Finite Group Databasebeta

← #49170607

Subgroups — #49170607

Order 37504.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036834C211yesmaximal, characteristic, complemented
2#34036763C211yesmaximal, characteristic
3#34036474C211yesmaximal, characteristic, complemented
4#34036449C211yesmaximal, characteristic, complemented
5#34036418C211yesmaximal, characteristic, complemented
6#34036359C211yesmaximal, characteristic, complemented
7#34036356C211yesmaximal, characteristic, complemented
8#34036328C211yesmaximal, characteristic, complemented
9#34036282C211yesmaximal, characteristic, complemented
10#34036281C211yesmaximal, characteristic, complemented
11#34036216C211yesmaximal, characteristic, complemented
12#34036192C211yesmaximal, characteristic, complemented
13C2×#22627544C211yesmaximal, characteristic, complemented
14#102×D293C211yesmaximal, characteristic, complemented
15#481×C293C211yesmaximal, characteristic, complemented
16C22×D1172C2211yescharacteristic
17C2×D4×D293—21no—
18C2×D4×D293C2211yescharacteristic
19C2×#17989791C2211yescharacteristic
20C2×#17989791C2211yescharacteristic, complemented
21C2×#17989790C2211yescharacteristic, complemented
22C22×#14556873C2211yescharacteristic, complemented
23C2×C8×D293C2211yescharacteristic, complemented
24#22627623C2211yescharacteristic, complemented
25#22627620C2211yescharacteristic
26#22627614—21no—
27#22627604C2211yescharacteristic, complemented
28#22627603—21no—
29#22627603C2211yescharacteristic, complemented
30#22627602—21no—
31#22627599C2211yescharacteristic
32#22627597—21no—
33#22627594—21no—
34#22627592—21no—
35#22627589C2211yescharacteristic, complemented
36#22627586C2211yescharacteristic
37C4×C22⋊D293—21no—
38C4×C22⋊D293C2211yescharacteristic
39C4⋊C4×D293C2211yescharacteristic, complemented
40#22627555C2211yescharacteristic, complemented
41#22627553C2211yescharacteristic, complemented
42#22627547C2211yescharacteristic, complemented
43#22627546C2211yescharacteristic
44#22627546C2211yescharacteristic, complemented
45#22627545C2211yescharacteristic, complemented
46#22627544C2211yescharacteristic
47#22627544C2211yescharacteristic
48#22627544C2211yescharacteristic, complemented
49#22627544C2211yescharacteristic, complemented
50#22627536C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52#121×C293C2211yescharacteristic
53C2×C4⋊C4×C293C2211yescharacteristic, complemented
54#106×C293C2211yescharacteristic, complemented
55#106×C293C2211yescharacteristic, complemented
56#102×C293C2211yescharacteristic, complemented
57#102×C293C2211yescharacteristic, complemented
58C22⋊C8×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60C23×D293—21no—
61C2×Dic586—21no—
62C2×Dic586C2311yescharacteristic
63C2×D1172—22no—
64C2×D1172—21no—
65C2×D1172—21no—
66C2×D1172—41no—
67C2×D1172C2311yescharacteristic
68C2×D1172C2311yescharacteristic
69C2×C22⋊D293—21no—
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C22⋊D293D411yescharacteristic
73D4×D293—22no—
74D4×D293—41no—
75D4×D293—44no—
76C2×C4×D293—21no—
77C2×C4×D293—21no—
78C2×C4×D293C2311yescharacteristic
79#17989791—22no—
80#17989791—41no—
81#17989790—41no—
82C2×#14556873—41no—
83C2×#14556873C2311yescharacteristic
84C2×#14556873C2311yescharacteristic
85C8×D293—41no—
86C4⋊Dic293—21no—
87C4⋊Dic293C2311yescharacteristic
88#17989781—21no—
89#17989781—21no—
90#17989781—21no—
91#17989781—21no—
92#17989781D411yescharacteristic
93C22⋊Dic293—21no—
94C22⋊Dic293D411yescharacteristic
95#17989779—21no—
96#17989779—21no—
97#17989779—21no—
98#17989779D411yescharacteristic
99C4×Dic293—21no—
100C4×Dic293C2311yescharacteristic
101C2×D4×C293—21no—
102C2×D4×C293C2311yescharacteristic
103C22×C4×C293—21no—
104C22×C4×C293C2311yescharacteristic
