Finite Group Databasebeta

← #49169270

Subgroups — #49169270

Order 37504.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22627652C211yesmaximal, characteristic, complemented
2C2×#22627644C211yesmaximal, characteristic, complemented
3C22×#17989787C211yesmaximal, characteristic, complemented
4#34036696C211yesmaximal, characteristic, complemented
5#34036693C211yesmaximal, characteristic, complemented
6#34036517C212yesmaximal, complemented
7#34036515C212yesmaximal, complemented
8#34036510C212yesmaximal, complemented
9#34036504C212yesmaximal, complemented
10#34036087C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C22×D1172C2211yescharacteristic
14#22627652—24no—
15#22627644—24no—
16C2×#17989792C2211yescharacteristic, complemented
17C2×#17989791C2212yescomplemented
18C2×#17989790C2211yescharacteristic, complemented
19C2×#17989789C2211yescharacteristic, complemented
20C2×D2344C2212yescomplemented
21C2×#17989787—24no—
22C2×#17989787C2212yes—
23C2×#17989787C2212yes—
24C2×#17989787C2211yescharacteristic, complemented
25C22×#14556873C2211yescharacteristic
26#22627620C2212yescomplemented
27#22627620C2212yescomplemented
28#22627613C2212yescomplemented
29#22627611C2212yescomplemented
30#22627598—22no—
31#22627595—22no—
32#22627593—22no—
33#22627592—22no—
34#22627583C2211yescharacteristic
35#22627534C2212yescomplemented
36C8×Dic293C2212yescomplemented
37C2×#41×C293C2211yescharacteristic
38C22×D4×C293C2211yescharacteristic
39AGL(1,ℤ/8)×C293—24no—
40C2×SD16×C293C2212yescomplemented
41C2×D8×C293C2212yescomplemented
42C2×M16×C293C2211yescharacteristic
43C23×D293—21no—
44#17989802—22no—
45#17989802—22no—
46C2×Dic586C2311yescharacteristic
47C2×D1172—22no—
48C2×D1172—22no—
49C2×D1172—21no—
50C2×D1172C2312yes—
51C2×D1172C2311yescharacteristic
52C2×C22⋊D293—24no—
53C2×C22⋊D293—24no—
54C2×C22⋊D293—21no—
55C2×C4×D293—21no—
56#17989792—22no—
57#17989791—24no—
58#17989790—22no—
59#17989789—22no—
60D2344—24no—
61#17989787—24no—
62#17989787—24no—
63#17989787—22no—
64C2×#14556873D414yescomplemented
65C2×#14556873C2311yescharacteristic
66C2×#14556873C2311yescharacteristic
67C4⋊Dic293D412yes—
68C22⋊Dic293—24no—
69C22⋊Dic293D412yes—
70#17989779—24no—
71C4×Dic293C2312yes—
72C24×C293—21no—
73#41×C293—22no—
74#41×C293—22no—
75C2×Q8×C293C2311yescharacteristic
76C2×D4×C293—21no—
77C2×D4×C293—22no—
78C2×D4×C293—22no—
79C2×D4×C293—21no—
80C2×D4×C293C2312yes—
81C2×D4×C293C2311yescharacteristic
82C22×C4×C293—21no—
83C22×C4×C293C2311yescharacteristic
84SD16×C293—24no—
85D8×C293—24no—
86M16×C293D414yescomplemented
87C2×C8×C293C2312yes—
88C22×D293—22no—
89C22×D293—21no—
90C22×D293—42no—
91C22×D293—82no—
92Dic586—21no—
93Dic586—21no—
94D1172—22no—
95D1172—21no—
96D1172—21no—
97D1172—22no—
98D1172—42no—
99C22⋊D293—42no—
100C22⋊D293—84no—
101C22⋊D293—84no—
102C2×Dic293—21no—
103C2×Dic293C2×D414yes—
104C4×D293—42no—
105#14556873C2×D412yes—
106#14556873C2×D412yes—
107C23×C293—21no—
108C23×C293—22no—
109C23×C293—42no—
110C23×C293—82no—
111C23×C293C2×D411yescharacteristic
112Q8×C293—21no—
113Q8×C293—21no—
114D4×C293—21no—
115D4×C293—22no—
116D4×C293—22no—
117D4×C293—21no—
118D4×C293—42no—
119D4×C293—42no—
120C2×C4×C293—21no—
121C2×C4×C293—42no—
122C2×C4×C293C2×D411yescharacteristic
123C2×C4×C293C2×D412yes—
124C2×C4×C293C2×D412yes—
125C2×C4×C293C2411yescharacteristic
126C8×C293C2×D414yes—
127C22×C293—22no—
128C22×C293—41no—
129C22×C293—42no—
130C22×C293—44no—
131C22×C293—42no—
132C22×C293—84no—
133C22×C293#12712yes—
134C22×C293C22×D411yescharacteristic
135C2×D293—41no—
136C2×D293—42no—
137C2×D293—42no—
138C2×D293—44no—
139C2×D293—84no—
140C4×C293—41no—
141C4×C293#12712yes—
142C4×C293C22×D411yescharacteristic
143C4×C293C22×D411yescharacteristic
144Dic293—24no—
145Dic293—41no—
146C2×C293—22no—
147C2×C293—81no—
148C2×C293—82no—
149C2×C293C2×#12711yescharacteristic
150C2×C293#57612yes—
151D293—81no—
152D293—82no—
153C293#315511yescharacteristic, complemented, Sylow
154#3155—2931nomaximal, Sylow
