Finite Group Databasebeta

← #49169233

Subgroups — #49169233

Order 37504.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#22627650C211yesmaximal, characteristic, complemented
3C2×#22627649C211yesmaximal, characteristic, complemented
4#34036675C211yesmaximal, characteristic, complemented
5C2×#22627620C211yesmaximal, characteristic, complemented
6#34036517C212yesmaximal, complemented
7#34036515C212yesmaximal, complemented
8#34036504C212yesmaximal, complemented
9#34036500C212yesmaximal, complemented
10M16×Dic293C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C22×D1172C2211yescharacteristic
14C22×C22⋊D293—22no—
15#22627652—24no—
16#22627650—24no—
17#22627649—24no—
18C2×#17989791C2212yescomplemented
19C2×#17989790C2212yescomplemented
20C2×#17989789C2212yescomplemented
21C2×D2344C2212yescomplemented
22C2×#17989787C2212yescomplemented
23C2×#17989787C2212yescomplemented
24C2×#17989786C2211yescharacteristic
25#22627622C2211yescharacteristic, complemented
26#22627620—22no—
27#22627620C2212yes—
28#22627620C2212yes—
29#22627620C2211yescharacteristic, complemented
30#22627613C2211yescharacteristic, complemented
31#22627611C2211yescharacteristic, complemented
32#22627596—22no—
33C4×C22⋊D293—22no—
34C2×C4×Dic293C2211yescharacteristic
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×D1172C2311yescharacteristic
51C2×D1172C2312yes—
52C2×C22⋊D293—22no—
53C2×C22⋊D293—24no—
54C2×C22⋊D293—24no—
55C2×C22⋊D293—21no—
56C2×C22⋊D293—48no—
57C22×Dic293D412yes—
58C2×C4×D293—21no—
59#17989791—24no—
60#17989790—24no—
61#17989789—24no—
62D2344—24no—
63#17989787—24no—
64#17989787—24no—
65#17989786D414yescomplemented
66C2×#14556873C2312yes—
67#17989781—22no—
68C22⋊Dic293—22no—
69#17989779—22no—
70C4×Dic293D412yes—
71C4×Dic293C2311yescharacteristic
72C4×Dic293C2311yescharacteristic
73C24×C293—21no—
74#41×C293—22no—
75#41×C293—22no—
76C2×Q8×C293C2311yescharacteristic
77C2×D4×C293—21no—
78C2×D4×C293—22no—
79C2×D4×C293—22no—
80C2×D4×C293—21no—
81C2×D4×C293C2311yescharacteristic
82C2×D4×C293C2312yes—
83C22×C4×C293—21no—
84C22×C4×C293C2311yescharacteristic
85SD16×C293—24no—
86D8×C293—24no—
87M16×C293D414yescomplemented
88C2×C8×C293C2312yes—
89C22×D293—21no—
90C22×D293—22no—
91C22×D293—44no—
92C22×D293—42no—
93Dic586—21no—
94Dic586—21no—
95D1172—21no—
96D1172—22no—
97D1172—21no—
98D1172—22no—
99D1172—42no—
100C22⋊D293—48no—
101C22⋊D293—42no—
102C22⋊D293—44no—
103C22⋊D293—48no—
104C2×Dic293—24no—
105C2×Dic293—21no—
106C2×Dic293C2×D412yes—
107C2×Dic293C2×D412yes—
108C4×D293—42no—
109#14556873C2×D414yescomplemented
110C23×C293—21no—
111C23×C293—22no—
112C23×C293—44no—
113C23×C293—42no—
114C23×C293C2×D411yescharacteristic
115Q8×C293—21no—
116Q8×C293—21no—
117D4×C293—21no—
118D4×C293—22no—
119D4×C293—22no—
120D4×C293—21no—
121D4×C293—42no—
122D4×C293—42no—
123C2×C4×C293—21no—
124C2×C4×C293—42no—
125C2×C4×C293C2×D411yescharacteristic
126C2×C4×C293C2×D412yes—
127C2×C4×C293C2×D412yes—
128C2×C4×C293C2411yescharacteristic
129C8×C293C2×D414yescomplemented
130C22×C293—22no—
131C22×C293—41no—
132C22×C293—42no—
133C22×C293—44no—
134C22×C293—42no—
135C22×C293—84no—
136C22×C293#12712yes—
137C22×C293C22×D411yescharacteristic
138C2×D293—41no—
139C2×D293—42no—
140C2×D293—44no—
141C2×D293—42no—
142C2×D293—84no—
143C4×C293—41no—
144C4×C293#12712yes—
145C4×C293C22×D411yescharacteristic
146C4×C293C22×D411yescharacteristic
147Dic293—22no—
148Dic293—41no—
149Dic293AGL(1,ℤ/8)14yescomplemented
150C2×C293—22no—
151C2×C293—81no—
152C2×C293—82no—
153C2×C293C2×#12711yescharacteristic
154C2×C293C2×AGL(1,ℤ/8)12yes—
155D293—81no—
156D293—82no—
157C293#315111yescharacteristic, complemented, Sylow
158#3151—2931nomaximal, Sylow
159C2×AGL(1,ℤ/8)—2931no—
160C2×AGL(1,ℤ/8)—2931no—
161C2×AGL(1,ℤ/8)—2931no—
162C2×AGL(1,ℤ/8)D29311yescharacteristic, complemented
163#532—2931no—
164C2×#127—2931no—
165#496—2932no—
166#496—2932no—
167#493—2932no—
