Finite Group Databasebeta

← #31790416

Subgroups — #31790416

Order 17536.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984120C211yesmaximal, characteristic, complemented
2C22×#14555493C211yesmaximal, characteristic, complemented
3#22577916C211yesmaximal, characteristic
4C22⋊Q8×D137C211yesmaximal, characteristic, complemented
5#22577660C211yesmaximal, characteristic, complemented
6#22577658C211yesmaximal, characteristic, complemented
7#22577652C211yesmaximal, characteristic, complemented
8#22577650C211yesmaximal, characteristic, complemented
9#22577642C211yesmaximal, characteristic, complemented
10#22577634C211yesmaximal, characteristic, complemented
11#22577632C211yesmaximal, characteristic, complemented
12#22577444C211yesmaximal, characteristic, complemented
13#22577410C211yesmaximal, characteristic, complemented
14#22577403C211yesmaximal, characteristic, complemented
15#467×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19#17984120—24no—
20C2×#14555500C2211yescharacteristic, complemented
21C2×#14555499C2211yescharacteristic, complemented
22C2×#14555497C2211yescharacteristic, complemented
23C2×#14555493—24no—
24C2×#14555493C2211yescharacteristic
25C2×#14555493C2211yescharacteristic
26C2×#14555493C2211yescharacteristic
27C2×#14555493C2211yescharacteristic
28C22×#11764040C2211yescharacteristic
29#17984073—21no—
30#17984068C2211yescharacteristic
31#17984067C2211yescharacteristic
32#17984066—21no—
33#17984065—21no—
34#17984064C2211yescharacteristic
35#17984064C2211yescharacteristic
36#17984063C2211yescharacteristic
37#17984062C2211yescharacteristic
38#17984058—21no—
39#17984050—21no—
40#17984048—21no—
41#17984046C2211yescharacteristic
42C4⋊C4×D137—21no—
43C4⋊C4×D137C2211yescharacteristic
44C22⋊C4×D137—21no—
45#17984021C2211yescharacteristic, complemented
46#17984020C2211yescharacteristic, complemented
47#17984019C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984012C2211yescharacteristic, complemented
50#17984012C2211yescharacteristic, complemented
51#17984011C2211yescharacteristic, complemented
52#17984010C2211yescharacteristic, complemented
53#17984008C2211yescharacteristic, complemented
54C2×Q16×C137C2211yescharacteristic, complemented
55C22×C8×C137C2211yescharacteristic
56C22⋊Q8×C137C2211yescharacteristic
57C22⋊Q8×C137C2211yescharacteristic
58#107×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#103×C137C2211yescharacteristic, complemented
61C23×D137D411yescharacteristic
62#14555511—22no—
63#14555511—22no—
64C2×Dic274—21no—
65C2×Dic274C2311yescharacteristic
66Q8×D137—22no—
67Q8×D137—22no—
68C2×D548—21no—
69C2×D548—21no—
70C2×D548C2311yescharacteristic
71C2×C22⋊D137—21no—
72C22×Dic137D411yescharacteristic
73C2×C4×D137—22no—
74C2×C4×D137—22no—
75C2×C4×D137—21no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137D411yescharacteristic
79C2×C4×D137D411yescharacteristic
80C2×C4×D137C2311yescharacteristic
81C2×C4×D137C2311yescharacteristic
82#14555500—22no—
83#14555499—22no—
84#14555497—22no—
85#14555493—22no—
86#14555493—22no—
87#14555493—22no—
88#14555493D414yescomplemented
89C2×#11764040—22no—
90C2×#11764040C2311yescharacteristic
91C2×#11764040C2311yescharacteristic
92C4⋊Dic137—21no—
93C4⋊Dic137C2311yescharacteristic
94C4⋊Dic137C2311yescharacteristic
95#14555489—21no—
96#14555489—21no—
97#14555489—21no—
98#14555489—21no—
99C22⋊Dic137—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106C4×Dic137—21no—
107C2×Q8×C137C2311yescharacteristic
108C2×Q8×C137C2311yescharacteristic
109C22×C4×C137C2311yescharacteristic
