Finite Group Databasebeta

← #31790251

Subgroups — #31790251

Order 17536.

548 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984115C211yesmaximal, characteristic, complemented
2C2×#17984112C211yesmaximal, characteristic, complemented
3#22577952C211yesmaximal, characteristic, complemented
4C4×D4×D137C211yesmaximal, characteristic, complemented
5#22577619C211yesmaximal, characteristic, complemented
6#22577618C211yesmaximal, characteristic, complemented
7#22577615C211yesmaximal, characteristic, complemented
8#22577607C211yesmaximal, characteristic, complemented
9#22577606C211yesmaximal, characteristic, complemented
10#22577597C211yesmaximal, characteristic, complemented
11#22577596C211yesmaximal, characteristic, complemented
12#22577425C211yesmaximal, characteristic, complemented
13#22577403C211yesmaximal, characteristic, complemented
14#22577401C211yesmaximal, characteristic, complemented
15#463×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×#14555509C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19C22×C4×D137—21no—
20#17984115—24no—
21#17984112—24no—
22C2×#14555499C2211yescharacteristic
23C2×#14555498C2211yescharacteristic
24C2×#14555497C2211yescharacteristic
25C2×D1096C2211yescharacteristic
26C2×#14555495C2211yescharacteristic
27C2×#14555495C2211yescharacteristic
28C2×#14555493C2211yescharacteristic
29C2×#14555493C2211yescharacteristic
30#17984086C2211yescharacteristic, complemented
31#17984084C2211yescharacteristic, complemented
32#17984076C2211yescharacteristic, complemented
33#17984061—21no—
34#17984061—21no—
35C4×D548C2211yescharacteristic
36D4×Dic137C2211yescharacteristic
37#17984050C2211yescharacteristic
38C4×C22⋊D137—21no—
39#17984048—21no—
40#17984048—21no—
41#17984048—21no—
42C4⋊C4×D137C2211yescharacteristic
43C22⋊C4×D137—21no—
44C42×D137C2211yescharacteristic
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984017C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984012C2211yescharacteristic, complemented
50#17984011C2211yescharacteristic, complemented
51#17984010C2211yescharacteristic, complemented
52#17984009C2211yescharacteristic, complemented
53#17984008C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic
55C2×D8×C137C2211yescharacteristic
56#124×C137C2211yescharacteristic, complemented
57C4×D4×C137C2211yescharacteristic
58C4⋊C8×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555511—22no—
63#14555511—22no—
64#14555509—22no—
65#14555509—22no—
66C2×Dic274C2311yescharacteristic
67C2×D548—21no—
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74D4×D137—22no—
75D4×D137D414yescomplemented
76C22×Dic137—21no—
77C22×Dic137—21no—
78C2×C4×D137—21no—
79C2×C4×D137—21no—
80C2×C4×D137—21no—
81C2×C4×D137—21no—
82C2×C4×D137—44no—
83C2×C4×D137D411yescharacteristic
84C2×C4×D137D411yescharacteristic
85C2×C4×D137C2311yescharacteristic
86#14555499—22no—
87#14555498—22no—
88#14555497—22no—
89D1096—22no—
90#14555495—22no—
91#14555495—22no—
92#14555493—22no—
93#14555493—22no—
94C2×#11764040C2311yescharacteristic
95C2×#11764040C2311yescharacteristic
96C4⋊Dic137C2311yescharacteristic
97#14555489—21no—
98#14555489—21no—
99#14555489—21no—
100C22⋊Dic137—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105C4×Dic137D411yescharacteristic
106C4×Dic137D411yescharacteristic
107C4×Dic137C2311yescharacteristic
108C2×Q8×C137C2311yescharacteristic
109C2×D4×C137C2311yescharacteristic
110C2×D4×C137C2311yescharacteristic
111C22×C4×C137—21no—
112SD16×C137—22no—
113D8×C137—22no—
114C2×C8×C137C2311yescharacteristic
