Finite Group Databasebeta

← #31792329

Subgroups — #31792329

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×#17984115C211yesmaximal, characteristic, complemented
2#22578072C211yesmaximal, characteristic, complemented
3#22578041C211yesmaximal, characteristic, complemented
4#22577789C211yesmaximal, characteristic, complemented
5#22577646C211yesmaximal, characteristic, complemented
6#22577639C211yesmaximal, characteristic, complemented
7#22577627C211yesmaximal, characteristic, complemented
8#22577596C211yesmaximal, characteristic, complemented
9#22577564C211yesmaximal, characteristic, complemented
10#22577548C211yesmaximal, characteristic, complemented
11#22577548C211yesmaximal, characteristic, complemented
12#22577535C211yesmaximal, characteristic, complemented
13#22577504C211yesmaximal, characteristic, complemented
14#22577474C211yesmaximal, characteristic, complemented
15C2×#135×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18#17984115—24no—
19C2×#14555497C2211yescharacteristic
20C2×#14555497C2211yescharacteristic, complemented
21C2×D1096C2211yescharacteristic
22C2×#14555495C2211yescharacteristic
23C2×#14555495C2211yescharacteristic
24C2×#14555493C2211yescharacteristic
25#17984084—21no—
26#17984084C2211yescharacteristic, complemented
27#17984077C2211yescharacteristic
28#17984071—21no—
29#17984064C2211yescharacteristic
30#17984062—21no—
31#17984062C2211yescharacteristic
32#17984062C2211yescharacteristic, complemented
33#17984060—21no—
34#17984059—21no—
35#17984059C2211yescharacteristic
36#17984056—21no—
37Q8×Dic137C2211yescharacteristic, complemented
38D4×Dic137C2211yescharacteristic
39C4×C22⋊D137—21no—
40C4×C22⋊D137C2211yescharacteristic
41C2×#14555487—21no—
42#17984022C2211yescharacteristic, complemented
43#17984017C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984012C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic
49#17984009C2211yescharacteristic, complemented
50C22⋊#11764040C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic
52#17983998C2211yescharacteristic, complemented
53C2×#41×C137C2211yescharacteristic
54C2×#41×C137C2211yescharacteristic, complemented
55#135×C137—24no—
56C2×Q16×C137C2211yescharacteristic, complemented
57C2×SD16×C137C2211yescharacteristic
58C2×SD16×C137C2211yescharacteristic, complemented
59C2×D8×C137C2211yescharacteristic
60C22×C8×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555511—22no—
63#14555511—41no—
64C2×D548—21no—
65C2×D548C2311yescharacteristic
66C2×D548C2311yescharacteristic
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137D411yescharacteristic
74D4×D137—22no—
75D4×D137—41no—
76C22×Dic137—21no—
77C2×C4×D137—21no—
78C2×C4×D137C2311yescharacteristic
79#14555497—22no—
80#14555497—41no—
81D1096—22no—
82#14555495—22no—
83#14555495—41no—
84#14555493—22no—
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C4⋊Dic137—21no—
88C4⋊Dic137C2311yescharacteristic
89C4⋊Dic137C2311yescharacteristic
90#14555489—21no—
91#14555489D411yescharacteristic
92C22⋊Dic137—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137D411yescharacteristic
95#14555487—21no—
96#14555487—21no—
97#14555487—21no—
98#14555487—21no—
99#14555487—21no—
100#14555487D411yescharacteristic
101C4×Dic137—21no—
102C4×Dic137C2311yescharacteristic
103#41×C137—22no—
104#41×C137—22no—
105#41×C137—41no—
106#41×C137—41no—
107C2×Q8×C137C2311yescharacteristic
108C2×Q8×C137C2311yescharacteristic
109C2×D4×C137—21no—
110C2×D4×C137—21no—
111C2×D4×C137C2311yescharacteristic
112C2×D4×C137C2311yescharacteristic
