Finite Group Databasebeta

← #49170622

Subgroups — #49170622

Order 37504.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036819C211yesmaximal, characteristic, complemented
2#34036775C211yesmaximal, characteristic
3#34036484C211yesmaximal, characteristic, complemented
4#34036460C211yesmaximal, characteristic, complemented
5#34036454C211yesmaximal, characteristic, complemented
6#34036430C211yesmaximal, characteristic, complemented
7#34036370C211yesmaximal, characteristic, complemented
8#34036358C211yesmaximal, characteristic, complemented
9#34036352C211yesmaximal, characteristic, complemented
10#34036293C211yesmaximal, characteristic, complemented
11#34036282C211yesmaximal, characteristic, complemented
12C4×#17989791C211yesmaximal, characteristic, complemented
13#34036126C211yesmaximal, characteristic, complemented
14C2×#22627546C211yesmaximal, characteristic, complemented
15C8×D4×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C2×Dic1172C2211yescharacteristic, complemented
19C2×#17989791C2211yescharacteristic
20C2×#17989791C2211yescharacteristic
21#22627617—21no—
22#22627616C2211yescharacteristic, complemented
23#22627614—21no—
24#22627612C2211yescharacteristic
25#22627609—21no—
26#22627605—21no—
27#22627603C2211yescharacteristic
28#22627603C2211yescharacteristic
29#22627603C2211yescharacteristic, complemented
30#22627598C2211yescharacteristic
31#22627597—21no—
32#22627592—21no—
33C4×Dic586C2211yescharacteristic, complemented
34C4×D1172C2211yescharacteristic
35#22627583C2211yescharacteristic, complemented
36#22627580—21no—
37C2×C4⋊Dic293C2211yescharacteristic
38C22⋊C4×D293—21no—
39#22627558C2211yescharacteristic, complemented
40#22627556C2211yescharacteristic, complemented
41#22627556C2211yescharacteristic, complemented
42#22627549C2211yescharacteristic, complemented
43#22627549C2211yescharacteristic, complemented
44#22627549C2211yescharacteristic, complemented
45#22627549C2211yescharacteristic, complemented
46#22627546C2211yescharacteristic
47#22627546C2211yescharacteristic
48#22627546C2211yescharacteristic
49#22627546C2211yescharacteristic
50#22627546C2211yescharacteristic
51#22627546C2211yescharacteristic
52C22×C8×C293C2211yescharacteristic
53C22×C8×C293C2211yescharacteristic, complemented
54C4×D4×C293C2211yescharacteristic
55C4⋊C8×C293C2211yescharacteristic, complemented
56C22⋊C8×C293C2211yescharacteristic
57C22⋊C8×C293C2211yescharacteristic, complemented
58C4×C8×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60#17989802—22no—
61#17989802—41no—
62C2×Dic586C2311yescharacteristic
63C2×Dic586C2311yescharacteristic
64C2×D1172—22no—
65C2×D1172—21no—
66C2×D1172—41no—
67C2×D1172C2311yescharacteristic
68C2×D1172C2311yescharacteristic
69C2×C22⋊D293—21no—
70C2×C22⋊D293—21no—
71C22×Dic293—21no—
72C2×C4×D293—21no—
73C2×C4×D293—21no—
74Dic1172—41no—
75#17989791—22no—
76#17989791—41no—
77C4⋊Dic293—21no—
78C4⋊Dic293C2311yescharacteristic
79C4⋊Dic293C2311yescharacteristic
80C4⋊Dic293C2311yescharacteristic
81C4⋊Dic293C2311yescharacteristic
82#17989781—21no—
83#17989781—21no—
84#17989781—21no—
85#17989781—21no—
86#17989781—21no—
87#17989781—21no—
88C22⋊Dic293—21no—
89C22⋊Dic293—21no—
90C22⋊Dic293—21no—
91C22⋊Dic293—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96C4×Dic293—21no—
97C4×Dic293—21no—
98C2×D4×C293D411yescharacteristic
99C22×C4×C293C2311yescharacteristic
100C22×C4×C293C2311yescharacteristic
101C2×C8×C293—41no—
102C2×C8×C293—41no—
