Finite Group Databasebeta

← #31792280

Subgroups — #31792280

Order 17536.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984114C211yesmaximal, characteristic, complemented
2C22×#14555498C211yesmaximal, characteristic, complemented
3#22578062C211yesmaximal, characteristic, complemented
4#22577790C211yesmaximal, characteristic, complemented
5#22577647C211yesmaximal, characteristic, complemented
6#22577637C211yesmaximal, characteristic, complemented
7#22577618C211yesmaximal, characteristic, complemented
8#22577600C211yesmaximal, characteristic, complemented
9#22577554C211yesmaximal, characteristic, complemented
10#22577546C211yesmaximal, characteristic, complemented
11#22577541C211yesmaximal, characteristic, complemented
12#22577537C211yesmaximal, characteristic, complemented
13#22577503C211yesmaximal, characteristic, complemented
14#22577475C211yesmaximal, characteristic, complemented
15D42×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic, complemented
17C22×Dic274C2211yescharacteristic, complemented
18#17984114—24no—
19C2×#14555498—24no—
20C2×#14555498C2211yescharacteristic
21C2×#14555498C2211yescharacteristic
22C2×#14555498C2211yescharacteristic, complemented
23C2×#14555498C2211yescharacteristic, complemented
24C2×#14555498C2211yescharacteristic, complemented
25C2×#14555498C2211yescharacteristic, complemented
26C2×#14555495C2211yescharacteristic
27C2×#14555495C2211yescharacteristic, complemented
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555494C2211yescharacteristic, complemented
30C22×#11764040C2211yescharacteristic, complemented
31#17984076C2211yescharacteristic, complemented
32#17984074—21no—
33#17984067C2211yescharacteristic, complemented
34#17984066—21no—
35#17984062C2211yescharacteristic, complemented
36#17984060—21no—
37C4×Dic274C2211yescharacteristic, complemented
38#17984048—21no—
39#17984021C2211yescharacteristic, complemented
40C4⋊#11764040C2211yescharacteristic, complemented
41#17984011C2211yescharacteristic, complemented
42#17984011C2211yescharacteristic, complemented
43#17984009C2211yescharacteristic
44#17984009C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic, complemented
47C22⋊#11764040C2211yescharacteristic, complemented
48C22⋊#11764040C2211yescharacteristic, complemented
49#17983999C2211yescharacteristic, complemented
50C22×D4×C137—21no—
51C22×D4×C137C2211yescharacteristic
52C22×D4×C137C2211yescharacteristic
53#127×C137C2211yescharacteristic
54#121×C137—21no—
55#121×C137C2211yescharacteristic
56#121×C137C2211yescharacteristic
57#120×C137—21no—
58#120×C137—21no—
59C4×D4×C137C2211yescharacteristic
60C4×D4×C137C2211yescharacteristic
61#14555510—22no—
62#14555510—41no—
63C2×Dic274—22no—
64C2×Dic274—21no—
65C2×Dic274—41no—
66C2×Dic274C2311yescharacteristic
67C2×Dic274C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73#14555498—22no—
74#14555498—22no—
75#14555498—22no—
76#14555498—22no—
77#14555498—41no—
78#14555498D414yescomplemented
79#14555495—22no—
80#14555495—22no—
81#14555495—41no—
82#14555494—22no—
83C2×#11764040—22no—
84C2×#11764040C2311yescharacteristic
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C2×#11764040C2311yescharacteristic
88C4⋊Dic137C2311yescharacteristic
89#14555489—21no—
90#14555489—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95C4×Dic137—21no—
96C24×C137—21no—
97C24×C137—21no—
98C2×D4×C137—22no—
99C2×D4×C137—22no—
100C2×D4×C137—21no—
101C2×D4×C137—21no—
102C2×D4×C137—21no—
103C2×D4×C137—21no—
104C2×D4×C137—21no—
105C2×D4×C137—21no—
106C2×D4×C137—41no—
107C2×D4×C137—41no—
108C2×D4×C137—44no—
109C2×D4×C137D411yescharacteristic
110C2×D4×C137C2311yescharacteristic