113C22×C4×C137—21no—
114C22×C4×C137—21no—
115C22×C4×C137C2311yescharacteristic
116Q16×C137—22no—
117SD16×C137—22no—
118SD16×C137D414yescomplemented
119D8×C137—22no—
120C2×C8×C137—22no—
121C2×C8×C137C2311yescharacteristic
122C2×C8×C137C2311yescharacteristic
123C22×D137—21no—
124C22×D137—21no—
125C22×D137—42no—
126C22×D137—42no—
127C22×D137—81no—
128C22×D137C2×D411yescharacteristic
129D548—21no—
130D548—21no—
131D548—21no—
132D548—21no—
133D548—42no—
134C22⋊D137—41no—
135C22⋊D137—42no—
136C22⋊D137—42no—
137C22⋊D137—81no—
138C22⋊D137—81no—
139C22⋊D137—81no—
140C22⋊D137—81no—
141C2×Dic137—21no—
142C2×Dic137—21no—
143C2×Dic137—81no—
144C2×Dic137C2×D411yescharacteristic
145C2×Dic137C2×D411yescharacteristic
146C2×Dic137C2×D411yescharacteristic
147C4×D137—21no—
148C4×D137—21no—
149C4×D137—42no—
150#11764040—21no—
151#11764040—21no—
152C23×C137—21no—
153C23×C137—21no—
154C23×C137C2×D411yescharacteristic
155Q8×C137—21no—
156Q8×C137—21no—
157Q8×C137—21no—
158Q8×C137C2×D412yes—
159D4×C137—21no—
160D4×C137—21no—
161D4×C137—21no—
162D4×C137—42no—
163D4×C137—42no—
164D4×C137C2×D412yes—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—21no—
168C2×C4×C137—21no—
169C2×C4×C137—42no—
170C2×C4×C137—42no—
171C2×C4×C137C2×D411yescharacteristic
172C2×C4×C137C2411yescharacteristic
173C8×C137—21no—
174C8×C137C2×D412yes—
175C22×C137—21no—
176C22×C137—22no—
177C22×C137—21no—
178C22×C137—21no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137C22×D411yescharacteristic
185C2×D137—21no—
186C2×D137—21no—
187C2×D137—21no—
188C2×D137—41no—
189C2×D137—41no—
190C2×D137—41no—
191C2×D137—41no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—84no—
195C4×C137—21no—
196C4×C137—22no—
197C4×C137—41no—
198C4×C137C22×D411yescharacteristic
199C4×C137H(5,𝔽2)11yescharacteristic
200Dic137—21no—
201Dic137—21no—
202Dic137—21no—
203Dic137—41no—
204Dic137—41no—
205C2×C137—42no—
206C2×C137—41no—
207C2×C137—81no—
208C2×C137D4211yescharacteristic
209C2×C137#57511yescharacteristic
210C2×C137#57611yescharacteristic
211D137—41no—
212D137—81no—
213D137—81no—
214C137#326811yescharacteristic, complemented, Sylow
215#3268—1371nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1371no—
217C2×#135D13711yescharacteristic, complemented
218#550—1371no—
219#546—1371no—
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×#41C2×D13711yescharacteristic
233C2×#41C2×D13711yescharacteristic, complemented
234C22×D4—1371no—
235AGL(1,ℤ/8)—2744no—
236#135—24no—
237C2×Q16C2×D13711yescharacteristic, complemented
238C2×SD16—1371no—
239C2×SD16—1371no—
240C2×SD16C2×D13711yescharacteristic
241C2×SD16C2×D13711yescharacteristic, complemented
242C2×D8—1371no—
243C2×D8—1371no—
244C2×D8—1371no—
245C2×D8C2×D13711yescharacteristic
246C2×M16—1371no—
247C22×C8C2×D13711yescharacteristic, complemented
248#127—2741no—
249#127—1371no—
250#124—1371no—
251#123—2741no—
252C22⋊Q8—1371no—
253#121—2741no—
254#121—2741no—
255#121—2741no—
256#121—1371no—
257#121—1371no—
258#121—1371no—
259#120—2741no—
260C4×Q8—1371no—
261C4×D4—2741no—
262C4×D4—1371no—
263C4×D4—1371no—
264C2×C22⋊C4—2741no—
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—2741no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—2742no—
282#41—5481no—
283C2×Q8C22×D13711yescharacteristic
284C2×Q8C22×D13711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2741no—
288C2×D4—2742no—
289C2×D4—2741no—
290C2×D4—2741no—
291C2×D4—2741no—
292C2×D4—2741no—
293C2×D4—2741no—
294C2×D4—2741no—
295C2×D4—5481no—
296C2×D4—1371no—
297C2×D4—1371no—
298C2×D4—1371no—