105C2×C8×C293C2311yescharacteristic
106C2×C8×C293C2311yescharacteristic
107C4⋊C4×C293—21no—
108C4⋊C4×C293C2311yescharacteristic
109C4⋊C4×C293C2311yescharacteristic
110C4⋊C4×C293C2311yescharacteristic
111C22⋊C4×C293—21no—
112C22×D293—21no—
113C22×D293—21no—
114C22×D293—42no—
115C22×D293—42no—
116C22×D293—42no—
117C22×D293—42no—
118C22×D293—81no—
119C22×D293—81no—
120C22×D293C2×D411yescharacteristic
121Dic586—21no—
122Dic586—21no—
123Dic586—81no—
124D1172—21no—
125D1172—21no—
126D1172—21no—
127D1172—42no—
128D1172—42no—
129D1172#4112yes—
130C22⋊D293—42no—
131C22⋊D293—42no—
132C22⋊D293—81no—
133C22⋊D293SD1614yescomplemented
134C2×Dic293—21no—
135C2×Dic293—21no—
136C2×Dic293C2×D411yescharacteristic
137C2×Dic293C2×D411yescharacteristic
138C2×Dic293C2×D411yescharacteristic
139C4×D293—21no—
140C4×D293—21no—
141C4×D293—42no—
142C4×D293—81no—
143#14556873—21no—
144#14556873—21no—
145C23×C293—21no—
146C23×C293C2×D411yescharacteristic
147D4×C293—21no—
148D4×C293—21no—
149D4×C293—42no—
150C2×C4×C293—21no—
151C2×C4×C293—21no—
152C2×C4×C293—21no—
153C2×C4×C293—21no—
154C2×C4×C293—21no—
155C2×C4×C293—81no—
156C2×C4×C293C2×D411yescharacteristic
157C2×C4×C293C2411yescharacteristic
158C8×C293—21no—
159C8×C293—21no—
160C22×C293—21no—
161C22×C293—21no—
162C22×C293—41no—
163C22×C293—41no—
164C22×C293—41no—
165C22×C293C2×SD1612yes—
166C22×C293C22×D411yescharacteristic
167C2×D293—21no—
168C2×D293—21no—
169C2×D293—22no—
170C2×D293—41no—
171C2×D293—41no—
172C2×D293—41no—
173C2×D293—41no—
174C2×D293—41no—
175C2×D293—84no—
176C2×D293—84no—
177C2×D293C2×SD1612yes—
178C4×C293—21no—
179C4×C293—41no—
180C4×C293—41no—
181C4×C293—41no—
182C4×C293C2×#4111yescharacteristic
183C4×C293H(5,𝔽2)11yescharacteristic
184Dic293—21no—
185Dic293—21no—
186Dic293—41no—
187Dic293—41no—
188Dic293C2×SD1612yes—
189C2×C293—22no—
190C2×C293—81no—
191C2×C293#54611yescharacteristic
192C2×C293C22×SD1611yescharacteristic
193C2×C293#57611yescharacteristic
194D293—22no—
195D293—42no—
196D293—81no—
197C293#329511yescharacteristic, complemented, Sylow
198#3295—2931nomaximal, Sylow
199#554—2931no—
200D42—2931no—
201#486—2931no—
202#481—2931no—
203#481D29311yescharacteristic, complemented
204#476—2931no—
205#465—2931no—
206#465—2931no—
207#460—2931no—
208#450—2931no—
209#450—2931no—
210C4×SD16—2931no—
211C8×D4—2931no—
212C2×#102—2931no—
213C2×#102—2931no—
214C22×D4—2931no—
215C22×D4—2931no—
216C22×D4—5861no—
217C2×SD16—2931no—
218C2×SD16—2931no—
219C2×SD16—2931no—
220C22×C8—2931no—
221C22×C8—2931no—
222C4⋊Q8—2931no—
223#127—2931no—
224#125—5861no—
225C22⋊Q8—2931no—
226C22⋊Q8—2931no—
227C22⋊Q8—5861no—
228C22⋊Q8—5861no—
229#121—2931no—
230#121—5861no—
231#121C2×D29311yescharacteristic
232#120—5861no—
233#120—5861no—
234C4×Q8—2931no—
235C4×D4—2931no—
236C4×D4—2931no—
237C4×D4—5861no—
238C2×C4⋊C4—2931no—
239C2×C4⋊C4C2×D29311yescharacteristic, complemented
240#106—2931no—
241#106C2×D29311yescharacteristic, complemented
242#106C2×D29311yescharacteristic, complemented
243C4⋊C8—2931no—
244#103—2931no—
245#102—2931no—
246#102—2931no—
247#102—2931no—
248#102—2931no—
249#102—2931no—
250#102—2931no—
251#102—2931no—
252#102C2×D29311yescharacteristic, complemented
253#102C2×D29311yescharacteristic, complemented
254C22⋊C8—2931no—
255C22⋊C8C2×D29311yescharacteristic, complemented
256C4×C8—2931no—
257C24—5861no—
258C24—5861no—
259C2×Q8—2931no—
260C2×Q8—5861no—
261C2×D4—21no—
262C2×D4—2931no—
263C2×D4—2931no—
264C2×D4—2931no—
265C2×D4—5862no—
266C2×D4—5861no—
267C2×D4—5861no—
268C2×D4—5861no—
269C2×D4—5861no—
270C2×D4—5862no—
271C2×D4—5861no—
272C2×D4—11721no—
273C2×D4—11721no—
274C2×D4—11724no—
275C2×D4C22×D29311yescharacteristic
276C22×C4—21no—
277C22×C4—2931no—
278C22×C4—5861no—
279C22×C4—5861no—