110Q16×C137—22no—
111C2×C8×C137—22no—
112C2×C8×C137C2311yescharacteristic
113C2×C8×C137C2311yescharacteristic
114C4⋊C4×C137—21no—
115C4⋊C4×C137—21no—
116C4⋊C4×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22×D137—22no—
121C22×D137—22no—
122C22×D137—21no—
123C22×D137—41no—
124C22×D137C2×D411yescharacteristic
125C22×D137C2×D411yescharacteristic
126Dic274—21no—
127Dic274—21no—
128Dic274—42no—
129D548—21no—
130D548—21no—
131D548—42no—
132D548—42no—
133C22⋊D137—42no—
134C2×Dic137—22no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137—21no—
138C2×Dic137C2×D411yescharacteristic
139C2×Dic137C2×D411yescharacteristic
140C4×D137—22no—
141C4×D137—21no—
142C4×D137—21no—
143C4×D137—22no—
144C4×D137—42no—
145C4×D137—42no—
146C4×D137—42no—
147C4×D137C2×D412yes—
148C4×D137#4112yes—
149#11764040—21no—
150#11764040C2×D412yes—
151C23×C137C2×D411yescharacteristic
152Q8×C137—21no—
153Q8×C137—21no—
154Q8×C137—21no—
155Q8×C137—21no—
156C2×C4×C137—21no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137C2×D411yescharacteristic
163C2×C4×C137C2411yescharacteristic
164C8×C137—21no—
165C8×C137C2×D412yes—
166C22×C137—21no—
167C22×C137—21no—
168C22×C137#13712yescomplemented
169C22×C137C22×D411yescharacteristic
170C2×D137—21no—
171C2×D137—21no—
172C2×D137—21no—
173C2×D137—21no—
174C2×D137—21no—
175C2×D137—44no—
176C2×D137—41no—
177C2×D137—41no—
178C2×D137—41no—
179C2×D137#12112yes—
180C4×C137—21no—
181C4×C137—41no—
182C4×C137—41no—
183C4×C137—41no—
184C4×C137—41no—
185C4×C137C22×D411yescharacteristic
186C4×C137C2×#4111yescharacteristic
187Dic137—21no—
188Dic137—41no—
189Dic137—41no—
190Dic137—41no—
191Dic137#12112yes—
192C2×C137—22no—
193C2×C137C2×#12111yescharacteristic
194C2×C137C2×#13711yescharacteristic
195C2×C137#57511yescharacteristic
196D137—22no—
197D137—41no—
198D137—81no—
199C137#305611yescharacteristic, complemented, Sylow
200#3056—1371nomaximal, Sylow
201C2×#137—1371no—
202C22×M16—1371no—
203#525—1371no—
204C2×C22⋊Q8—1371no—
205#470—1371no—
206#470—1371no—
207#468—1371no—
208#468—1371no—
209#467—1371no—
210#467D13711yescharacteristic, complemented
211#465—1371no—
212#465—1371no—
213#428—1371no—
214#419—1371no—
215#417—1371no—
216C2×#41—1371no—
217C22×Q8—1371no—
218C23×C4—1371no—
219#137—2744no—
220C2×Q16—1371no—
221C2×Q16C2×D13711yescharacteristic, complemented
222C2×SD16—1371no—
223C2×SD16—1371no—
224C2×M16—2744no—
225C2×M16—1371no—
226C2×M16—1371no—
227C2×M16—1371no—
228C2×M16—1371no—
229C22×C8—1371no—
230C22×C8C2×D13711yescharacteristic
231#123—2741no—
232C22⋊Q8—2741no—
233C22⋊Q8—2741no—
234C22⋊Q8—1371no—
235C22⋊Q8—1371no—
236C22⋊Q8—1371no—
237C22⋊Q8—1371no—
238C22⋊Q8C2×D13711yescharacteristic
239C22⋊Q8C2×D13711yescharacteristic
240#121—1371no—
241#121—1371no—
242#120—2741no—
243C4×D4—2741no—
244C4×D4—2741no—
245#117—1371no—
246C2×C4⋊C4—2741no—
247C2×C4⋊C4—1371no—
248C2×C22⋊C4—2741no—
249#107—1371no—
250#107C2×D13711yescharacteristic, complemented
251#106—1371no—
252#106—1371no—
253#103—1371no—
254#103—1371no—
255#103—1371no—
256#103—1371no—
257#103C2×D13711yescharacteristic, complemented
258#103C2×D13711yescharacteristic, complemented
259#102—1371no—
260#102—1371no—
261C24—1371no—
262#41—2742no—
263#41—2742no—
264C2×Q8—2741no—
265C2×Q8—2742no—
266C2×Q8—2742no—
267C2×Q8—1371no—
268C2×Q8C22×D13711yescharacteristic
269C2×Q8C22×D13711yescharacteristic
270C2×D4—2741no—
271C2×D4—2741no—
272C2×D4—2741no—
273C2×D4—1371no—
274C22×C4—2742no—
275C22×C4—2742no—
276C22×C4—2741no—
277C22×C4—2741no—
278C22×C4—2741no—
279C22×C4—1371no—
280C22×C4—1371no—
