Finite Group Databasebeta

← #31790323

Subgroups — #31790323

Order 17536.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984112C211yesmaximal, characteristic, complemented
2C2×#17984107C211yesmaximal, characteristic, complemented
3#22577966C211yesmaximal, characteristic, complemented
4#22577882C211yesmaximal, characteristic, complemented
5#22577626C211yesmaximal, characteristic, complemented
6#22577624C211yesmaximal, characteristic, complemented
7#22577618C211yesmaximal, characteristic, complemented
8#22577616C211yesmaximal, characteristic, complemented
9#22577602C211yesmaximal, characteristic, complemented
10#22577600C211yesmaximal, characteristic, complemented
11#22577596C211yesmaximal, characteristic, complemented
12#22577428C211yesmaximal, characteristic, complemented
13#22577401C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#459×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×#14555509C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984112—24no—
20#17984107—24no—
21C2×Dic548C2211yescharacteristic, complemented
22C2×#14555499C2211yescharacteristic
23C2×#14555498C2211yescharacteristic
24C2×#14555498C2211yescharacteristic, complemented
25C2×#14555498C2211yescharacteristic, complemented
26C2×#14555495C2211yescharacteristic
27C2×#14555493C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic
29#17984084C2211yescharacteristic
30#17984076C2211yescharacteristic
31#17984075C2211yescharacteristic
32#17984060—21no—
33#17984060—21no—
34#17984056—21no—
35#17984056—21no—
36C4×Dic274C2211yescharacteristic, complemented
37#17984053C2211yescharacteristic, complemented
38D4×Dic137C2211yescharacteristic, complemented
39C4×C22⋊D137—21no—
40#17984048—21no—
41#17984045C2211yescharacteristic
42#17984044—21no—
43C2×#14555489—21no—
44C4⋊C4×D137C2211yescharacteristic
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984017C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984011C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic
51#17984009C2211yescharacteristic
52#17984009C2211yescharacteristic, complemented
53#17984008C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic
55C2×D8×C137C2211yescharacteristic, complemented
56#127×C137C2211yescharacteristic
57C4×D4×C137C2211yescharacteristic, complemented
58C4⋊C8×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555509—22no—
63#14555509—22no—
64#14555509—22no—
65#14555509D414yescomplemented
66C2×Dic274C2311yescharacteristic
67C2×Dic274C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74D4×D137—22no—
75D4×D137—22no—
76C22×Dic137—21no—
77C22×Dic137—21no—
78C22×Dic137—21no—
79C2×C4×D137—21no—
80C2×C4×D137C2311yescharacteristic
81Dic548—22no—
82#14555499—22no—
83#14555498—22no—
84#14555498—22no—
85#14555498—22no—
86#14555495—22no—
87#14555493—22no—
88C2×#11764040C2311yescharacteristic
89C2×#11764040C2311yescharacteristic
90C8×D137—22no—
91C4⋊Dic137C2311yescharacteristic
92#14555489—21no—
93#14555489—21no—
94#14555489—21no—
95#14555489D411yescharacteristic
96#14555489D411yescharacteristic
97C22⋊Dic137—21no—
98C22⋊Dic137—21no—
99C22⋊Dic137—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487D411yescharacteristic
103#14555487D411yescharacteristic
104C4×Dic137—21no—
105C4×Dic137—21no—
106C4×Dic137C2311yescharacteristic
107C2×D4×C137—21no—
108C2×D4×C137—21no—
109C2×D4×C137C2311yescharacteristic
110C2×D4×C137C2311yescharacteristic
111C2×D4×C137C2311yescharacteristic
112C22×C4×C137—21no—