115C2×C8×C137C2311yescharacteristic
116C4⋊C4×C137C2311yescharacteristic
117C22⋊C4×C137—21no—
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C42×C137C2311yescharacteristic
121C22×D137—21no—
122C22×D137—24no—
123C22×D137—21no—
124C22×D137—42no—
125C22×D137—42no—
126C22×D137C2×D411yescharacteristic
127Dic274—21no—
128Dic274—21no—
129D548—21no—
130D548—21no—
131D548—21no—
132D548—42no—
133D548C2×D412yes—
134C22⋊D137—24no—
135C22⋊D137—42no—
136C22⋊D137—42no—
137C22⋊D137—42no—
138C2×Dic137—21no—
139C2×Dic137—21no—
140C2×Dic137—42no—
141C2×Dic137—42no—
142C2×Dic137C2×D411yescharacteristic
143C2×Dic137C2×D411yescharacteristic
144C2×Dic137C2×D411yescharacteristic
145C4×D137—22no—
146C4×D137—41no—
147C4×D137—42no—
148C4×D137—42no—
149C4×D137—42no—
150C4×D137—42no—
151C4×D137C2×D412yes—
152C4×D137#4112yes—
153#11764040—21no—
154#11764040—21no—
155C23×C137—21no—
156C23×C137—21no—
157Q8×C137—21no—
158Q8×C137—21no—
159D4×C137—21no—
160D4×C137—21no—
161D4×C137—21no—
162D4×C137C2×D412yes—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—42no—
166C2×C4×C137C2×D411yescharacteristic
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C8×C137—21no—
170C8×C137—21no—
171C22×C137—22no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137C22×D411yescharacteristic
178C2×D137—21no—
179C2×D137—21no—
180C2×D137—22no—
181C2×D137—48no—
182C2×D137—41no—
183C2×D137—41no—
184C2×D137—41no—
185C2×D137—41no—
186C2×D137—41no—
187C2×D137#12112yes—
188C4×C137—21no—
189C4×C137—21no—
190C4×C137—41no—
191C4×C137—41no—
192C4×C137C22×D411yescharacteristic
193C4×C137C2×#4111yescharacteristic
194Dic137—21no—
195Dic137—41no—
196Dic137—41no—
197Dic137#12112yes—
198Dic137AGL(1,ℤ/8)12yescomplemented
199C2×C137—42no—
200C2×C137—81no—
201C2×C137C2×#12111yescharacteristic
202C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C137#57511yescharacteristic
204D137—22no—
205D137—42no—
206D137—81no—
207C137#303611yescharacteristic, complemented, Sylow
208#3036—1371nomaximal, Sylow
209C2×AGL(1,ℤ/8)—1371no—
210C2×AGL(1,ℤ/8)—1371no—
211#532—1371no—
212C2×C4×D4—1371no—
213#463—1371no—
214#463—1371no—
215#463—1371no—
216#463D13711yescharacteristic, complemented
217#461—1371no—
218#461—1371no—
219#459—1371no—
220#459—1371no—
221#423—1371no—
222#417—1371no—
223#417—1371no—
224C2×#41—1371no—
225C2×#41—1371no—
226C22×D4—1371no—
227C23×C4—2741no—
228AGL(1,ℤ/8)—2744no—
229AGL(1,ℤ/8)—2744no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×SD16—1371no—
233C2×SD16C2×D13711yescharacteristic
234C2×D8—1371no—
235C2×D8—1371no—
236C2×D8—1371no—
237C2×D8C2×D13711yescharacteristic
238C2×M16—1371no—
239C2×M16—1371no—
240C4⋊Q8—1371no—
241#127—1371no—
242#124—1371no—
243#124C2×D13711yescharacteristic, complemented
244#121—2741no—
245#121—2741no—
246C4×D4—2741no—
247C4×D4—2741no—
248C4×D4—2741no—
249C4×D4—2741no—
250C4×D4—1371no—
251C4×D4—1371no—
252C4×D4—1371no—
253C4×D4C2×D13711yescharacteristic
254C2×C4⋊C4—1371no—
255C2×C22⋊C4—2741no—
256C2×C42—1371no—
257C4⋊C8—1371no—
258C4⋊C8—1371no—
259C4⋊C8—1371no—
260C4⋊C8C2×D13711yescharacteristic, complemented
261#103—1371no—
262#103—1371no—
263#103—1371no—
264#103C2×D13711yescharacteristic, complemented
265#102—1371no—
266#102—1371no—
267#102—1371no—
268#102C2×D13711yescharacteristic, complemented
269C24—2741no—
270#41—2742no—
271#41—2742no—
272#41—2742no—
273#41—2742no—
274C2×Q8—1371no—
275C2×Q8C22×D13711yescharacteristic
276C2×D4—2741no—
277C2×D4—2741no—
278C2×D4—2742no—
279C2×D4—2741no—
280C2×D4—2741no—
281C2×D4—2741no—
282C2×D4—1374no—
283C2×D4—1371no—
