Finite Group Databasebeta

← #31792307

Subgroups — #31792307

Order 17536.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984112C211yesmaximal, characteristic, complemented
2C2×#17984108C211yesmaximal, characteristic, complemented
3#22578052C211yesmaximal, characteristic, complemented
4#22578050C211yesmaximal, characteristic, complemented
5#22577794C211yesmaximal, characteristic, complemented
6#22577660C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577616C211yesmaximal, characteristic, complemented
9#22577607C211yesmaximal, characteristic, complemented
10#22577567C211yesmaximal, characteristic, complemented
11#22577566C211yesmaximal, characteristic, complemented
12#22577552C211yesmaximal, characteristic, complemented
13#22577512C211yesmaximal, characteristic, complemented
14#22577471C211yesmaximal, characteristic, complemented
15#449×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555509C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984112—24no—
20#17984108—24no—
21C2×Dic548C2211yescharacteristic, complemented
22C2×#14555500C2211yescharacteristic, complemented
23C2×#14555499C2211yescharacteristic
24C2×#14555498C2211yescharacteristic
25C2×#14555498C2211yescharacteristic, complemented
26C2×#14555497C2211yescharacteristic
27C2×#14555495C2211yescharacteristic
28C2×#14555493C2211yescharacteristic
29C22×#11764040C2211yescharacteristic, complemented
30#17984086C2211yescharacteristic, complemented
31#17984075C2211yescharacteristic
32#17984071—21no—
33#17984070—21no—
34#17984069—21no—
35#17984068C2211yescharacteristic
36#17984067C2211yescharacteristic, complemented
37#17984064C2211yescharacteristic
38#17984060—21no—
39#17984059—21no—
40#17984056—21no—
41D4×Dic137C2211yescharacteristic, complemented
42#17984050C2211yescharacteristic
43C4×C22⋊D137C2211yescharacteristic
44C2×#14555489—21no—
45C22⋊C4×D137—21no—
46#17984019C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984011C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic
51#17984009C2211yescharacteristic
52#17984000C2211yescharacteristic
53#17983998C2211yescharacteristic, complemented
54C2×#41×C137C2211yescharacteristic, complemented
55C2×SD16×C137C2211yescharacteristic
56C2×D8×C137C2211yescharacteristic
57#121×C137C2211yescharacteristic
58#103×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60C22⋊C8×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555510—22no—
63#14555510—41no—
64#14555509—22no—
65#14555509—41no—
66C2×Dic274C2311yescharacteristic
67C2×Dic274C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137D411yescharacteristic
73D4×D137—22no—
74D4×D137—41no—
75C22×Dic137—21no—
76C22×Dic137—21no—
77C2×C4×D137—21no—
78C2×C4×D137—21no—
79C2×C4×D137C2311yescharacteristic
80Dic548—41no—
81#14555500—22no—
82#14555499—22no—
83#14555498—22no—
84#14555498—22no—
85#14555497—22no—
86#14555495D414yescomplemented
87#14555493—22no—
88C2×#11764040—22no—
89C2×#11764040C2311yescharacteristic
90C2×#11764040C2311yescharacteristic
91C4⋊Dic137—21no—
92C4⋊Dic137C2311yescharacteristic
93#14555489—21no—
94#14555489—21no—
95#14555489—21no—
96#14555489—21no—
97#14555489D411yescharacteristic
98C22⋊Dic137—21no—
99C22⋊Dic137—21no—
100C22⋊Dic137—21no—
101C22⋊Dic137—21no—
102C22⋊Dic137D411yescharacteristic
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487D411yescharacteristic
107C4×Dic137C2311yescharacteristic
108#41×C137—22no—
109#41×C137—41no—
110C2×Q8×C137C2311yescharacteristic
111C2×D4×C137—21no—
112C2×D4×C137—21no—
