Finite Group Databasebeta

← #49169262

Subgroups — #49169262

Order 37504.

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

normal subgroups only

This group has 564 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×#22627654C211yesmaximal, characteristic, complemented
2C2×#22627649C211yesmaximal, characteristic, complemented
3C22×#17989791C211yesmaximal, characteristic, complemented
4#34036700C211yesmaximal, characteristic, complemented
5#34036693C211yesmaximal, characteristic, complemented
6#34036522C212yesmaximal, complemented
7#34036517C212yesmaximal, complemented
8#34036510C212yesmaximal, complemented
9#34036500C212yesmaximal, complemented
10#34036084C211yesmaximal, characteristic, complemented
11C2×#135×C293C211yesmaximal, characteristic, complemented
12C22×Dic586C2211yescharacteristic
13C22×D1172C2211yescharacteristic
14#22627654—24no—
15#22627649—24no—
16C2×#17989792C2212yescomplemented
17C2×#17989791—24no—
18C2×#17989791C2212yes—
19C2×#17989791C2212yes—
20C2×#17989790C2212yescomplemented
21C2×#17989789C2212yes—
22C2×#17989787C2212yes—
23C2×#17989786C2211yescharacteristic
24#22627622C2212yescomplemented
25#22627620C2212yes—
26#22627613C2212yes—
27#22627611C2212yescomplemented
28#22627605—22no—
29#22627603—22no—
30#22627598—22no—
31#22627592—22no—
32#22627583C2211yescharacteristic
33#22627534C2212yescomplemented
34C8×Dic293C2212yescomplemented
35C2×#41×C293C2211yescharacteristic
36C2×#41×C293C2211yescharacteristic
37#135×C293—24no—
38C2×Q16×C293C2211yescharacteristic, complemented
39C2×SD16×C293C2211yescharacteristic, complemented
40C2×SD16×C293C2211yescharacteristic, complemented
41C2×D8×C293C2211yescharacteristic, complemented
42C22×C8×C293C2211yescharacteristic
43C23×D293—21no—
44C2×Dic586—21no—
45C2×Dic586—22no—
46C2×Dic586—22no—
47C2×Dic586C2312yes—
48C2×D1172—22no—
49C2×D1172—22no—
50C2×D1172—21no—
51C2×D1172C2312yes—
52C2×C22⋊D293—24no—
53C22×Dic293—21no—
54#17989792—24no—
55#17989791—24no—
56#17989791—24no—
57#17989790—24no—
58#17989789—24no—
59#17989787—24no—
60#17989786D414yescomplemented
61C2×#14556873C2312yes—
62C4⋊Dic293D412yes—
63#17989781—24no—
64C22⋊Dic293—24no—
65C22⋊Dic293D412yes—
66#17989779—24no—
67C4×Dic293C2312yes—
68#41×C293—22no—
69#41×C293—22no—
70#41×C293—22no—
71#41×C293—22no—
72C2×Q8×C293C2311yescharacteristic
73C2×Q8×C293C2311yescharacteristic
74C2×D4×C293—21no—
75C2×D4×C293—21no—
76C2×D4×C293C2311yescharacteristic
77C2×D4×C293C2311yescharacteristic
78C22×C4×C293—21no—
79C22×C4×C293—21no—
80C22×C4×C293C2311yescharacteristic
81Q16×C293—22no—
82SD16×C293—22no—
83SD16×C293—22no—
84D8×C293—22no—
85C2×C8×C293D414yescomplemented
86C2×C8×C293C2311yescharacteristic
87C2×C8×C293C2311yescharacteristic
88C22×D293—22no—
89C22×D293—42no—
90C22×D293—82no—
91Dic586—22no—
92Dic586—22no—
93Dic586—42no—
94D1172—22no—
95D1172—22no—
96D1172—42no—
97C22⋊D293—84no—
98C2×Dic293—22no—
99C2×Dic293—42no—
100C2×Dic293C2×D414yes—
101#14556873C2×D414yes—
102C23×C293—21no—
103C23×C293—21no—
104C23×C293C2×D411yescharacteristic
105Q8×C293—21no—
106Q8×C293—21no—
107Q8×C293—21no—
108Q8×C293—21no—
109D4×C293—21no—
110D4×C293—21no—
111D4×C293—21no—
112D4×C293—21no—
113D4×C293—42no—
114D4×C293—42no—
115C2×C4×C293—21no—
116C2×C4×C293—21no—
117C2×C4×C293—42no—
118C2×C4×C293—42no—
119C2×C4×C293C2×D411yescharacteristic
120C2×C4×C293C2×D412yes—
121C2×C4×C293C2×D412yes—
122C2×C4×C293C2411yescharacteristic
123C8×C293C2×D412yes—
124C8×C293C2×D412yes—
125C22×C293—22no—
126C22×C293—41no—
127C22×C293—41no—
128C22×C293—42no—
129C22×C293—42no—
130C22×C293#12712yes—
131C22×C293C22×D411yescharacteristic
132C2×D293—42no—
133C2×D293—44no—
134C2×D293—84no—
135C4×C293—41no—
136C4×C293—41no—
137C4×C293#12712yes—
138C4×C293C22×D411yescharacteristic
139C4×C293C22×D411yescharacteristic
140Dic293—24no—
141Dic293—42no—
142C2×C293—22no—
143C2×C293—81no—
144C2×C293—81no—
145C2×C293C2×#12711yescharacteristic
146C2×C293#57712yes—
147D293—82no—
148C293#315411yescharacteristic, complemented, Sylow