103C2×C8×C293C2311yescharacteristic
104C2×C8×C293C2311yescharacteristic
105C2×C8×C293C2311yescharacteristic
106C2×C8×C293C2311yescharacteristic
107C4⋊C4×C293D411yescharacteristic
108C22⋊C4×C293D411yescharacteristic
109C22⋊C4×C293D411yescharacteristic
110C42×C293C2311yescharacteristic
111C22×D293—21no—
112C22×D293—21no—
113C22×D293—42no—
114C22×D293—42no—
115C22×D293—81no—
116Dic586—21no—
117Dic586—21no—
118Dic586—21no—
119Dic586—21no—
120D1172—21no—
121D1172—21no—
122D1172—21no—
123D1172—42no—
124D1172#4112yes—
125C22⋊D293—42no—
126C22⋊D293—81no—
127C2×Dic293—21no—
128C2×Dic293—21no—
129C2×Dic293—21no—
130C2×Dic293—21no—
131C2×Dic293—21no—
132C2×Dic293—21no—
133C2×Dic293—42no—
134C4×D293—42no—
135C4×D293—81no—
136C23×C293C2×D411yescharacteristic
137C23×C293C2×D411yescharacteristic
138D4×C293—22no—
139C2×C4×C293—21no—
140C2×C4×C293—21no—
141C2×C4×C293—21no—
142C2×C4×C293—21no—
143C2×C4×C293C2×D411yescharacteristic
144C2×C4×C293C2×D411yescharacteristic
145C2×C4×C293C2×D411yescharacteristic
146C2×C4×C293C2×D411yescharacteristic
147C2×C4×C293C2411yescharacteristic
148C8×C293—21no—
149C8×C293—21no—
150C8×C293—21no—
151C8×C293—21no—
152C22×C293—21no—
153C22×C293—21no—
154C22×C293—21no—
155C22×C293—21no—
156C22×C293—21no—
157C22×C293#13512yes—
158C22×C293C22×D411yescharacteristic
159C2×D293—22no—
160C2×D293—41no—
161C2×D293—41no—
162C2×D293—41no—
163C2×D293—41no—
164C2×D293—41no—
165C2×D293—84no—
166C4×C293—21no—
167C4×C293—21no—
168C4×C293—21no—
169C4×C293—21no—
170C4×C293C2×#4111yescharacteristic
171C4×C293H(5,𝔽2)11yescharacteristic
172Dic293—41no—
173Dic293—41no—
174Dic293—41no—
175Dic293—41no—
176Dic293—41no—
177Dic293—41no—
178C2×C293—22no—
179C2×C293—41no—
180C2×C293#54611yescharacteristic
181C2×C293C2×#13511yescharacteristic
182C2×C293#57711yescharacteristic
183D293—42no—
184D293—81no—
185C293#329611yescharacteristic, complemented, Sylow
186#3296—2931nomaximal, Sylow
187#552—2931no—
188#546—2931no—
189#488—2931no—
190#483—2931no—
191#482—2931no—
192#478—2931no—
193#467—2931no—
194#465—2931no—
195#464—2931no—
196#452—2931no—
197#450—2931no—
198C4×SD16—2931no—
199C8×D4D29311yescharacteristic, complemented
200#416—2931no—
201C2×#102—2931no—
202C2×#41—2931no—
203C22×D4—2931no—
204C2×Q16—2931no—
205C2×SD16—2931no—
206C2×SD16—2931no—
207C22×C8C2×D29311yescharacteristic
208C22×C8C2×D29311yescharacteristic, complemented
209#126—5861no—
210#125—2931no—
211#125—5861no—
212#124—2931no—
213#123—5861no—
214#123—5861no—
215C22⋊Q8—2931no—
216C22⋊Q8—2931no—
217C22⋊Q8—2931no—
218#121—2931no—
219#121—5861no—
220#120—5861no—
221C4×Q8—2931no—
222C4×D4—2931no—
223C4×D4C2×D29311yescharacteristic
224#117—2931no—
225#117—5861no—
226C2×C4⋊C4—2931no—
227C2×C22⋊C4—5861no—
228#107—2931no—
229#106—2931no—
230#106—2931no—
231C4⋊C8C2×D29311yescharacteristic, complemented
232#103—2931no—
233#103—2931no—
234#103—2931no—
235#103—2931no—
236#102—2931no—
237#102—2931no—
238#102—2931no—
239#102—2931no—
240#102—2931no—
241#102—2931no—
242C22⋊C8C2×D29311yescharacteristic
243C22⋊C8C2×D29311yescharacteristic, complemented
244C4×C8C2×D29311yescharacteristic, complemented
245C24—5861no—
246#41—5862no—
247#41—11721no—
248C2×Q8—2931no—
249C2×Q8—2931no—
250C2×D4—2931no—
251C2×D4—2931no—
252C2×D4—5862no—
253C2×D4—5861no—
254C2×D4—5861no—
255C2×D4—5861no—
256C2×D4—11721no—
257C2×D4D117211yescharacteristic
258C22×C4—5861no—
259C22×C4—5861no—
260C22×C4—5861no—
261C22×C4C22×D29311yescharacteristic