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115C22×C4×C137—21no—
116C22×C4×C137C2311yescharacteristic
117SD16×C137—41no—
118D8×C137—22no—
119C2×C8×C137C2311yescharacteristic
120C2×C8×C137C2311yescharacteristic
121C4⋊C4×C137C2311yescharacteristic
122C22⋊C4×C137—21no—
123C22×D137—21no—
124C22×D137—42no—
125C22×D137—42no—
126C22×D137—81no—
127C22×D137C2×D411yescharacteristic
128Dic274—21no—
129Dic274—21no—
130Dic274—21no—
131Dic274—21no—
132D548—21no—
133D548C2×D412yes—
134C22⋊D137—41no—
135C22⋊D137—42no—
136C22⋊D137—42no—
137C22⋊D137—42no—
138C2×Dic137—21no—
139C2×Dic137—21no—
140C2×Dic137—21no—
141C2×Dic137—42no—
142C2×Dic137—42no—
143C2×Dic137C2×D411yescharacteristic
144C2×Dic137C2×D411yescharacteristic
145C2×Dic137C2×D411yescharacteristic
146C4×D137—21no—
147C4×D137—21no—
148C4×D137—42no—
149C4×D137—81no—
150#11764040—21no—
151#11764040C2×D412yes—
152C23×C137—21no—
153C23×C137—21no—
154C23×C137C2×D411yescharacteristic
155Q8×C137—21no—
156Q8×C137—21no—
157D4×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—42no—
161D4×C137—81no—
162D4×C137C2×D412yes—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—42no—
168C2×C4×C137C2×D411yescharacteristic
169C2×C4×C137C2411yescharacteristic
170C8×C137—21no—
171C8×C137—21no—
172C22×C137—21no—
173C22×C137—22no—
174C22×C137—21no—
175C22×C137—21no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137C22×D411yescharacteristic
182C2×D137—21no—
183C2×D137—22no—
184C2×D137—21no—
185C2×D137—21no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—84no—
189C4×C137—21no—
190C4×C137—41no—
191C4×C137—41no—
192C4×C137C22×D411yescharacteristic
193C4×C137H(5,𝔽2)11yescharacteristic
194Dic137—21no—
195Dic137—21no—
196Dic137—21no—
197Dic137—41no—
198Dic137—41no—
199Dic137—41no—
200C2×C137—41no—
201C2×C137—42no—
202C2×C137—81no—
203C2×C137D4211yescharacteristic
204C2×C137#57511yescharacteristic
205C2×C137#57711yescharacteristic
206D137—41no—
207D137—42no—
208C137#326711yescharacteristic, complemented, Sylow
209#3267—1371nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1371no—
211C2×#135—1371no—
212#547—1371no—
213#546—1371no—
214#497—1371no—
215#470—1371no—
216#465—1371no—
217#463—1371no—
218#461—1371no—
219#452—1371no—
220#452—1371no—
221#450—1371no—
222#449D13711yescharacteristic, complemented
223#442—1371no—
224#435—1371no—
225C2×#41—1371no—
226C2×#41—1371no—
227C2×#41C2×D13711yescharacteristic, complemented
228C22×D4—1371no—
229AGL(1,ℤ/8)—2744no—
230#135—2744no—
231C2×Q16—1371no—
232C2×Q16—1371no—
233C2×SD16—1371no—
234C2×SD16—1371no—
235C2×SD16—1371no—
236C2×SD16—1371no—
237C2×SD16C2×D13711yescharacteristic
238C2×D8—1371no—
239C2×D8C2×D13711yescharacteristic
240C2×M16—1371no—
241C22×C8—1371no—
242C4⋊Q8—1371no—
243#124—1371no—
244#123—2741no—
245#123—2741no—
246#123—2741no—
247C22⋊Q8—1371no—
248C22⋊Q8—1371no—
249C22⋊Q8—1371no—
250#121—2741no—
251#121—2741no—
252#121C2×D13711yescharacteristic
253#120—2741no—
254C4×D4—1371no—
255C4×D4—1371no—
256C4×D4—1371no—
257C2×C4⋊C4—2741no—
258C2×C22⋊C4—2741no—
259#106—1371no—
260C4⋊C8—1371no—
261#103—1371no—
262#103—1371no—
263#103C2×D13711yescharacteristic, complemented
264#102—1371no—
265#102—1371no—
266#102C2×D13711yescharacteristic, complemented
267C22⋊C8—1371no—
268C22⋊C8C2×D13711yescharacteristic, complemented
269#97—1371no—
270C24—2741no—
271#41—22no—
272#41—41no—
273#41—2742no—
274#41—2742no—
275#41—5481no—
276#41—5481no—
277C2×Q8—1371no—
278C2×Q8—1371no—
279C2×Q8C22×D13711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2742no—
283C2×D4—2741no—
