Finite Group Databasebeta

← #31790787

Subgroups — #31790787

Order 17536.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577950C211yesmaximal, characteristic, complemented
2C4×D4×D137C211yesmaximal, characteristic, complemented
3#22577718C211yesmaximal, characteristic, complemented
4#22577715C211yesmaximal, characteristic, complemented
5#22577682C211yesmaximal, characteristic, complemented
6#22577681C211yesmaximal, characteristic, complemented
7#22577597C211yesmaximal, characteristic, complemented
8#22577596C211yesmaximal, characteristic, complemented
9#22577538C211yesmaximal, characteristic, complemented
10#22577535C211yesmaximal, characteristic, complemented
11C4×D1096C211yesmaximal, characteristic, complemented
12C4×#14555495C211yesmaximal, characteristic, complemented
13C4×#14555495C211yesmaximal, characteristic, complemented
14#22577454C211yesmaximal, characteristic, complemented
15C4×D8×C137C211yesmaximal, characteristic, complemented
16C2×#14555510—21no—
17C2×D4×D137C2211yescharacteristic
18C22×C4×D137—21no—
19C2×D1096C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21C2×#14555495C2211yescharacteristic
22#17984087C2211yescharacteristic, complemented
23#17984084C2211yescharacteristic, complemented
24#17984074—21no—
25#17984063C2211yescharacteristic
26#17984061—21no—
27#17984059C2211yescharacteristic
28C4×D548C2211yescharacteristic
29C4×D548C2211yescharacteristic, complemented
30D4×Dic137C2211yescharacteristic
31#17984050C2211yescharacteristic
32C4×C22⋊D137—21no—
33C4×C22⋊D137—21no—
34#17984048—21no—
35C4⋊C4×D137C2211yescharacteristic
36C22⋊C4×D137—21no—
37C42×D137C2211yescharacteristic
38#17984022C2211yescharacteristic, complemented
39#17984021C2211yescharacteristic, complemented
40#17984021C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic, complemented
49#17984000C2211yescharacteristic
50#17984000C2211yescharacteristic
51C4×#11764040C2211yescharacteristic, complemented
52C2×D8×C137C2211yescharacteristic
53C4×D4×C137C2211yescharacteristic
54C4×D4×C137C2211yescharacteristic, complemented
55#107×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic, complemented
58C4×C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—44no—
61C2×Dic274—21no—
62C2×D548—21no—
63C2×D548C2311yescharacteristic
64C2×D548C2311yescharacteristic
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69D4×D137—22no—
70D4×D137—22no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137—44no—
77C2×C4×D137D411yescharacteristic
78C2×C4×D137D411yescharacteristic
79C2×C4×D137C2311yescharacteristic
80D1096—22no—
81#14555495—22no—
82#14555495—22no—
83C2×#11764040C2311yescharacteristic
84C2×#11764040C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95C4×Dic137D411yescharacteristic
96C4×Dic137D411yescharacteristic
97C4×Dic137C2311yescharacteristic
98C2×D4×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C22×C4×C137—21no—
101C22×C4×C137—21no—
102D8×C137—22no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108C22⋊C4×C137—21no—
109C42×C137C2311yescharacteristic
110C22×D137—24no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137C2×D411yescharacteristic
116Dic274—42no—
117D548—21no—
118D548—21no—
119D548—21no—
120D548—42no—
121D548#4112yescomplemented
122C22⋊D137—24no—
123C22⋊D137—42no—
124C22⋊D137—42no—
125C22⋊D137—42no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—42no—
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—41no—
135C4×D137—42no—
136C4×D137—42no—
137C4×D137—42no—
138C4×D137—42no—
139C4×D137D814yescomplemented
140#11764040—21no—
141#11764040—21no—
142C23×C137—21no—
143C23×C137—21no—
144D4×C137—21no—
145D4×C137—21no—
146D4×C137—21no—
147D4×C137#4112yescomplemented
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—42no—
151C2×C4×C137—42no—
152C2×C4×C137C2×D411yescharacteristic
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2411yescharacteristic
155C8×C137—21no—
156C8×C137—21no—
157C22×C137—22no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137C22×D411yescharacteristic
164C2×D137—21no—
165C2×D137—21no—
166C2×D137—22no—
167C2×D137—48no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137C2×D812yes—
174C4×C137—21no—
175C4×C137—41no—
176C4×C137—41no—
177C4×C137C2×D812yes—
178C4×C137C2×#4111yescharacteristic
179C4×C137C2×#4111yescharacteristic
180Dic137—21no—
181Dic137—21no—
182Dic137—41no—
183Dic137—41no—
184Dic137C2×D812yes—
185C2×C137—42no—
186C2×C137—81no—
187C2×C137#52511yescharacteristic
188C2×C137C22×D811yescharacteristic
189C2×C137#57511yescharacteristic
190D137—22no—
191D137—42no—
192D137—81no—
193C137#310011yescharacteristic, complemented, Sylow
194#3100—1371nomaximal, Sylow
195#532—1371no—
196C2×C4×D4—1371no—
197#480—1371no—
198#480—1371no—
199#474—1371no—
200#474—1371no—
201#459—1371no—
202#459—1371no—
