Finite Group Databasebeta

← #31792325

Subgroups — #31792325

Order 17536.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984114C211yesmaximal, characteristic, complemented
2C2×#17984108C211yesmaximal, characteristic, complemented
3#22578074C211yesmaximal, characteristic, complemented
4#22577793C211yesmaximal, characteristic, complemented
5#22577651C211yesmaximal, characteristic, complemented
6#22577637C211yesmaximal, characteristic, complemented
7#22577622C211yesmaximal, characteristic, complemented
8#22577598C211yesmaximal, characteristic, complemented
9#22577563C211yesmaximal, characteristic, complemented
10#22577548C211yesmaximal, characteristic, complemented
11#22577546C211yesmaximal, characteristic, complemented
12#22577537C211yesmaximal, characteristic, complemented
13#22577503C211yesmaximal, characteristic, complemented
14#22577475C211yesmaximal, characteristic, complemented
15#546×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555510C2211yescharacteristic, complemented
18#17984114—24no—
19#17984108—24no—
20C2×#14555500C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic
22C2×#14555498C2211yescharacteristic
23C2×#14555497C2211yescharacteristic, complemented
24C2×#14555497C2211yescharacteristic, complemented
25C2×#14555495C2211yescharacteristic
26C2×#14555495C2211yescharacteristic
27C2×#14555495C2211yescharacteristic
28C2×#14555495C2211yescharacteristic
29C2×#14555494C2211yescharacteristic
30C22×#11764040C2211yescharacteristic, complemented
31#17984085C2211yescharacteristic, complemented
32#17984084—21no—
33#17984062C2211yescharacteristic
34#17984062C2211yescharacteristic, complemented
35#17984061—21no—
36#17984061—21no—
37C4×Dic274C2211yescharacteristic, complemented
38#17984050—21no—
39#17984021C2211yescharacteristic, complemented
40C4⋊#11764040C2211yescharacteristic, complemented
41#17984012C2211yescharacteristic, complemented
42#17984011C2211yescharacteristic, complemented
43#17984009C2211yescharacteristic
44#17984009C2211yescharacteristic
45#17984008C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic, complemented
47C22⋊#11764040C2211yescharacteristic
48C22⋊#11764040C2211yescharacteristic, complemented
49#17983999C2211yescharacteristic, complemented
50C2×#41×C137C2211yescharacteristic
51C22×D4×C137C2211yescharacteristic
52#124×C137C2211yescharacteristic
53#123×C137—21no—
54C22⋊Q8×C137C2211yescharacteristic
55#121×C137—21no—
56#121×C137C2211yescharacteristic
57#120×C137—21no—
58C4×D4×C137C2211yescharacteristic
59C4×D4×C137C2211yescharacteristic
60C2×C22⋊C4×C137—21no—
61#14555510—22no—
62#14555510—22no—
63#14555510—41no—
64#14555510—41no—
65C2×Dic274C2311yescharacteristic
66C2×D548—21no—
67C2×D548—21no—
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76#14555500—22no—
77#14555498—22no—
78#14555498D414yescomplemented
79#14555497—22no—
80#14555497—41no—
81#14555495—22no—
82#14555495—22no—
83#14555495—22no—
84#14555495—41no—
85#14555494—22no—
86C2×#11764040—22no—
87C2×#11764040C2311yescharacteristic
88C2×#11764040C2311yescharacteristic
89C2×#11764040C2311yescharacteristic
90C2×#11764040C2311yescharacteristic
91C4⋊Dic137C2311yescharacteristic
92#14555489—21no—
93#14555487—21no—
94#14555487—21no—
95C4×Dic137—21no—
96C24×C137—21no—
97#41×C137—22no—
98#41×C137—41no—
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137—22no—
101C2×D4×C137—21no—
102C2×D4×C137—21no—
103C2×D4×C137—21no—
104C2×D4×C137—41no—
105C2×D4×C137D411yescharacteristic
106C2×D4×C137C2311yescharacteristic
107C2×D4×C137C2311yescharacteristic
108C22×C4×C137—21no—