113D8×C137—22no—
114D8×C137—22no—
115C2×C8×C137C2311yescharacteristic
116C2×C8×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C42×C137C2311yescharacteristic
120C22×D137—21no—
121C22×D137—42no—
122C22×D137—81no—
123C22×D137—81no—
124C22×D137C2×D411yescharacteristic
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128Dic274C2×D412yes—
129D548—21no—
130D548—21no—
131C22⋊D137—24no—
132C22⋊D137—42no—
133C22⋊D137—42no—
134C22⋊D137—81no—
135C22⋊D137—81no—
136C2×Dic137—21no—
137C2×Dic137—24no—
138C2×Dic137—21no—
139C2×Dic137—21no—
140C2×Dic137—42no—
141C2×Dic137—81no—
142C2×Dic137C2×D411yescharacteristic
143C2×Dic137C2×D411yescharacteristic
144C2×Dic137C2×D411yescharacteristic
145C4×D137—81no—
146C4×D137C2×D412yes—
147C4×D137#4112yes—
148#11764040—21no—
149#11764040—21no—
150C23×C137—21no—
151C23×C137—21no—
152C23×C137—21no—
153D4×C137—21no—
154D4×C137—21no—
155D4×C137—21no—
156D4×C137—21no—
157D4×C137—21no—
158D4×C137—81no—
159D4×C137—81no—
160D4×C137C2×D412yes—
161C2×C4×C137—21no—
162C2×C4×C137—81no—
163C2×C4×C137C2×D411yescharacteristic
164C2×C4×C137C2×D411yescharacteristic
165C2×C4×C137C2411yescharacteristic
166C8×C137—21no—
167C8×C137—21no—
168C22×C137—22no—
169C22×C137—41no—
170C22×C137—41no—
171C22×C137—41no—
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—41no—
181C2×D137—41no—
182C2×D137—41no—
183C2×D137—84no—
184C2×D137#12112yes—
185C4×C137—21no—
186C4×C137—21no—
187C4×C137—41no—
188C4×C137C22×D411yescharacteristic
189C4×C137C2×#4111yescharacteristic
190Dic137—22no—
191Dic137—21no—
192Dic137—21no—
193Dic137—41no—
194Dic137—41no—
195Dic137#12112yes—
196C2×C137—42no—
197C2×C137—81no—
198C2×C137—81no—
199C2×C137C2×#12111yescharacteristic
200C2×C137#57611yescharacteristic
201C2×C137#57711yescharacteristic
202D137—22no—
203D137—81no—
204C137#304511yescharacteristic, complemented, Sylow
205#3045—1371nomaximal, Sylow
206C2×AGL(1,ℤ/8)—1371no—
207C2×#135—1371no—
208#535—1371no—
209#520—1371no—
210#464—1371no—
211#464—1371no—
212#463—1371no—
213#463—1371no—
214#460—1371no—
215#460—1371no—
216#459—1371no—
217#459D13711yescharacteristic, complemented
218#424—1371no—
219#417—1371no—
220C2×#102—1371no—
221C2×#41—1371no—
222C2×#41—1371no—
223C22×D4—1371no—
224AGL(1,ℤ/8)—2744no—
225#135—2744no—
226C2×Q16—1371no—
227C2×SD16—1371no—
228C2×SD16—1371no—
229C2×SD16—1371no—
230C2×SD16—1371no—
231C2×D8—1371no—
232C2×D8C2×D13711yescharacteristic
233C2×D8C2×D13711yescharacteristic, complemented
234C2×M16—1371no—
235C22×C8—1371no—
236#127—1371no—
237#127C2×D13711yescharacteristic
238#124—1371no—
239#124—1371no—
240#121—2741no—
241#121—2741no—
242#120—2741no—
243#120—2741no—
244C4×Q8—1371no—
245C4×Q8—1371no—
246C4×D4—2741no—
247C4×D4—2741no—
248C4×D4—1371no—
249C4×D4C2×D13711yescharacteristic, complemented
250#117—2741no—
251#117—1371no—
252C2×C4⋊C4—2741no—
253C2×C4⋊C4—1371no—
254C4⋊C8—1371no—
255C4⋊C8—1371no—
256C4⋊C8—1371no—
257C4⋊C8C2×D13711yescharacteristic, complemented
258#103—1371no—
259#103—1371no—
260#102—1371no—
261#102—1371no—
262#102—1371no—
263#102—1371no—
264#102C2×D13711yescharacteristic
265#102C2×D13711yescharacteristic, complemented
266C24—2741no—
267#41—2742no—
268#41—2742no—
269#41—2742no—
270#41—1374no—
271C2×Q8—1371no—
272C2×Q8—1371no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—2741no—
276C2×D4—2741no—
277C2×D4—2742no—
278C2×D4—2742no—
279C2×D4—2741no—
280C2×D4—2741no—
281C2×D4—2741no—
282C2×D4—1371no—
283C2×D4C22×D13711yescharacteristic