168#492—2932no—
169C4×M16—2931no—
170C2×#41—2931no—
171C2×#41C2×D29311yescharacteristic
172C22×D4—2931no—
173C22×D4—5862no—
174C22×D4C2×D29311yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5864no—
177AGL(1,ℤ/8)—5864no—
178AGL(1,ℤ/8)—5864no—
179C2×SD16—2932no—
180C2×SD16—2932no—
181C2×SD16—2932no—
182C2×SD16C2×D29312yescomplemented
183C2×D8—2932no—
184C2×D8—2932no—
185C2×D8—2932no—
186C2×D8C2×D29312yescomplemented
187C2×M16—2931no—
188C2×M16C2×D29311yescharacteristic
189C4⋊Q8—2931no—
190#127—2932no—
191#127—2932no—
192#127—2931no—
193#127—5862no—
194#124—2931no—
195#124—2931no—
196#121—5862no—
197C4×D4—5862no—
198C2×C42—2931no—
199#97—2932no—
200C4×C8—2932no—
201C24—21no—
202C24—5861no—
203#41—22no—
204#41—22no—
205#41—5862no—
206#41—5862no—
207C2×Q8—2931no—
208C2×Q8C22×D29311yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2931no—
214C2×D4—2932no—
215C2×D4—5862no—
216C2×D4—5864no—
217C2×D4—5864no—
218C2×D4—5861no—
219C2×D4—5862no—
220C2×D4—5861no—
221C2×D4—5862no—
222C2×D4—11728no—
223C2×D4C22×D29311yescharacteristic
224C2×D4C22×D29312yes—
225C22×C4—21no—
226C22×C4—2932no—
227C22×C4—5861no—
228C22×C4C22×D29311yescharacteristic
229SD16—24no—
230SD16—5864no—
231SD16—5864no—
232SD16—5864no—
233D8—24no—
234D8—5864no—
235D8—5864no—
236D8—5864no—
237M16—2934no—
238M16C22⋊D29314yescomplemented
239C2×C8—2932no—
240C2×C8C22×D29312yes—
241C4⋊C4—5862no—
242C22⋊C4—5862no—
243C22⋊C4—5862no—
244C42—2932no—
245C42—2931no—
246C42—2931no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—5861no—
252C23—5862no—
253C23—11724no—
254C23—11722no—
255C23D4×D29311yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—5861no—
259Q8—5861no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—5861no—
267D4—5862no—
268D4—5861no—
269D4—5862no—
270D4—11728no—
271D4—11724no—
272D4—11728no—
273D4—11722no—
274D4—11722no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2932no—
278C2×C4—2932no—
279C2×C4—5864no—
280C2×C4—5861no—
281C2×C4—11722no—
282C2×C4D4×D29311yescharacteristic
283C2×C4D4×D29312yes—
284C2×C4C2×C22⋊D29312yes—
285C2×C4C23×D29311yescharacteristic
286C8—2934no—
287C8C2×C22⋊D29314yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—23444no—
295C22—11724no—
296C22—11722no—
297C22—11721no—
298C22—11722no—
299C22#2262762012yes—
300C22C2×D4×D29311yescharacteristic
301C4—41no—
302C4—2934no—
303C4—5862no—
304C4—11721no—
305C4#2262762012yes—
306C4C2×D4×D29311yescharacteristic
307C4C22×C22⋊D29311yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—23442no—
312C2—23441no—
313C2C2×#2262762011yescharacteristic
314C2AGL(1,ℤ/8)×D29312yes—
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#3151
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×#22627650C2semidirect
3. C2×#22627649C2semidirect
4. #34036675C2semidirect
5. C2×#22627620C2semidirect
6. #34036517C2semidirect
7. #34036515C2semidirect
8. #34036504C2semidirect
9. #34036500C2semidirect
10. M16×Dic293C2semidirect
11. C2×AGL(1,ℤ/8)×C293C2semidirect
18. C2×#17989791C22semidirect
19. C2×#17989790C22semidirect
20. C2×#17989789C22semidirect
21. C2×D2344C22semidirect
22. C2×#17989787C22semidirect
23. C2×#17989787C22semidirect
25. #22627622C22semidirect
29. #22627620C22semidirect
30. #22627613C22semidirect
31. #22627611C22semidirect
35. #22627534C22semidirect
36. C8×Dic293C22semidirect
40. C2×SD16×C293C22semidirect
41. C2×D8×C293C22semidirect
65. #17989786D4semidirect
87. M16×C293D4semidirect
109. #14556873C2×D4semidirect
129. C8×C293C2×D4semidirect
149. Dic293AGL(1,ℤ/8)semidirect
157. C293#3151semidirect
162. C2×AGL(1,ℤ/8)D293semidirect
182. C2×SD16C2×D293semidirect
186. C2×D8C2×D293semidirect
238. M16C22⋊D293semidirect
287. C8C2×C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276522
C2×#226276502
C2×#226276492
#340366752
C2×#226276202
#340365172
#340365152
#340365042
#340365002
M16×Dic2932
C2×AGL(1,ℤ/8)×C2932
#3151293