D8×C293—24no—
78C2×C8×C293C2312yes—
79C22⋊C4×C293—22no—
80C42×C293C2311yescharacteristic
81C22×D293—21no—
82C22×D293—22no—
83C22×D293—48no—
84C22×D293—44no—
85C22×D293C2×D411yescharacteristic
86Dic586—22no—
87D1172—22no—
88D1172—22no—
89D1172—22no—
90D1172—44no—
91D1172—42no—
92C22⋊D293—42no—
93C22⋊D293—44no—
94C22⋊D293—48no—
95C22⋊D293—44no—
96C2×Dic293—21no—
97C2×Dic293—42no—
98C2×Dic293C2×D411yescharacteristic
99C2×Dic293C2×D412yes—
100C4×D293—24no—
101C4×D293—42no—
102C4×D293C2×D414yes—
103#14556873C2×D414yescomplemented
104C23×C293—21no—
105C23×C293—22no—
106Q8×C293—22no—
107D4×C293—22no—
108D4×C293—22no—
109D4×C293—22no—
110D4×C293—44no—
111C2×C4×C293—21no—
112C2×C4×C293C2×D412yes—
113C2×C4×C293C2411yescharacteristic
114C8×C293C2×D414yescomplemented
115C22×C293—41no—
116C22×C293—42no—
117C22×C293—44no—
118C22×C293—42no—
119C22×C293C22×D411yescharacteristic
120C2×D293—22no—
121C2×D293—41no—
122C2×D293—42no—
123C2×D293—44no—
124C2×D293—42no—
125C2×D293—88no—
126C2×D293#12712yes—
127C4×C293—22no—
128C4×C293—41no—
129C4×C293C22×D412yes—
130Dic293—41no—
131Dic293#12712yes—
132Dic293AGL(1,ℤ/8)14yescomplemented
133C2×C293—82no—
134C2×C293—81no—
135C2×C293C2×#12711yescharacteristic
136C2×C293C2×AGL(1,ℤ/8)12yes—
137D293—22no—
138D293—81no—
139D293—82no—
140C293#315111yescharacteristic, complemented, Sylow
141#3151—2931nomaximal, Sylow
142C2×AGL(1,ℤ/8)—2932no—
143C2×AGL(1,ℤ/8)—2932no—
144#532—2931no—
145C2×#127—2931no—
146#496—2931no—
147#496—2931no—
148#496—2931no—
149#496D29311yescharacteristic, complemented
150#493—2932no—
151#492—2932no—
152C4×M16—2931no—
153C2×#41—2931no—
154C2×#41—2931no—
155C22×D4—2932no—
156C22×D4—5862no—
157AGL(1,ℤ/8)—5868no—
158AGL(1,ℤ/8)—5868no—
159C2×SD16—2932no—
160C2×SD16—2932no—
161C2×SD16—2932no—
162C2×SD16C2×D29312yescomplemented
163C2×D8—2932no—
164C2×D8—2932no—
165C2×D8—2932no—
166C2×D8C2×D29312yescomplemented
167C2×M16—2932no—
168C4⋊Q8—2931no—
169#127—2932no—
170#127—2931no—
171#127—2931no—
172#127—5862no—
173#127C2×D29311yescharacteristic
174#124—2931no—
175#124C2×D29311yescharacteristic, complemented
176#121—5862no—
177C4×D4—5862no—
178C2×C42—2931no—
179#97—2931no—
180#97C2×D29311yescharacteristic, complemented
181C4×C8—2932no—
182C24—5862no—
183#41—5864no—
184#41—5864no—
185C2×Q8—2931no—
186C2×Q8C22×D29311yescharacteristic
187C2×D4—22no—
188C2×D4—2931no—
189C2×D4—2932no—
190C2×D4—5861no—
191C2×D4—5864no—
192C2×D4—5864no—
193C2×D4—5862no—
194C2×D4—5864no—
195C2×D4—5862no—
196C2×D4—5861no—
197C2×D4—5862no—
198C2×D4—11728no—
199C2×D4C22×D29311yescharacteristic
200C2×D4C22×D29312yes—
201C22×C4—2931no—
202C22×C4—2932no—
203C22×C4—5861no—
204C22×C4—5861no—
205SD16—24no—
206SD16—5864no—
207SD16—5864no—
208SD16—5864no—
209D8—24no—
210D8—5864no—
211D8—5864no—
212D8—5864no—
213M16—2938no—
214C2×C8—2932no—
215C2×C8C22×D29312yes—
216C4⋊C4—5862no—
217C22⋊C4—22no—
218C22⋊C4—5862no—
219C42—2932no—
220C42—2931no—
221C42C22×D29311yescharacteristic
222C23—21no—
223C23—22no—
224C23—2931no—
225C23—5861no—
226C23—5862no—
227C23—11724no—
228C23—11728no—
229Q8—22no—
230Q8—5862no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—5862no—
236D4—5862no—
237D4—5862no—
238D4—11724no—
239D4—11722no—
240D4—11724no—
241D4—11728no—
242D4—11724no—
243D4—11722no—
244C2×C4—21no—
245C2×C4—2932no—
246C2×C4—2931no—
247C2×C4—2934no—
248C2×C4—5864no—
249C2×C4—5861no—
250C2×C4—11722no—
251C2×C4—11722no—
252C2×C4D4×D29312yes—
253C2×C4C23×D29311yescharacteristic
254C8—2934no—
255C8D4×D29314yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2932no—
261C22—23448no—
262C22—5862no—
263C22—11724no—
264C22—11722no—
265C22—11721no—
266C22—11722no—
267C22C2×D4×D29311yescharacteristic
268C4—22no—
269C4—41no—
270C4—2934no—
271C4—2932no—
272C4—11721no—
273C4C2×D4×D29312yes—
274C2—82no—
275C2—81no—
276C2—23442no—
277C2—23441no—
278C2—5862no—
279C2#127×D29311yescharacteristic
280C2AGL(1,ℤ/8)×D29312yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC22×C293C22×D4
hypercenter#496D293
hypocenterC293#3151
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C2×#22627648C2semidirect
3. #34036678C2semidirect
4. #127×D293C2semidirect
5. #34036519C2semidirect
6. #34036518C2semidirect
7. #34036516C2semidirect
8. #34036504C2semidirect
9. #34036500C2semidirect
10. #34036081C2semidirect
11. #496×C293C2semidirect
18. C2×#17989791C22semidirect
19. C2×#17989790C22semidirect
20. C2×#17989789C22semidirect
21. C2×D2344C22semidirect
22. C2×#17989787C22semidirect
23. C2×#17989787C22semidirect
25. #22627622C22semidirect
29. #22627620C22semidirect
30. #22627612C22semidirect
34. #22627535C22semidirect
35. C8×Dic293C22semidirect
36. C2×SD16×C293C22semidirect
37. C2×D8×C293C22semidirect
39. #124×C293C22semidirect
40. #97×C293C22semidirect
66. #17989785D4semidirect
103. #14556873C2×D4semidirect
114. C8×C293C2×D4semidirect
132. Dic293AGL(1,ℤ/8)semidirect
140. C293#3151semidirect
149. #496D293semidirect
162. C2×SD16C2×D293semidirect
166. C2×D8C2×D293semidirect
175. #124C2×D293semidirect
180. #97C2×D293semidirect
255. C8D4×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×#226276482
#340366782
#127×D2932
#340365192
#340365182
#340365162
#340365042
#340365002
#340360812
#496×C2932
#3151293