280C22×C4C22×D29311yescharacteristic
281SD16—5862no—
282SD16—11721no—
283SD16—11721no—
284C2×C8—2931no—
285C2×C8—2931no—
286C2×C8—11721no—
287C2×C8—11721no—
288C2×C8C22×D29311yescharacteristic
289C2×C8C22×D29311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2931no—
292C4⋊C4—2931no—
293C4⋊C4—5861no—
294C4⋊C4—5861no—
295C4⋊C4—5861no—
296C4⋊C4—5861no—
297C4⋊C4—5861no—
298C4⋊C4C22×D29311yescharacteristic
299C4⋊C4C22×D29311yescharacteristic
300C4⋊C4C22×D29311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—2931no—
303C22⋊C4—2931no—
304C22⋊C4—5861no—
305C22⋊C4—5861no—
306C22⋊C4—5861no—
307C22⋊C4—5861no—
308C42—2931no—
309C42—5861no—
310C23—21no—
311C23—2931no—
312C23—23441no—
313C23—23441no—
314C23—5861no—
315C23—5861no—
316C23—11722no—
317C23—11722no—
318C23—11722no—
319C23—11722no—
320C23D4×D29311yescharacteristic
321Q8—23441no—
322Q8—5861no—
323Q8—5861no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—2934no—
328D4—2932no—
329D4—23441no—
330D4—5861no—
331D4—5861no—
332D4—5861no—
333D4—11722no—
334D4—11722no—
335D4—11722no—
336D4—11722no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—81no—
343C2×C4—2931no—
344C2×C4—2931no—
345C2×C4—2931no—
346C2×C4—23441no—
347C2×C4—5861no—
348C2×C4—5861no—
349C2×C4—5861no—
350C2×C4—5861no—
351C2×C4—11722no—
352C2×C4D4×D29311yescharacteristic
353C2×C4C23×D29311yescharacteristic
354C8—21no—
355C8—21no—
356C8—5861no—
357C8—5861no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2932no—
364C22—23444no—
365C22—23444no—
366C22—5861no—
367C22—5861no—
368C22—5862no—
369C22—11721no—
370C22—11721no—
371C22—11721no—
372C22—11721no—
373C22—11721no—
374C22SD16×D29312yes—
375C22C2×D4×D29311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2932no—
381C4—5861no—
382C4—5861no—
383C4—11721no—
384C4—11721no—
385C4#41×D29311yescharacteristic
386C4#2262767011yescharacteristic
387C2—22no—
388C2—81no—
389C2—23441no—
390C2—5862no—
391C2—11722no—
392C2#3403677111yescharacteristic
393C2C2×SD16×D29311yescharacteristic
394C2#3403693311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#481×C293C2
socleC22×C293C22×D4
prime radical#34036763C2
hypercenter#481D293
hypocenterC293#3295
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #34036834C2semidirect
3. #34036474C2semidirect
4. #34036449C2semidirect
5. #34036418C2semidirect
6. #34036359C2semidirect
7. #34036356C2semidirect
8. #34036328C2semidirect
9. #34036282C2semidirect
10. #34036281C2semidirect
11. #34036216C2semidirect
12. #34036192C2semidirect
13. C2×#22627544C2semidirect
14. #102×D293C2semidirect
15. #481×C293C2semidirect
20. C2×#17989791C22semidirect
21. C2×#17989790C22semidirect
22. C22×#14556873C22semidirect
23. C2×C8×D293C22semidirect
24. #22627623C22semidirect
27. #22627604C22semidirect
29. #22627603C22semidirect
35. #22627589C22semidirect
39. C4⋊C4×D293C22semidirect
40. #22627555C22semidirect
41. #22627553C22semidirect
42. #22627547C22semidirect
44. #22627546C22semidirect
45. #22627545C22semidirect
48. #22627544C22semidirect
49. #22627544C22semidirect
50. #22627536C22semidirect
51. C8×Dic293C22semidirect
53. C2×C4⋊C4×C293C22semidirect
54. #106×C293C22semidirect
55. #106×C293C22semidirect
56. #102×C293C22semidirect
57. #102×C293C22semidirect
58. C22⋊C8×C293C22semidirect
133. C22⋊D293SD16semidirect
197. C293#3295semidirect
203. #481D293semidirect
239. C2×C4⋊C4C2×D293semidirect
241. #106C2×D293semidirect
242. #106C2×D293semidirect
252. #102C2×D293semidirect
253. #102C2×D293semidirect
255. C22⋊C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368342
#340367632
#340364742
#340364492
#340364182
#340363592
#340363562
#340363282
#340362822
#340362812
#340362162
#340361922
C2×#226275442
#102×D2932
#481×C2932
#3295293