299C2×D4C22×D13711yescharacteristic
300C2×D4C22×D13711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—2741no—
304C22×C4—2741no—
305C22×C4—1371no—
306C22×C4C22×D13711yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—2742no—
310SD16—5481no—
311SD16C22⋊D13714yescomplemented
312D8—22no—
313D8—2742no—
314D8—2742no—
315D8—5481no—
316M16—2742no—
317C2×C8—22no—
318C2×C8—1371no—
319C2×C8—1371no—
320C2×C8C22×D13711yescharacteristic
321C2×C8C22×D13711yescharacteristic
322C4⋊C4—2741no—
323C4⋊C4—2741no—
324C4⋊C4—1371no—
325C4⋊C4—1371no—
326C4⋊C4—1371no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C22⋊C4—2741no—
330C22⋊C4—2741no—
331C22⋊C4—2741no—
332C22⋊C4—2741no—
333C22⋊C4—2741no—
334C22⋊C4—1371no—
335C22⋊C4—1371no—
336C42—2741no—
337C42—1371no—
338C23—21no—
339C23—21no—
340C23—2741no—
341C23—2741no—
342C23—5482no—
343C23—5482no—
344C23—10961no—
345C23—1371no—
346C23D4×D13711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D13712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—2741no—
357D4—2741no—
358D4—2741no—
359D4—2741no—
360D4—5482no—
361D4—5481no—
362D4—5482no—
363D4—5482no—
364D4—10961no—
365D4—10961no—
366D4—10961no—
367D4—10961no—
368D4C2×C22⋊D13712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—2741no—
376C2×C4—2741no—
377C2×C4—2741no—
378C2×C4—2741no—
379C2×C4—5482no—
380C2×C4—10961no—
381C2×C4—1371no—
382C2×C4—1371no—
383C2×C4—1371no—
384C2×C4D4×D13711yescharacteristic
385C2×C4C23×D13711yescharacteristic
386C8—21no—
387C8—2741no—
388C8—2741no—
389C8C2×C22⋊D13712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—2741no—
400C22—2741no—
401C22—2741no—
402C22—5481no—
403C22—5481no—
404C22—5481no—
405C22—5481no—
406C22—5481no—
407C22—5481no—
408C22—10964no—
409C22C2×D4×D13711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—2741no—
414C4—2741no—
415C4—2741no—
416C4—5481no—
417C4—5481no—
418C4C22×C22⋊D13711yescharacteristic
419C4#1798413311yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—5481no—
424C2—10961no—
425C2—10961no—
426C2D4×C22⋊D13711yescharacteristic
427C2#2257820011yescharacteristic
428C2#2257820511yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#135×C137C2
socleC22×C137C22×D4
hypercenterC2×#135D137
hypocenterC137#3268
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. #22578072C2semidirect
3. #22578041C2semidirect
4. #22577789C2semidirect
5. #22577646C2semidirect
6. #22577639C2semidirect
7. #22577627C2semidirect
8. #22577596C2semidirect
9. #22577564C2semidirect
10. #22577548C2semidirect
11. #22577548C2semidirect
12. #22577535C2semidirect
13. #22577504C2semidirect
14. #22577474C2semidirect
15. C2×#135×C137C2semidirect
20. C2×#14555497C22semidirect
26. #17984084C22semidirect
32. #17984062C22semidirect
37. Q8×Dic137C22semidirect
42. #17984022C22semidirect
43. #17984017C22semidirect
44. #17984012C22semidirect
45. #17984012C22semidirect
47. #17984010C22semidirect
49. #17984009C22semidirect
50. C22⋊#11764040C22semidirect
52. #17983998C22semidirect
54. C2×#41×C137C22semidirect
56. C2×Q16×C137C22semidirect
58. C2×SD16×C137C22semidirect
60. C22×C8×C137C22semidirect
118. SD16×C137D4semidirect
214. C137#3268semidirect
217. C2×#135D137semidirect
233. C2×#41C2×D137semidirect
237. C2×Q16C2×D137semidirect
241. C2×SD16C2×D137semidirect
247. C22×C8C2×D137semidirect
311. SD16C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
#225780722
#225780412
#225777892
#225776462
#225776392
#225776272
#225775962
#225775642
#225775482
#225775352
#225775042
#225774742
C2×#135×C1372
#3268137