203#447—1371no—
204#447—1371no—
205C4×D8—1371no—
206C4×D8—1371no—
207C4×D8—1371no—
208C4×D8D13711yescharacteristic, complemented
209#431—1371no—
210C2×#41—2741no—
211C22×D4—1371no—
212C23×C4—2741no—
213C2×D8—1371no—
214C2×D8—1371no—
215C2×D8—1371no—
216C2×D8C2×D13711yescharacteristic
217C4⋊Q8—1371no—
218#127—1371no—
219#124—2741no—
220#121—2741no—
221#121—1371no—
222#121—1371no—
223C4×D4—2741no—
224C4×D4—2741no—
225C4×D4—2741no—
226C4×D4—1371no—
227C4×D4—1371no—
228C4×D4—1371no—
229C4×D4—1371no—
230C4×D4C2×D13711yescharacteristic
231C4×D4C2×D13711yescharacteristic, complemented
232C2×C4⋊C4—1371no—
233C2×C22⋊C4—2741no—
234C2×C42—1371no—
235#107—1371no—
236#107—1371no—
237#107—1371no—
238#107C2×D13711yescharacteristic, complemented
239C4⋊C8—1371no—
240C4⋊C8—1371no—
241#102—1371no—
242#102—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102C2×D13711yescharacteristic
248#102C2×D13711yescharacteristic, complemented
249C22⋊C8—1371no—
250C22⋊C8—1371no—
251C4×C8—1371no—
252C4×C8C2×D13711yescharacteristic, complemented
253C24—2741no—
254#41—5484no—
255C2×Q8—2741no—
256C2×D4—2741no—
257C2×D4—2742no—
258C2×D4—2742no—
259C2×D4—2741no—
260C2×D4—2741no—
261C2×D4—2741no—
262C2×D4—2741no—
263C2×D4—1371no—
264C2×D4—1371no—
265C2×D4C22×D13711yescharacteristic
266C2×D4C22×D13711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2741no—
270C22×C4—2741no—
271C22×C4—2741no—
272C22×C4—2741no—
273C22×C4—2741no—
274C22×C4—5484no—
275C22×C4—1371no—
276C22×C4—1371no—
277C22×C4—1371no—
278D8—22no—
279D8—2742no—
280D8—2742no—
281D8—2742no—
282C2×C8—1371no—
283C2×C8—1371no—
284C2×C8C22×D13711yescharacteristic
285C2×C8C22×D13711yescharacteristic
286C4⋊C4—2741no—
287C4⋊C4—2741no—
288C4⋊C4—1371no—
289C4⋊C4—1371no—
290C4⋊C4C22×D13711yescharacteristic
291C4⋊C4C22×D13711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4—2741no—
300C42—1371no—
301C42—1371no—
302C42—1371no—
303C42C22×D13711yescharacteristic
304C23—21no—
305C23—21no—
306C23—2744no—
307C23—2741no—
308C23—2741no—
309C23—5482no—
310C23—5482no—
311C23—1371no—
312Q8—5482no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—2744no—
317D4—2741no—
318D4—2741no—
319D4—2741no—
320D4—5482no—
321D4—5482no—
322D4—5482no—
323D4—5482no—
324D4—1372no—
325D4#1455551012yescomplemented
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—5481no—
335C2×C4—5482no—
336C2×C4—5482no—
337C2×C4—5482no—
338C2×C4—5482no—
339C2×C4—5482no—
340C2×C4—1371no—
341C2×C4—1371no—
342C2×C4—1371no—
343C2×C4—1374no—
344C2×C4D4×D13711yescharacteristic
345C2×C4D4×D13711yescharacteristic
346C2×C4C23×D13711yescharacteristic
347C8—21no—
348C8—21no—
349C8—2741no—
350C8—2741no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—2741no—
358C22—2741no—
359C22—2742no—
360C22—5488no—
361C22—5481no—
362C22—5481no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—1372no—
367C22C2×D4×D13711yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—2741no—
372C4—2741no—
373C4—5481no—
374C4—5481no—
375C4—1372no—
376C4D8×D13712yes—
377C4#41×D13711yescharacteristic
378C4C2×#1455551011yescharacteristic
379C2—42no—
380C2—81no—
381C2—2742no—
382C2—5482no—
383C2—10961no—
384C2#2257791211yescharacteristic
385C2C2×D8×D13711yescharacteristic
386C2#2257820011yescharacteristic
subgroupG/H
centerC2×C4D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC4×D8×C137C2
socleC22×C137C22×D4
hypercenterC4×D8D137
hypocenterC137#3100
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22577950C2semidirect
2. C4×D4×D137C2semidirect
3. #22577718C2semidirect
4. #22577715C2semidirect
5. #22577682C2semidirect
6. #22577681C2semidirect
7. #22577597C2semidirect
8. #22577596C2semidirect
9. #22577538C2semidirect
10. #22577535C2semidirect
11. C4×D1096C2semidirect
12. C4×#14555495C2semidirect
13. C4×#14555495C2semidirect
14. #22577454C2semidirect
15. C4×D8×C137C2semidirect
22. #17984087C22semidirect
23. #17984084C22semidirect
29. C4×D548C22semidirect
38. #17984022C22semidirect
39. #17984021C22semidirect
40. #17984021C22semidirect
41. #17984017C22semidirect
42. #17984017C22semidirect
44. #17984010C22semidirect
46. #17984009C22semidirect
48. #17984008C22semidirect
51. C4×#11764040C22semidirect
54. C4×D4×C137C22semidirect
55. #107×C137C22semidirect
57. #102×C137C22semidirect
58. C4×C8×C137C22semidirect
121. D548#41semidirect
139. C4×D137D8semidirect
147. D4×C137#41semidirect
193. C137#3100semidirect
208. C4×D8D137semidirect
231. C4×D4C2×D137semidirect
238. #107C2×D137semidirect
248. #102C2×D137semidirect
252. C4×C8C2×D137semidirect
325. D4#14555510semidirect

Maximal subgroups

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

subgroupindex
#225779502
C4×D4×D1372
#225777182
#225777152
#225776822
#225776812
#225775972
#225775962
#225775382
#225775352
C4×D10962
C4×#145554952
#225774542
C4×D8×C1372
#3100137