284C2×D4C22×D13711yescharacteristic
285C2×D4C22×D13711yescharacteristic
286C22×C4—21no—
287C22×C4—2741no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4—1371no—
292Q16—2742no—
293SD16—2742no—
294SD16—2742no—
295SD16—2742no—
296SD16—2742no—
297D8—22no—
298D8—22no—
299D8—2742no—
300M16—2742no—
301C2×C8—2742no—
302C2×C8—1371no—
303C2×C8—1371no—
304C2×C8C22×D13711yescharacteristic
305C2×C8C22×D13711yescharacteristic
306C4⋊C4—2741no—
307C4⋊C4—2741no—
308C4⋊C4—2741no—
309C4⋊C4—1371no—
310C4⋊C4—1371no—
311C4⋊C4—1371no—
312C4⋊C4C22×D13711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—2741no—
315C22⋊C4—2741no—
316C22⋊C4—2741no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C22⋊C4—1371no—
320C22⋊C4—1371no—
321C42—2741no—
322C42—2741no—
323C42—1371no—
324C42C22×D13711yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—2741no—
329C23—5482no—
330C23—10961no—
331C23—10961no—
332C23—1371no—
333Q8—2741no—
334Q8—2741no—
335Q8—2741no—
336Q8—1372no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—81no—
343D4—81no—
344D4—2744no—
345D4—2741no—
346D4—2741no—
347D4—5482no—
348D4—5482no—
349D4—10961no—
350D4—10961no—
351D4D4×D13712yes—
352C2×C4—21no—
353C2×C4—81no—
354C2×C4—2741no—
355C2×C4—2744no—
356C2×C4—2741no—
357C2×C4—2741no—
358C2×C4—5482no—
359C2×C4—10961no—
360C2×C4—10961no—
361C2×C4—1371no—
362C2×C4—1371no—
363C2×C4—1371no—
364C2×C4—1372no—
365C2×C4—1372no—
366C2×C4D4×D13711yescharacteristic
367C2×C4D4×D13711yescharacteristic
368C2×C4C23×D13711yescharacteristic
369C8—21no—
370C8—21no—
371C8—2741no—
372C8—2741no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—2741no—
383C22—2741no—
384C22—5481no—
385C22—5481no—
386C22—5481no—
387C22—10964no—
388C22—1372no—
389C22C2×D4×D13711yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—2741no—
394C4—2741no—
395C4—2742no—
396C4—5481no—
397C4—5481no—
398C4—1372no—
399C4C2×D4×D13711yescharacteristic
400C4#41×D13711yescharacteristic
401C2—42no—
402C2—81no—
403C2—81no—
404C2—2742no—
405C2—10961no—
406C2#121×D13711yescharacteristic
407C2#2257820411yescharacteristic
408C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#459×C137C2
socleC22×C137C22×D4
hypercenter#459D137
hypocenterC137#3045
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×#17984107C2semidirect
3. #22577966C2semidirect
4. #22577882C2semidirect
5. #22577626C2semidirect
6. #22577624C2semidirect
7. #22577618C2semidirect
8. #22577616C2semidirect
9. #22577602C2semidirect
10. #22577600C2semidirect
11. #22577596C2semidirect
12. #22577428C2semidirect
13. #22577401C2semidirect
14. #102×D137C2semidirect
15. #459×C137C2semidirect
21. C2×Dic548C22semidirect
24. C2×#14555498C22semidirect
25. C2×#14555498C22semidirect
36. C4×Dic274C22semidirect
37. #17984053C22semidirect
38. D4×Dic137C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984017C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984011C22semidirect
52. #17984009C22semidirect
55. C2×D8×C137C22semidirect
57. C4×D4×C137C22semidirect
58. C4⋊C8×C137C22semidirect
60. #102×C137C22semidirect
65. #14555509D4semidirect
204. C137#3045semidirect
217. #459D137semidirect
233. C2×D8C2×D137semidirect
249. C4×D4C2×D137semidirect
257. C4⋊C8C2×D137semidirect
265. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×#179841072
#225779662
#225778822
#225776262
#225776242
#225776182
#225776162
#225776022
#225776002
#225775962
#225774282
#225774012
#102×D1372
#459×C1372
#3045137