262C22×C4C22×D29311yescharacteristic
263Q16—11721no—
264SD16—5862no—
265SD16—11721no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D29311yescharacteristic
269C2×C8C22×D29311yescharacteristic
270C2×C8C22×D29311yescharacteristic
271C2×C8C22×D29311yescharacteristic
272C4⋊C4—2931no—
273C4⋊C4—2931no—
274C4⋊C4—2931no—
275C4⋊C4—2931no—
276C4⋊C4—5861no—
277C4⋊C4—5861no—
278C4⋊C4—5861no—
279C4⋊C4—5861no—
280C4⋊C4—5861no—
281C4⋊C4—5861no—
282C4⋊C4—5861no—
283C4⋊C4D117211yescharacteristic
284C22⋊C4—5861no—
285C22⋊C4—5861no—
286C22⋊C4—5861no—
287C22⋊C4—5861no—
288C22⋊C4—5861no—
289C22⋊C4—5861no—
290C22⋊C4—5861no—
291C22⋊C4—5861no—
292C22⋊C4D117211yescharacteristic
293C22⋊C4D117211yescharacteristic
294C42—5861no—
295C42—5861no—
296C42C22×D29311yescharacteristic
297C23—23441no—
298C23—5861no—
299C23—5861no—
300C23—11722no—
301C23—11722no—
302C23C2×D117211yescharacteristic
303C23C2×D117211yescharacteristic
304Q8—5861no—
305Q8—5861no—
306Q8—5861no—
307Q8—5861no—
308D4—22no—
309D4—2932no—
310D4—23441no—
311D4—5861no—
312D4—5861no—
313D4—5861no—
314D4—11722no—
315D4—11722no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—23441no—
321C2×C4—5861no—
322C2×C4—5861no—
323C2×C4—5861no—
324C2×C4—5861no—
325C2×C4—5861no—
326C2×C4—5861no—
327C2×C4—11722no—
328C2×C4—11722no—
329C2×C4C2×D117211yescharacteristic
330C2×C4C2×D117211yescharacteristic
331C2×C4C2×D117211yescharacteristic
332C2×C4C2×D117211yescharacteristic
333C2×C4C23×D29311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—23444no—
344C22—5862no—
345C22—11721no—
346C22—11721no—
347C22—11721no—
348C22—11721no—
349C22—11721no—
350C22#2262764612yes—
351C22C22×D117211yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—11721no—
357C4—11721no—
358C4—11721no—
359C4—11721no—
360C4—11721no—
361C4—11721no—
362C4#41×D29311yescharacteristic
363C4#2262766911yescharacteristic
364C2—22no—
365C2—41no—
366C2—23441no—
367C2—11722no—
368C2#3403677411yescharacteristic
369C2C2×#2262764611yescharacteristic
370C2#3403694111yescharacteristic
subgroupG/H
centerC22C22×D1172
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC8×D4×C293C2
socleC22×C293C22×D4
prime radical#34036775C2
hypercenterC8×D4D293
hypocenterC293#3296
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #34036819C2semidirect
3. #34036484C2semidirect
4. #34036460C2semidirect
5. #34036454C2semidirect
6. #34036430C2semidirect
7. #34036370C2semidirect
8. #34036358C2semidirect
9. #34036352C2semidirect
10. #34036293C2semidirect
11. #34036282C2semidirect
12. C4×#17989791C2semidirect
13. #34036126C2semidirect
14. C2×#22627546C2semidirect
15. C8×D4×C293C2semidirect
18. C2×Dic1172C22semidirect
22. #22627616C22semidirect
29. #22627603C22semidirect
33. C4×Dic586C22semidirect
35. #22627583C22semidirect
39. #22627558C22semidirect
40. #22627556C22semidirect
41. #22627556C22semidirect
42. #22627549C22semidirect
43. #22627549C22semidirect
44. #22627549C22semidirect
45. #22627549C22semidirect
53. C22×C8×C293C22semidirect
55. C4⋊C8×C293C22semidirect
57. C22⋊C8×C293C22semidirect
58. C4×C8×C293C22semidirect
185. C293#3296semidirect
199. C8×D4D293semidirect
208. C22×C8C2×D293semidirect
231. C4⋊C8C2×D293semidirect
243. C22⋊C8C2×D293semidirect
244. C4×C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368192
#340367752
#340364842
#340364602
#340364542
#340364302
#340363702
#340363582
#340363522
#340362932
#340362822
C4×#179897912
#340361262
C2×#226275462
C8×D4×C2932
#3296293