149#3154—2931nomaximal, Sylow
150C2×#137—2931no—
151C2×AGL(1,ℤ/8)—2931no—
152C2×#135D29311yescharacteristic, complemented
153C22×SD16—2931no—
154#536—2931no—
155#535—2931no—
156#497—2932no—
157#496—2932no—
158#495—2932no—
159#492—2932no—
160#405—2931no—
161C2×#41C2×D29311yescharacteristic
162C2×#41C2×D29311yescharacteristic
163C22×Q8—2931no—
164C22×D4—2931no—
165#137—5864no—
166AGL(1,ℤ/8)—5864no—
167#135—24no—
168C2×Q16—2932no—
169C2×Q16C2×D29311yescharacteristic, complemented
170C2×SD16—2932no—
171C2×SD16—2932no—
172C2×SD16—2932no—
173C2×SD16—2932no—
174C2×SD16—5864no—
175C2×SD16C2×D29311yescharacteristic, complemented
176C2×SD16C2×D29311yescharacteristic, complemented
177C2×D8—2932no—
178C2×D8C2×D29311yescharacteristic, complemented
179C2×M16—2931no—
180C22×C8C2×D29311yescharacteristic
181C4⋊Q8—2932no—
182#127—2932no—
183#124—2932no—
184#124—2932no—
185#123—5862no—
186C22⋊Q8—5862no—
187#121—5862no—
188#120—5862no—
189#117—2931no—
190#97—2932no—
191C4×C8—2932no—
192C24—5861no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—2932no—
198C2×Q8—5861no—
199C2×Q8—5862no—
200C2×Q8—5862no—
201C2×Q8C22×D29311yescharacteristic
202C2×Q8C22×D29311yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—2932no—
206C2×D4—5864no—
207C2×D4—5862no—
208C2×D4—5861no—
209C2×D4—5862no—
210C2×D4C22×D29311yescharacteristic
211C2×D4C22×D29311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—5861no—
215C22×C4C22×D29311yescharacteristic
216Q16—22no—
217Q16—5864no—
218SD16—22no—
219SD16—22no—
220SD16—5864no—
221SD16—5864no—
222SD16—5864no—
223SD16—5864no—
224D8—22no—
225D8—5864no—
226M16—2934no—
227C2×C8—2932no—
228C2×C8C22⋊D29314yescomplemented
229C2×C8C22×D29311yescharacteristic
230C2×C8C22×D29311yescharacteristic
231C4⋊C4—2932no—
232C4⋊C4—5864no—
233C22⋊C4—2932no—
234C22⋊C4—5864no—
235C22⋊C4—5864no—
236C42—2932no—
237C23—21no—
238C23—21no—
239C23—23442no—
240C23—5862no—
241C23—11722no—
242C23D4×D29311yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—5862no—
248Q8—5862no—
249Q8—11722no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—23444no—
257D4—5862no—
258D4—5862no—
259D4—11722no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—2934no—
265C2×C4—5862no—
266C2×C4—11722no—
267C2×C4D4×D29311yescharacteristic
268C2×C4D4×D29312yes—
269C2×C4C2×C22⋊D29312yes—
270C2×C4C23×D29311yescharacteristic
271C8—2934no—
272C8C2×C22⋊D29312yes—
273C8C2×C22⋊D29312yes—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—23444no—
280C22—11722no—
281C22—11724no—
282C22#2262762012yes—
283C22C2×D4×D29311yescharacteristic
284C4—41no—
285C4—41no—
286C4—5864no—
287C4—11722no—
288C4#2262762012yes—
289C4C2×D4×D29311yescharacteristic
290C4C22×C22⋊D29311yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—23442no—
295C2C2×#2262762011yescharacteristic
296C2#3403693612yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×#135×C293C2
socleC22×C293C22×D4
hypercenterC2×#135D293
hypocenterC293#3154
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22627654C2semidirect
2. C2×#22627649C2semidirect
3. C22×#17989791C2semidirect
4. #34036700C2semidirect
5. #34036693C2semidirect
6. #34036522C2semidirect
7. #34036517C2semidirect
8. #34036510C2semidirect
9. #34036500C2semidirect
10. #34036084C2semidirect
11. C2×#135×C293C2semidirect
16. C2×#17989792C22semidirect
20. C2×#17989790C22semidirect
24. #22627622C22semidirect
27. #22627611C22semidirect
33. #22627534C22semidirect
34. C8×Dic293C22semidirect
38. C2×Q16×C293C22semidirect
39. C2×SD16×C293C22semidirect
40. C2×SD16×C293C22semidirect
41. C2×D8×C293C22semidirect
60. #17989786D4semidirect
85. C2×C8×C293D4semidirect
148. C293#3154semidirect
152. C2×#135D293semidirect
169. C2×Q16C2×D293semidirect
175. C2×SD16C2×D293semidirect
176. C2×SD16C2×D293semidirect
178. C2×D8C2×D293semidirect
228. C2×C8C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276542
C2×#226276492
C22×#179897912
#340367002
#340366932
#340365222
#340365172
#340365102
#340365002
#340360842
C2×#135×C2932
#3154293