284C2×D4—2741no—
285C2×D4—2741no—
286C2×D4—5481no—
287C2×D4—1371no—
288C2×D4—1371no—
289C2×D4C22×D13711yescharacteristic
290C2×D4C22×D13711yescharacteristic
291C22×C4—21no—
292C22×C4—2741no—
293C22×C4—2741no—
294C22×C4—2741no—
295C22×C4—2741no—
296C22×C4—1371no—
297C22×C4C22×D13711yescharacteristic
298Q16—2742no—
299Q16—5481no—
300SD16—41no—
301SD16—2742no—
302SD16—2742no—
303SD16—2742no—
304SD16—2742no—
305D8—22no—
306D8—1374no—
307M16—2742no—
308C2×C8—2742no—
309C2×C8—1371no—
310C2×C8—1371no—
311C2×C8C22×D13711yescharacteristic
312C2×C8C22×D13711yescharacteristic
313C4⋊C4—2741no—
314C4⋊C4—2741no—
315C4⋊C4—2741no—
316C4⋊C4—2741no—
317C4⋊C4—2741no—
318C4⋊C4—1371no—
319C4⋊C4—1371no—
320C4⋊C4C22×D13711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C22⋊C4—2741no—
326C22⋊C4—2741no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C22⋊C4—1371no—
330C22⋊C4—1371no—
331C42—1371no—
332C23—21no—
333C23—21no—
334C23—2741no—
335C23—5482no—
336C23—5482no—
337C23—10961no—
338C23—1371no—
339C23D4×D13711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2741no—
343Q8—2741no—
344Q8—2741no—
345Q8—2741no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—2741no—
352D4—5481no—
353D4—5482no—
354D4—5482no—
355D4—5482no—
356D4—1372no—
357D4D4×D13712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—2741no—
364C2×C4—2741no—
365C2×C4—2741no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—5482no—
369C2×C4—5482no—
370C2×C4—5482no—
371C2×C4—10961no—
372C2×C4—1371no—
373C2×C4—1371no—
374C2×C4—1371no—
375C2×C4D4×D13711yescharacteristic
376C2×C4C23×D13711yescharacteristic
377C8—21no—
378C8—21no—
379C8—2741no—
380C8—1372no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—2741no—
391C22—2741no—
392C22—2741no—
393C22—2742no—
394C22—5481no—
395C22—5481no—
396C22—10964no—
397C22C2×D4×D13711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—2741no—
402C4—2741no—
403C4—2741no—
404C4—5481no—
405C4—5481no—
406C4—5481no—
407C4C2×D4×D13711yescharacteristic
408C4#1798413311yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—5481no—
413C2—5482no—
414C2#2257803511yescharacteristic
415C2#2257820011yescharacteristic
416C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#449×C137C2
socleC22×C137C22×D4
hypercenter#449D137
hypocenterC137#3267
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×#17984108C2semidirect
3. #22578052C2semidirect
4. #22578050C2semidirect
5. #22577794C2semidirect
6. #22577660C2semidirect
7. #22577630C2semidirect
8. #22577616C2semidirect
9. #22577607C2semidirect
10. #22577567C2semidirect
11. #22577566C2semidirect
12. #22577552C2semidirect
13. #22577512C2semidirect
14. #22577471C2semidirect
15. #449×C137C2semidirect
21. C2×Dic548C22semidirect
22. C2×#14555500C22semidirect
25. C2×#14555498C22semidirect
29. C22×#11764040C22semidirect
30. #17984086C22semidirect
36. #17984067C22semidirect
41. D4×Dic137C22semidirect
46. #17984019C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984011C22semidirect
53. #17983998C22semidirect
54. C2×#41×C137C22semidirect
58. #103×C137C22semidirect
59. #102×C137C22semidirect
60. C22⋊C8×C137C22semidirect
86. #14555495D4semidirect
208. C137#3267semidirect
222. #449D137semidirect
227. C2×#41C2×D137semidirect
263. #103C2×D137semidirect
266. #102C2×D137semidirect
268. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×#179841082
#225780522
#225780502
#225777942
#225776602
#225776302
#225776162
#225776072
#225775672
#225775662
#225775522
#225775122
#225774712
#449×C1372
#3267137