109C22×C4×C137—21no—
110C22×C4×C137C2311yescharacteristic
111C22×C4×C137C2311yescharacteristic
112C4⋊C4×C137—21no—
113C4⋊C4×C137D411yescharacteristic
114C4⋊C4×C137C2311yescharacteristic
115C22⋊C4×C137—21no—
116C22⋊C4×C137—21no—
117C22⋊C4×C137—21no—
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22⋊C4×C137D411yescharacteristic
121C22⋊C4×C137D411yescharacteristic
122C42×C137C2311yescharacteristic
123C22×D137—21no—
124C22×D137—21no—
125Dic274—21no—
126Dic274C2×D412yes—
127D548—21no—
128D548—21no—
129D548—21no—
130D548—21no—
131D548—42no—
132D548—81no—
133C22⋊D137—42no—
134C22⋊D137—42no—
135C22⋊D137—81no—
136C22⋊D137—81no—
137C2×Dic137—21no—
138C2×Dic137—21no—
139C4×D137—42no—
140C4×D137—42no—
141#11764040—21no—
142#11764040—21no—
143#11764040—21no—
144#11764040C2×D412yes—
145C23×C137—21no—
146C23×C137—21no—
147C23×C137—42no—
148C23×C137—42no—
149C23×C137—81no—
150C23×C137C2×D411yescharacteristic
151C23×C137C2×D411yescharacteristic
152Q8×C137—21no—
153Q8×C137—21no—
154D4×C137—21no—
155D4×C137—21no—
156D4×C137—21no—
157D4×C137—41no—
158D4×C137—42no—
159D4×C137—42no—
160D4×C137—81no—
161D4×C137C2×D412yes—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—21no—
168C2×C4×C137—42no—
169C2×C4×C137—81no—
170C2×C4×C137C2×D411yescharacteristic
171C2×C4×C137C2×D411yescharacteristic
172C2×C4×C137C2×D411yescharacteristic
173C2×C4×C137C2×D411yescharacteristic
174C2×C4×C137C2411yescharacteristic
175C22×C137—21no—
176C22×C137—21no—
177C22×C137—22no—
178C22×C137—21no—
179C22×C137—21no—
180C22×C137—21no—
181C22×C137—21no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—84no—
188C22×C137C22×D411yescharacteristic
189C2×D137—41no—
190C2×D137—41no—
191C2×D137—41no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—41no—
195C4×C137—21no—
196C4×C137—21no—
197C4×C137—21no—
198C4×C137—21no—
199C4×C137—41no—
200C4×C137—41no—
201C4×C137C22×D411yescharacteristic
202C4×C137H(5,𝔽2)11yescharacteristic
203Dic137—22no—
204Dic137—41no—
205C2×C137—41no—
206C2×C137—41no—
207C2×C137—42no—
208C2×C137—81no—
209C2×C137D4211yescharacteristic
210C2×C137#57511yescharacteristic
211C2×C137#57611yescharacteristic
212D137—81no—
213D137—81no—
214C137#326811yescharacteristic, complemented, Sylow
215#3268—1371nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1371no—
217C2×#135—1371no—
218#550—1371no—
219#546D13711yescharacteristic, complemented
220#496—1371no—
221#468—1371no—
222#466—1371no—
223#464—1371no—
224#459—1371no—
225#452—1371no—
226#449—1371no—
227#449—1371no—
228#447—1371no—
229#440—1371no—
230#435—1371no—
231C2×#41—1371no—
232C2×#41—1371no—
233C2×#41C2×D13711yescharacteristic
234C22×D4C2×D13711yescharacteristic
235AGL(1,ℤ/8)—2744no—
236#135—2744no—
237C2×Q16—1371no—
238C2×SD16—1371no—
239C2×SD16—1371no—
240C2×SD16—1371no—
241C2×SD16—1371no—
242C2×D8—1371no—
243C2×D8—1371no—
244C2×D8—1371no—
245C2×D8—1371no—
246C2×M16—1371no—
247C22×C8—1371no—
248#127—2741no—
249#127—1371no—
250#124C2×D13711yescharacteristic
251#123—21no—
252C22⋊Q8C2×D13711yescharacteristic
253#121—21no—
254#121—2741no—
255#121—2741no—
256#121—1371no—
257#121—1371no—
258#121C2×D13711yescharacteristic
259#120—21no—
260C4×Q8—1371no—
261C4×D4—2741no—
262C4×D4C2×D13711yescharacteristic
263C4×D4C2×D13711yescharacteristic
264C2×C22⋊C4—21no—
265#107—1371no—
266C4⋊C8—1371no—
267#103—1371no—
268#103—1371no—
269#102—1371no—
270#102—1371no—
271#102—1371no—
272#102—1371no—
273C22⋊C8—1371no—
274C22⋊C8—1371no—
275#97—1371no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—2742no—
280#41—2742no—
281#41—5481no—