111C2×D4×C137C2311yescharacteristic
112C2×D4×C137C2311yescharacteristic
113C22×C4×C137—21no—
114C22×C4×C137C2311yescharacteristic
115C22×C4×C137C2311yescharacteristic
116C4⋊C4×C137D411yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22⋊C4×C137—21no—
121C22⋊C4×C137D411yescharacteristic
122C22⋊C4×C137D411yescharacteristic
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128Dic274—42no—
129Dic274C2×D412yescomplemented
130D548—21no—
131D548—21no—
132C22⋊D137—42no—
133C22⋊D137—42no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137—42no—
138C4×D137—42no—
139#11764040—21no—
140#11764040—21no—
141#11764040—21no—
142#11764040C2×D412yescomplemented
143C23×C137—21no—
144C23×C137—21no—
145C23×C137—21no—
146C23×C137—42no—
147C23×C137—42no—
148C23×C137—42no—
149C23×C137—42no—
150C23×C137—81no—
151C23×C137—81no—
152C23×C137C2×D411yescharacteristic
153C23×C137C2×D411yescharacteristic
154D4×C137—22no—
155D4×C137—21no—
156D4×C137—21no—
157D4×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—42no—
161D4×C137—42no—
162D4×C137—42no—
163D4×C137—42no—
164D4×C137—42no—
165D4×C137—81no—
166D4×C137C2×D412yes—
167C2×C4×C137—21no—
168C2×C4×C137—21no—
169C2×C4×C137—21no—
170C2×C4×C137—21no—
171C2×C4×C137—21no—
172C2×C4×C137—42no—
173C2×C4×C137C2×D411yescharacteristic
174C2×C4×C137C2×D411yescharacteristic
175C2×C4×C137C2×D411yescharacteristic
176C2×C4×C137C2×D411yescharacteristic
177C2×C4×C137C2411yescharacteristic
178C22×C137—21no—
179C22×C137—21no—
180C22×C137—21no—
181C22×C137—22no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137—41no—
190C22×C137—41no—
191C22×C137—41no—
192C22×C137—84no—
193C22×C137—84no—
194C22×C137AGL(1,ℤ/8)12yescomplemented
195C22×C137C22×D411yescharacteristic
196C2×D137—41no—
197C2×D137—41no—
198C2×D137—41no—
199C4×C137—21no—
200C4×C137—21no—
201C4×C137—21no—
202C4×C137—21no—
203C4×C137—41no—
204C4×C137C22×D411yescharacteristic
205C4×C137H(5,𝔽2)11yescharacteristic
206Dic137—22no—
207Dic137—41no—
208Dic137—41no—
209C2×C137—22no—
210C2×C137—41no—
211C2×C137—42no—
212C2×C137—81no—
213C2×C137—81no—
214C2×C137D4211yescharacteristic
215C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C137#57711yescharacteristic
217D137—81no—
218C137#326511yescharacteristic, complemented, Sylow
219#3265—1371nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1371no—
221C22×SD16—1371no—
222#548—1371no—
223D42D13711yescharacteristic, complemented
224#496—1371no—
225#468—1371no—
226#466—1371no—
227#463—1371no—
228#460—1371no—
229#450—1371no—
230#449—1371no—
231#448—1371no—
232#447—1371no—
233#440—1371no—
234#435—1371no—
235C2×#41—1371no—
236C22×Q8—1371no—
237C22×D4—21no—
238C22×D4C2×D13711yescharacteristic
239C22×D4C2×D13711yescharacteristic
240AGL(1,ℤ/8)—2744no—
241C2×SD16—2744no—
242C2×SD16—1371no—
243C2×SD16—1371no—
244C2×SD16—1371no—
245C2×SD16—1371no—
246C2×SD16—1371no—
247C2×SD16—1371no—
248C2×D8—1371no—
249C2×D8—1371no—
250C2×D8—1371no—
251C2×M16—1371no—
252C22×C8—1371no—
253#127C2×D13711yescharacteristic
254#124—2741no—
255#124—1371no—
256C22⋊Q8—2741no—
257C22⋊Q8—1371no—
258#121—21no—
259#121—2741no—
260#121—1371no—
261#121C2×D13711yescharacteristic
262#121C2×D13711yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1371no—
266C4×D4—2741no—
267C4×D4C2×D13711yescharacteristic
268C4×D4C2×D13711yescharacteristic
269#107—1371no—
270C4⋊C8—1371no—
271#103—1371no—
272#103—1371no—
273#102—1371no—
274#102—1371no—
275#102—1371no—
276#102—1371no—
277C22⋊C8—1371no—
278C22⋊C8—1371no—
279#97—1371no—
280C24—21no—
281C24—21no—
282#41—2742no—
283#41—5481no—
284C2×Q8—2742no—
285C2×Q8—2741no—
286C2×Q8—5481no—
287C2×Q8—1371no—
288C2×Q8—1371no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—2741no—