284C2×D4—1371no—
285C2×D4C22×D13711yescharacteristic
286C2×D4C22×D13711yescharacteristic
287C22×C4—21no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4—2741no—
292C22×C4—2741no—
293C22×C4—2741no—
294C22×C4—5484no—
295C22×C4—1371no—
296C22×C4—1371no—
297C22×C4—1371no—
298SD16—22no—
299SD16—2742no—
300SD16—2742no—
301SD16—2742no—
302D8—22no—
303D8—2742no—
304D8—2742no—
305D8—2742no—
306M16—2742no—
307M16—2742no—
308C2×C8—1371no—
309C2×C8—1371no—
310C2×C8C22×D13711yescharacteristic
311C2×C8C22×D13711yescharacteristic
312C4⋊C4—2741no—
313C4⋊C4—2741no—
314C4⋊C4—2741no—
315C4⋊C4—1371no—
316C4⋊C4C22×D13711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C42—1371no—
326C42—1371no—
327C42—1371no—
328C42C22×D13711yescharacteristic
329C23—21no—
330C23—21no—
331C23—2744no—
332C23—2741no—
333C23—2741no—
334C23—5482no—
335C23—5482no—
336C23—1371no—
337Q8—21no—
338Q8—21no—
339Q8—2741no—
340Q8—2741no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—2744no—
345D4—2741no—
346D4—2741no—
347D4—2741no—
348D4—5482no—
349D4—5482no—
350D4—5482no—
351D4—5482no—
352D4—1372no—
353D4D4×D13712yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—2742no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—5481no—
361C2×C4—5482no—
362C2×C4—5482no—
363C2×C4—5482no—
364C2×C4—5482no—
365C2×C4—5482no—
366C2×C4—5482no—
367C2×C4—1371no—
368C2×C4—1371no—
369C2×C4—1371no—
370C2×C4—1372no—
371C2×C4—1372no—
372C2×C4D4×D13711yescharacteristic
373C2×C4D4×D13711yescharacteristic
374C2×C4C23×D13711yescharacteristic
375C8—21no—
376C8—21no—
377C8—2741no—
378C8—2741no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—2741no—
386C22—2741no—
387C22—2742no—
388C22—5488no—
389C22—5481no—
390C22—5481no—
391C22—5481no—
392C22—5481no—
393C22—5481no—
394C22—1372no—
395C22C2×D4×D13711yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—2741no—
401C4—5481no—
402C4—5481no—
403C4—1372no—
404C4—1372no—
405C4C2×D4×D13711yescharacteristic
406C4#41×D13711yescharacteristic
407C2—42no—
408C2—81no—
409C2—2742no—
410C2—5482no—
411C2—10961no—
412C2#121×D13711yescharacteristic
413C2AGL(1,ℤ/8)×D13711yescharacteristic
414C2#2257820011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#463×C137C2
socleC22×C137C22×D4
hypercenter#463D137
hypocenterC137#3036
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. C2×#17984112C2semidirect
3. #22577952C2semidirect
4. C4×D4×D137C2semidirect
5. #22577619C2semidirect
6. #22577618C2semidirect
7. #22577615C2semidirect
8. #22577607C2semidirect
9. #22577606C2semidirect
10. #22577597C2semidirect
11. #22577596C2semidirect
12. #22577425C2semidirect
13. #22577403C2semidirect
14. #22577401C2semidirect
15. #463×C137C2semidirect
30. #17984086C22semidirect
31. #17984084C22semidirect
32. #17984076C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984017C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984012C22semidirect
50. #17984011C22semidirect
51. #17984010C22semidirect
52. #17984009C22semidirect
53. #17984008C22semidirect
56. #124×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #103×C137C22semidirect
60. #102×C137C22semidirect
75. D4×D137D4semidirect
198. Dic137AGL(1,ℤ/8)semidirect
207. C137#3036semidirect
216. #463D137semidirect
243. #124C2×D137semidirect
260. C4⋊C8C2×D137semidirect
264. #103C2×D137semidirect
268. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
C2×#179841122
#225779522
C4×D4×D1372
#225776192
#225776182
#225776152
#225776072
#225776062
#225775972
#225775962
#225774252
#225774032
#225774012
#463×C1372
#3036137