281C22×C4—1371no—
282C22×C4—1371no—
283C22×C4—1371no—
284C22×C4C22×D13711yescharacteristic
285Q16—22no—
286Q16—2742no—
287SD16—2742no—
288SD16—2742no—
289M16—2742no—
290M16—2742no—
291M16—2742no—
292M16—1374no—
293C2×C8—22no—
294C2×C8—2742no—
295C2×C8—1371no—
296C2×C8—1371no—
297C2×C8C22×D13711yescharacteristic
298C2×C8C22×D13711yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—2741no—
302C4⋊C4—2741no—
303C4⋊C4—2741no—
304C4⋊C4—2741no—
305C4⋊C4—2741no—
306C4⋊C4—1371no—
307C4⋊C4—1371no—
308C4⋊C4C22×D13711yescharacteristic
309C4⋊C4C22×D13711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2741no—
313C22⋊C4—2741no—
314C22⋊C4—2741no—
315C22⋊C4—2741no—
316C22⋊C4—2741no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C42—2741no—
320C23—2742no—
321C23—2742no—
322C23—2741no—
323C23—5481no—
324C23—1371no—
325C23—1371no—
326C23D4×D13711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—2741no—
332Q8—2741no—
333Q8—5482no—
334D4—2741no—
335D4—2741no—
336D4—5482no—
337D4—5482no—
338D4—5482no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—2742no—
346C2×C4—2742no—
347C2×C4—2742no—
348C2×C4—2741no—
349C2×C4—2741no—
350C2×C4—2741no—
351C2×C4—2741no—
352C2×C4—2741no—
353C2×C4—5482no—
354C2×C4—5482no—
355C2×C4—5482no—
356C2×C4—1371no—
357C2×C4—1371no—
358C2×C4—1372no—
359C2×C4—1372no—
360C2×C4D4×D13711yescharacteristic
361C2×C4C23×D13711yescharacteristic
362C8—21no—
363C8—2741no—
364C8—1372no—
365C8D4×D13712yes—
366C22—21no—
367C22—21no—
368C22—2741no—
369C22—2741no—
370C22—2741no—
371C22—2741no—
372C22—2741no—
373C22—5484no—
374C22—5481no—
375C22—5481no—
376C22—5481no—
377C22—1372no—
378C22#1798412012yescomplemented
379C22C2×D4×D13711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—2741no—
386C4—5481no—
387C4—5481no—
388C4—5481no—
389C4—1372no—
390C4C2×D4×D13711yescharacteristic
391C4#41×D13711yescharacteristic
392C2—22no—
393C2—2742no—
394C2—5481no—
395C2—10961no—
396C2#121×D13711yescharacteristic
397C2C2×#1798412011yescharacteristic
398C2#2257820011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#467×C137C2
socleC22×C137C22×D4
prime radical#22577916C2
hypercenter#467D137
hypocenterC137#3056
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#17984120C2semidirect
2. C22×#14555493C2semidirect
4. C22⋊Q8×D137C2semidirect
5. #22577660C2semidirect
6. #22577658C2semidirect
7. #22577652C2semidirect
8. #22577650C2semidirect
9. #22577642C2semidirect
10. #22577634C2semidirect
11. #22577632C2semidirect
12. #22577444C2semidirect
13. #22577410C2semidirect
14. #22577403C2semidirect
15. #467×C137C2semidirect
20. C2×#14555500C22semidirect
21. C2×#14555499C22semidirect
22. C2×#14555497C22semidirect
45. #17984021C22semidirect
46. #17984020C22semidirect
47. #17984019C22semidirect
48. #17984013C22semidirect
49. #17984012C22semidirect
50. #17984012C22semidirect
51. #17984011C22semidirect
52. #17984010C22semidirect
53. #17984008C22semidirect
54. C2×Q16×C137C22semidirect
58. #107×C137C22semidirect
59. #103×C137C22semidirect
60. #103×C137C22semidirect
88. #14555493D4semidirect
168. C22×C137#137semidirect
199. C137#3056semidirect
210. #467D137semidirect
221. C2×Q16C2×D137semidirect
250. #107C2×D137semidirect
257. #103C2×D137semidirect
258. #103C2×D137semidirect
378. C22#17984120semidirect

Maximal subgroups

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

subgroupindex
C2×#179841202
C22×#145554932
#225779162
C22⋊Q8×D1372
#225776602
#225776582
#225776522
#225776502
#225776422
#225776342
#225776322
#225774442
#225774102
#225774032
#467×C1372
#3056137