155C2×AGL(1,ℤ/8)—2931no—
156C2×AGL(1,ℤ/8)D29311yescharacteristic, complemented
157C2×#135—2931no—
158C22×D8—2931no—
159#535—2931no—
160#535—2931no—
161#496—2932no—
162#496—2932no—
163#495—2932no—
164#493—2932no—
165#405—2931no—
166C2×#41—2931no—
167C2×#41C2×D29311yescharacteristic
168C22×D4—2931no—
169C22×D4C2×D29311yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—5864no—
172#135—5864no—
173C2×Q16—2931no—
174C2×SD16—2931no—
175C2×SD16—2931no—
176C2×SD16—2932no—
177C2×SD16C2×D29312yescomplemented
178C2×D8—2932no—
179C2×D8—2932no—
180C2×D8—2931no—
181C2×D8—2932no—
182C2×D8—5864no—
183C2×D8C2×D29312yescomplemented
184C2×M16C2×D29311yescharacteristic
185C22×C8—2931no—
186#127—2932no—
187#127—2932no—
188#124—2932no—
189#124—2932no—
190#121—5862no—
191#121—5862no—
192#120—5862no—
193#120—5862no—
194#117—2931no—
195#97—2932no—
196C4×C8—2932no—
197C24—21no—
198C24—5861no—
199#41—22no—
200#41—22no—
201#41—5862no—
202#41—5862no—
203C2×Q8—2931no—
204C2×Q8C22×D29311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2932no—
210C2×D4—2931no—
211C2×D4—5864no—
212C2×D4—5864no—
213C2×D4—5861no—
214C2×D4—5862no—
215C2×D4—5861no—
216C2×D4—5862no—
217C2×D4C22×D29312yes—
218C2×D4C22×D29311yescharacteristic
219C22×C4—21no—
220C22×C4—5861no—
221C22×C4C22×D29311yescharacteristic
222Q16—5862no—
223SD16—24no—
224SD16—5862no—
225SD16—5862no—
226SD16—5864no—
227D8—24no—
228D8—5864no—
229D8—5864no—
230D8—5862no—
231D8—5864no—
232M16C22⋊D29314yescomplemented
233C2×C8—2934no—
234C2×C8—2931no—
235C2×C8—2931no—
236C2×C8C22×D29312yes—
237C4⋊C4—2932no—
238C22⋊C4—2932no—
239C22⋊C4—5864no—
240C22⋊C4—5864no—
241C42—2932no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—23442no—
247C23—5862no—
248C23—5861no—
249C23—11722no—
250C23D4×D29311yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—5861no—
254Q8—5861no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—23444no—
262D4—23444no—
263D4—5862no—
264D4—5861no—
265D4—5861no—
266D4—5862no—
267D4—11722no—
268D4—11722no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—2934no—
272C2×C4—5861no—
273C2×C4—11722no—
274C2×C4D4×D29311yescharacteristic
275C2×C4D4×D29312yes—
276C2×C4C2×C22⋊D29312yes—
277C2×C4C23×D29311yescharacteristic
278C8—2932no—
279C8—2932no—
280C8C2×C22⋊D29314yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—23444no—
288C22—11721no—
289C22—11722no—
290C22—11722no—
291C22—11724no—
292C22#2262762012yes—
293C22C2×D4×D29311yescharacteristic
294C4—41no—
295C4—5864no—
296C4—11721no—
297C4#2262762012yes—
298C4C2×D4×D29311yescharacteristic
299C4C22×C22⋊D29311yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—23442no—
304C2—23441no—
305C2C2×#2262762011yescharacteristic
306C2#3403693312yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×AGL(1,ℤ/8)×C293C2
socleC22×C293C22×D4
hypercenterC2×AGL(1,ℤ/8)D293
hypocenterC293#3155
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22627652C2semidirect
2. C2×#22627644C2semidirect
3. C22×#17989787C2semidirect
4. #34036696C2semidirect
5. #34036693C2semidirect
6. #34036517C2semidirect
7. #34036515C2semidirect
8. #34036510C2semidirect
9. #34036504C2semidirect
10. #34036087C2semidirect
11. C2×AGL(1,ℤ/8)×C293C2semidirect
16. C2×#17989792C22semidirect
17. C2×#17989791C22semidirect
18. C2×#17989790C22semidirect
19. C2×#17989789C22semidirect
20. C2×D2344C22semidirect
24. C2×#17989787C22semidirect
26. #22627620C22semidirect
27. #22627620C22semidirect
28. #22627613C22semidirect
29. #22627611C22semidirect
35. #22627534C22semidirect
36. C8×Dic293C22semidirect
40. C2×SD16×C293C22semidirect
41. C2×D8×C293C22semidirect
64. C2×#14556873D4semidirect
86. M16×C293D4semidirect
153. C293#3155semidirect
156. C2×AGL(1,ℤ/8)D293semidirect
177. C2×SD16C2×D293semidirect
183. C2×D8C2×D293semidirect
232. M16C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276522
C2×#226276442
C22×#179897872
#340366962
#340366932
#340365172
#340365152
#340365102
#340365042
#340360872
C2×AGL(1,ℤ/8)×C2932
#3155293