282#41—5481no—
283C2×Q8—1371no—
284C2×Q8C22×D13711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2741no—
291C2×D4—2741no—
292C2×D4—2741no—
293C2×D4—2741no—
294C2×D4—2741no—
295C2×D4—2741no—
296C2×D4—1371no—
297C2×D4—1371no—
298C2×D4C22⋊D13711yescharacteristic
299C2×D4C22×D13711yescharacteristic
300C2×D4C22×D13711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—2741no—
304C22×C4—2741no—
305C22×C4C22×D13711yescharacteristic
306C22×C4C22×D13711yescharacteristic
307Q16—2742no—
308SD16—2742no—
309SD16—2742no—
310SD16—5481no—
311SD16—1374no—
312D8—2742no—
313D8—2742no—
314D8—2742no—
315D8—5481no—
316M16—2742no—
317C2×C8—2742no—
318C2×C8—1371no—
319C2×C8—1371no—
320C2×C8—1371no—
321C2×C8—1371no—
322C4⋊C4—21no—
323C4⋊C4—2741no—
324C4⋊C4—1371no—
325C4⋊C4C22⋊D13711yescharacteristic
326C4⋊C4C22×D13711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—2741no—
333C22⋊C4—2741no—
334C22⋊C4C22⋊D13711yescharacteristic
335C22⋊C4C22⋊D13711yescharacteristic
336C42—2741no—
337C42C22×D13711yescharacteristic
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—2741no—
344C23—2741no—
345C23C2×C22⋊D13711yescharacteristic
346C23C2×C22⋊D13711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—2741no—
350Q8—1372no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—2741no—
359D4—2741no—
360D4—2741no—
361D4—2741no—
362D4—5482no—
363D4—5482no—
364D4—5482no—
365D4—10961no—
366D4—10961no—
367D4—10961no—
368D4D4×D13712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—2741no—
378C2×C4—2741no—
379C2×C4—5482no—
380C2×C4—5482no—
381C2×C4C2×C22⋊D13711yescharacteristic
382C2×C4C2×C22⋊D13711yescharacteristic
383C2×C4C2×C22⋊D13711yescharacteristic
384C2×C4C2×C22⋊D13711yescharacteristic
385C2×C4C23×D13711yescharacteristic
386C8—2741no—
387C8—2741no—
388C8—2741no—
389C8—1372no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—5481no—
404C22—5481no—
405C22—5481no—
406C22—5481no—
407C22—5481no—
408C22—5481no—
409C22C22×C22⋊D13711yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—2742no—
417C4—5481no—
418C4C2×D4×D13711yescharacteristic
419C4#1798413311yescharacteristic
420C2—41no—
421C2—41no—
422C2—42no—
423C2—81no—
424C2—10961no—
425C2—10961no—
426C2D4×C22⋊D13711yescharacteristic
427C2#2257820111yescharacteristic
428C2#2257820611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#546×C137C2
socleC22×C137C22×D4
hypercenter#546D137
hypocenterC137#3268
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. C2×#17984108C2semidirect
3. #22578074C2semidirect
4. #22577793C2semidirect
5. #22577651C2semidirect
6. #22577637C2semidirect
7. #22577622C2semidirect
8. #22577598C2semidirect
9. #22577563C2semidirect
10. #22577548C2semidirect
11. #22577546C2semidirect
12. #22577537C2semidirect
13. #22577503C2semidirect
14. #22577475C2semidirect
15. #546×C137C2semidirect
17. C2×#14555510C22semidirect
20. C2×#14555500C22semidirect
23. C2×#14555497C22semidirect
24. C2×#14555497C22semidirect
30. C22×#11764040C22semidirect
31. #17984085C22semidirect
34. #17984062C22semidirect
37. C4×Dic274C22semidirect
39. #17984021C22semidirect
40. C4⋊#11764040C22semidirect
41. #17984012C22semidirect
42. #17984011C22semidirect
45. #17984008C22semidirect
46. #17984008C22semidirect
48. C22⋊#11764040C22semidirect
49. #17983999C22semidirect
78. #14555498D4semidirect
214. C137#3268semidirect
219. #546D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C2×#179841082
#225780742
#225777932
#225776512
#225776372
#225776222
#225775982
#225775632
#225775482
#225775462
#225775372
#225775032
#225774752
#546×C1372
#3268137