Finite Group Databasebeta

← #49169247

Subgroups — #49169247

Order 37504.

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

normal subgroups only

This group has 572 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×#22627657C211yesmaximal, characteristic, complemented
2C2×#22627654C211yesmaximal, characteristic, complemented
3C2×#22627650C211yesmaximal, characteristic, complemented
4#34036675C211yesmaximal, characteristic, complemented
5C2×#22627611C211yesmaximal, characteristic, complemented
6#34036520C212yesmaximal, complemented
7#34036517C212yesmaximal, complemented
8#34036512C212yesmaximal, complemented
9#34036510C212yesmaximal, complemented
10M16×Dic293C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C22×Dic586C2211yescharacteristic
14#22627657—24no—
15#22627654—24no—
16#22627650—24no—
17C2×Dic1172C2212yescomplemented
18C2×#17989792C2212yescomplemented
19C2×#17989791C2212yescomplemented
20C2×#17989790C2212yes—
21C2×#17989790C2212yescomplemented
22C2×#17989787C2212yes—
23C2×#17989786C2211yescharacteristic
24#22627622C2211yescharacteristic, complemented
25#22627620C2211yescharacteristic, complemented
26#22627613C2211yescharacteristic, complemented
27#22627611—22no—
28#22627611C2212yes—
29#22627611C2212yes—
30#22627611C2211yescharacteristic, complemented
31#22627596—22no—
32C4×C22⋊D293—22no—
33C2×C22⋊Dic293—22no—
34C2×C4×Dic293C2211yescharacteristic
35#22627534C2212yescomplemented
36C8×Dic293C2212yescomplemented
37C2×#41×C293C2211yescharacteristic
38C22×D4×C293C2211yescharacteristic
39AGL(1,ℤ/8)×C293—24no—
40C2×SD16×C293C2212yes—
41C2×D8×C293C2212yes—
42C2×M16×C293C2211yescharacteristic
43#17989802—22no—
44#17989802—22no—
45C2×Dic586—21no—
46C2×Dic586—22no—
47C2×Dic586—22no—
48C2×Dic586C2312yes—
49C2×Dic586C2311yescharacteristic
50C2×D1172C2311yescharacteristic
51C2×C22⋊D293—22no—
52C2×C22⋊D293—21no—
53C22×Dic293—21no—
54C22×Dic293D412yes—
55C2×C4×D293—21no—
56Dic1172—24no—
57#17989792—24no—
58#17989791—24no—
59#17989790—24no—
60#17989790—24no—
61#17989787—24no—
62#17989786D414yescomplemented
63C2×#14556873C2312yes—
64#17989781—22no—
65C22⋊Dic293—22no—
66C22⋊Dic293—24no—
67C22⋊Dic293—24no—
68#17989779—22no—
69C4×Dic293D412yes—
70C4×Dic293C2311yescharacteristic
71C4×Dic293C2311yescharacteristic
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—21no—
89Dic586—21no—
90Dic586—22no—
91Dic586—22no—
92Dic586—21no—
93Dic586—42no—
94D1172—21no—
95D1172—21no—
96C22⋊D293—44no—
97C22⋊D293—42no—
98C2×Dic293—22no—
99C2×Dic293—21no—
100C2×Dic293—42no—
101C2×Dic293—42no—
102C2×Dic293C2×D412yes—
103C2×Dic293C2×D412yes—
104C4×D293—42no—
105#14556873C2×D414yes—
106C23×C293—21no—
107C23×C293—22no—
108C23×C293—44no—
109C23×C293—42no—
110C23×C293C2×D411yescharacteristic
111Q8×C293—21no—
112Q8×C293—21no—
113D4×C293—21no—
114D4×C293—22no—
115D4×C293—22no—
116D4×C293—21no—
117D4×C293—42no—
118D4×C293—42no—
119C2×C4×C293—21no—
120C2×C4×C293—42no—
121C2×C4×C293C2×D411yescharacteristic
122C2×C4×C293C2×D412yes—
123C2×C4×C293C2×D412yes—
124C2×C4×C293C2411yescharacteristic
125C8×C293C2×D414yes—
126C22×C293—22no—
127C22×C293—41no—
128C22×C293—42no—
129C22×C293—44no—
130C22×C293—42no—
131C22×C293—84no—
132C22×C293#12712yes—
133C22×C293C22×D411yescharacteristic
134C2×D293—41no—
135C2×D293—42no—
136C4×C293—41no—
137C4×C293#12712yes—
138C4×C293C22×D411yescharacteristic
139C4×C293C22×D411yescharacteristic
140Dic293—22no—
141Dic293—22no—
142Dic293—41no—
143Dic293—42no—
144C2×C293—22no—
145C2×C293—81no—
146C2×C293—82no—
147C2×C293C2×#12711yescharacteristic
148C2×C293#57512yes—
149D293—81no—
150C293#315311yescharacteristic, complemented, Sylow
151#3153—2931nomaximal, Sylow
152C2×#137—2931no—