301C2×D4—2741no—
302C2×D4—1371no—
303C2×D4C22⋊D13711yescharacteristic
304C2×D4C22×D13711yescharacteristic
305C2×D4C22×D13711yescharacteristic
306C2×D4C22×D13711yescharacteristic
307C22×C4—21no—
308C22×C4—2741no—
309C22×C4—2741no—
310C22×C4C22×D13711yescharacteristic
311C22×C4C22×D13711yescharacteristic
312SD16—2742no—
313SD16—2742no—
314SD16—2742no—
315SD16—2742no—
316SD16—5481no—
317SD16—1374no—
318D8—2742no—
319D8—2742no—
320D8—5481no—
321M16—2742no—
322C2×C8—2742no—
323C2×C8—1371no—
324C2×C8—1371no—
325C2×C8—1371no—
326C2×C8—1371no—
327C4⋊C4—2741no—
328C4⋊C4—2741no—
329C4⋊C4—1371no—
330C4⋊C4C22⋊D13711yescharacteristic
331C4⋊C4C22×D13711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—2741no—
336C22⋊C4—2741no—
337C22⋊C4—2741no—
338C22⋊C4—2741no—
339C22⋊C4C22⋊D13711yescharacteristic
340C22⋊C4C22⋊D13711yescharacteristic
341C42—2741no—
342C42C22×D13711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—2741no—
353C23C2×C22⋊D13711yescharacteristic
354C23C2×C22⋊D13711yescharacteristic
355Q8—2741no—
356Q8—2741no—
357Q8—2741no—
358Q8—5482no—
359Q8—1372no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—2741no—
373D4—2741no—
374D4—5482no—
375D4—5482no—
376D4D4×D13712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—2741no—
386C2×C4—5482no—
387C2×C4—5482no—
388C2×C4C2×C22⋊D13711yescharacteristic
389C2×C4C2×C22⋊D13711yescharacteristic
390C2×C4C2×C22⋊D13711yescharacteristic
391C2×C4C2×C22⋊D13711yescharacteristic
392C2×C4C23×D13711yescharacteristic
393C8—2741no—
394C8—2741no—
395C8—2741no—
396C8—1372no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—5481no—
414C22—5481no—
415C22—5481no—
416C22#1798411412yescomplemented
417C22C22×C22⋊D13711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—2742no—
424C4—5481no—
425C4—5481no—
426C4C2×D4×D13711yescharacteristic
427C4#1798413311yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—10961no—
434C2D4×C22⋊D13711yescharacteristic
435C2C2×#1798411411yescharacteristic
436C2#2257821411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingD42×C137C2
socleC22×C137C22×D4
hypercenterD42D137
hypocenterC137#3265
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. C22×#14555498C2semidirect
3. #22578062C2semidirect
4. #22577790C2semidirect
5. #22577647C2semidirect
6. #22577637C2semidirect
7. #22577618C2semidirect
8. #22577600C2semidirect
9. #22577554C2semidirect
10. #22577546C2semidirect
11. #22577541C2semidirect
12. #22577537C2semidirect
13. #22577503C2semidirect
14. #22577475C2semidirect
15. D42×C137C2semidirect
16. C2×#14555510C22semidirect
17. C22×Dic274C22semidirect
22. C2×#14555498C22semidirect
23. C2×#14555498C22semidirect
24. C2×#14555498C22semidirect
25. C2×#14555498C22semidirect
27. C2×#14555495C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555494C22semidirect
30. C22×#11764040C22semidirect
31. #17984076C22semidirect
33. #17984067C22semidirect
35. #17984062C22semidirect
37. C4×Dic274C22semidirect
39. #17984021C22semidirect
40. C4⋊#11764040C22semidirect
41. #17984011C22semidirect
42. #17984011C22semidirect
44. #17984009C22semidirect
45. #17984009C22semidirect
46. #17984008C22semidirect
47. C22⋊#11764040C22semidirect
48. C22⋊#11764040C22semidirect
49. #17983999C22semidirect
78. #14555498D4semidirect
129. Dic274C2×D4semidirect
142. #11764040C2×D4semidirect
194. C22×C137AGL(1,ℤ/8)semidirect
218. C137#3265semidirect
223. D42D137semidirect
416. C22#17984114semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C22×#145554982
#225780622
#225777902
#225776472
#225776372
#225776182
#225776002
#225775542
#225775462
#225775412
#225775372
#225775032
#225774752
D42×C1372
#3265137