153C2×#137—2931no—
154C2×AGL(1,ℤ/8)—2931no—
155C2×AGL(1,ℤ/8)D29311yescharacteristic, complemented
156#532—2931no—
157C2×#124—2931no—
158#497—2932no—
159#496—2932no—
160#495—2932no—
161#495—2932no—
162C4×M16—2931no—
163C2×#41—2931no—
164C2×#41C2×D29311yescharacteristic
165C22×Q8—2931no—
166C22×D4C2×D29311yescharacteristic
167#137—5864no—
168#137—5864no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—5864no—
171C2×Q16—2932no—
172C2×Q16—2932no—
173C2×SD16—2932no—
174C2×SD16—2932no—
175C2×SD16—2932no—
176C2×SD16C2×D29312yes—
177C2×D8—2932no—
178C2×D8C2×D29312yes—
179C2×M16—2931no—
180C2×M16C2×D29311yescharacteristic
181C4⋊Q8—2931no—
182#127—2931no—
183#124—2931no—
184#124—2932no—
185#124—2932no—
186#124—2931no—
187#124—5862no—
188#121—5862no—
189C4×D4—5862no—
190C2×C22⋊C4—5862no—
191C2×C42—2931no—
192#97—2932no—
193C4×C8—2932no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—5862no—
198#41—5862no—
199C2×Q8—2932no—
200C2×Q8—2931no—
201C2×Q8—5861no—
202C2×Q8—5862no—
203C2×Q8—5862no—
204C2×Q8C22×D29311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2931no—
210C2×D4—5862no—
211C2×D4—5861no—
212C2×D4C22×D29312yes—
213C2×D4C22×D29311yescharacteristic
214C22×C4—21no—
215C22×C4—2932no—
216C22×C4—5861no—
217C22×C4—5861no—
218C22×C4C22×D29311yescharacteristic
219Q16—5864no—
220Q16—5864no—
221SD16—24no—
222SD16—5864no—
223SD16—5864no—
224SD16—5864no—
225D8—24no—
226D8—5864no—
227M16—2934no—
228M16C22⋊D29314yescomplemented
229C2×C8—2932no—
230C2×C8C22×D29312yes—
231C4⋊C4—5862no—
232C22⋊C4—5862no—
233C22⋊C4—5864no—
234C22⋊C4—5864no—
235C22⋊C4—5862no—
236C42—2932no—
237C42—2931no—
238C42—2931no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—5861no—
244C23D4×D29311yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—5861no—
248Q8—5862no—
249Q8—5862no—
250Q8—5861no—
251Q8—11722no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—5861no—
259D4—5861no—
260D4—11724no—
261D4—11722no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—2932no—
265C2×C4—2932no—
266C2×C4—5862no—
267C2×C4—5861no—
268C2×C4—11722no—
269C2×C4—11722no—
270C2×C4—11722no—
271C2×C4D4×D29311yescharacteristic
272C2×C4D4×D29312yes—
273C2×C4C2×C22⋊D29312yes—
274C2×C4C23×D29311yescharacteristic
275C8—2934no—
276C8C2×C22⋊D29314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—11721no—
284C22—11722no—
285C22#2262762012yes—
286C22C2×D4×D29311yescharacteristic
287C4—41no—
288C4—5862no—
289C4—5862no—
290C4—11721no—
291C4—11722no—
292C4#2262762012yes—
293C4C2×D4×D29311yescharacteristic
294C4C22×C22⋊D29311yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—23441no—
299C2C2×#2262762011yescharacteristic
300C2#3403692512yes—
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#3153
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×#22627657C2semidirect
2. C2×#22627654C2semidirect
3. C2×#22627650C2semidirect
4. #34036675C2semidirect
5. C2×#22627611C2semidirect
6. #34036520C2semidirect
7. #34036517C2semidirect
8. #34036512C2semidirect
9. #34036510C2semidirect
10. M16×Dic293C2semidirect
11. C2×AGL(1,ℤ/8)×C293C2semidirect
17. C2×Dic1172C22semidirect
18. C2×#17989792C22semidirect
19. C2×#17989791C22semidirect
21. C2×#17989790C22semidirect
24. #22627622C22semidirect
25. #22627620C22semidirect
26. #22627613C22semidirect
30. #22627611C22semidirect
35. #22627534C22semidirect
36. C8×Dic293C22semidirect
62. #17989786D4semidirect
86. M16×C293D4semidirect
150. C293#3153semidirect
155. C2×AGL(1,ℤ/8)D293semidirect
228. M16C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276572
C2×#226276542
C2×#226276502
#340366752
C2×#226276112
#340365202
#340365172
#340365122
#340365102
M16×Dic2932
C2×AGL(1,ℤ/8)×C2932
#3153293