Finite Group Databasebeta

← #31790414

Subgroups — #31790414

Order 17536.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984117C211yesmaximal, characteristic, complemented
2C2×M16×D137C211yesmaximal, characteristic, complemented
3#22577913C211yesmaximal, characteristic
4C22⋊Q8×D137C211yesmaximal, characteristic, complemented
5#22577661C211yesmaximal, characteristic, complemented
6#22577659C211yesmaximal, characteristic, complemented
7#22577647C211yesmaximal, characteristic, complemented
8#22577645C211yesmaximal, characteristic, complemented
9#22577643C211yesmaximal, characteristic, complemented
10#22577631C211yesmaximal, characteristic, complemented
11#22577629C211yesmaximal, characteristic, complemented
12#22577444C211yesmaximal, characteristic, complemented
13#22577410C211yesmaximal, characteristic, complemented
14#22577401C211yesmaximal, characteristic, complemented
15#468×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19#17984117—24no—
20C2×Dic548C2211yescharacteristic, complemented
21C2×#14555500C2211yescharacteristic, complemented
22C2×#14555498C2211yescharacteristic, complemented
23C2×#14555494C2211yescharacteristic
24C2×#14555493C2211yescharacteristic
25C2×#14555493C2211yescharacteristic
26M16×D137—24no—
27C2×C8×D137C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic
29#17984070—21no—
30#17984068C2211yescharacteristic
31#17984068C2211yescharacteristic
32#17984067C2211yescharacteristic
33#17984067C2211yescharacteristic
34#17984066—21no—
35#17984065—21no—
36#17984064C2211yescharacteristic
37#17984059C2211yescharacteristic
38#17984056—21no—
39D4×Dic137—21no—
40#17984048—21no—
41#17984046C2211yescharacteristic
42C4⋊C4×D137—21no—
43C4⋊C4×D137C2211yescharacteristic
44C22⋊C4×D137—21no—
45#17984021C2211yescharacteristic, complemented
46#17984020C2211yescharacteristic, complemented
47#17984019C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984013C2211yescharacteristic, complemented
50#17984012C2211yescharacteristic, complemented
51#17984011C2211yescharacteristic, complemented
52#17984011C2211yescharacteristic, complemented
53#17984009C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic, complemented
55C2×M16×C137C2211yescharacteristic
56C22⋊Q8×C137C2211yescharacteristic
57#121×C137C2211yescharacteristic
58#107×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137D411yescharacteristic
62#14555509—22no—
63#14555509—22no—
64C2×Dic274—21no—
65C2×Dic274C2311yescharacteristic
66C2×Dic274C2311yescharacteristic
67Q8×D137—22no—
68Q8×D137—22no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C22×Dic137—21no—
72C22×Dic137D411yescharacteristic
73C2×C4×D137—22no—
74C2×C4×D137—22no—
75C2×C4×D137—21no—
76C2×C4×D137—21no—
77C2×C4×D137D411yescharacteristic
78C2×C4×D137D411yescharacteristic
79C2×C4×D137C2311yescharacteristic
80C2×C4×D137C2311yescharacteristic
81Dic548—22no—
82#14555500—22no—
83#14555498—22no—
84#14555494—22no—
85#14555493—22no—
86#14555493D414yescomplemented
87C2×#11764040C2311yescharacteristic
88C2×#11764040C2311yescharacteristic
89C8×D137—22no—
90C8×D137—22no—
91C4⋊Dic137—21no—
92C4⋊Dic137—21no—
93C4⋊Dic137C2311yescharacteristic
94C4⋊Dic137C2311yescharacteristic
95#14555489—21no—
96#14555489—21no—
97#14555489—21no—
98#14555489—21no—
99C22⋊Dic137—21no—
100C22⋊Dic137—21no—
101C22⋊Dic137—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106C4×Dic137—21no—
107C2×Q8×C137C2311yescharacteristic
108C2×D4×C137—21no—
109C2×D4×C137C2311yescharacteristic
110C22×C4×C137C2311yescharacteristic
111SD16×C137—22no—
112M16×C137—22no—
113C2×C8×C137C2311yescharacteristic
114C2×C8×C137C2311yescharacteristic
115C4⋊C4×C137—21no—
116C4⋊C4×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22×D137—22no—
121C22×D137—22no—
122C22×D137—41no—
123C22×D137C2×D411yescharacteristic
124C22×D137C2×D411yescharacteristic
125Dic274—21no—
126Dic274—21no—
127Dic274—21no—
128Dic274—21no—
129Dic274—42no—
130C22⋊D137—42no—
131C22⋊D137—42no—
132C2×Dic137—22no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137—42no—
138C2×Dic137C2×D411yescharacteristic
139C2×Dic137C2×D411yescharacteristic
140C4×D137—22no—
141C4×D137—21no—
142C4×D137—21no—
143C4×D137—22no—
144C4×D137—42no—
145C4×D137—42no—
146C4×D137C2×D412yes—
147C4×D137#4112yes—
148#11764040—21no—
149#11764040C2×D412yes—
150C23×C137—21no—
151C23×C137C2×D411yescharacteristic
152Q8×C137—21no—
153Q8×C137—21no—
154D4×C137—21no—
155D4×C137—21no—
156D4×C137—42no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137C2×D411yescharacteristic
163C2×C4×C137C2411yescharacteristic
164C8×C137—21no—
165C8×C137C2×D412yes—
166C22×C137—21no—
167C22×C137—21no—
168C22×C137—21no—
169C22×C137—41no—
170C22×C137—41no—
171C22×C137—41no—
172C22×C137C22×D411yescharacteristic
173C2×D137—21no—
174C2×D137—21no—
175C2×D137—21no—
176C2×D137—21no—
177C2×D137—44no—
178C2×D137#12112yes—
179C2×D137#13712yescomplemented
180C4×C137—21no—
181C4×C137—41no—
182C4×C137—41no—
183C4×C137—41no—
184C4×C137C22×D411yescharacteristic
185C4×C137C2×#4111yescharacteristic
186Dic137—21no—
187Dic137—41no—
188Dic137—41no—
189Dic137—41no—
190Dic137—41no—
191Dic137#12112yes—
192C2×C137—41no—
193C2×C137—81no—
194C2×C137C2×#12111yescharacteristic
195C2×C137C2×#13711yescharacteristic
196C2×C137#57511yescharacteristic
197D137—22no—
198D137—22no—
199C137#305611yescharacteristic, complemented, Sylow
200#3056—1371nomaximal, Sylow
201C2×#137—1371no—
202C22×M16—1371no—
203#525—1371no—
204C2×C22⋊Q8—1371no—
205#470—1371no—
206#470—1371no—
207#468—1371no—
208#468D13711yescharacteristic, complemented
209#467—1371no—
210#467—1371no—
211#465—1371no—
212#465—1371no—
213#428—1371no—
214#419—1371no—
215#417—1371no—
216C2×#41—1371no—
217C22×Q8—1371no—
218C23×C4—1371no—
219#137—2744no—
220C2×Q16—1371no—
221C2×Q16—1371no—
222C2×SD16—1371no—
223C2×SD16C2×D13711yescharacteristic, complemented
224C2×M16—2744no—
225C2×M16—1371no—
226C2×M16—1371no—
227C2×M16—1371no—
228C2×M16C2×D13711yescharacteristic
229C22×C8—1371no—
230C22×C8—1371no—
231#123—2741no—
232C22⋊Q8—2741no—
233C22⋊Q8—2741no—
234C22⋊Q8—1371no—
235C22⋊Q8—1371no—
236C22⋊Q8—1371no—
237C22⋊Q8—1371no—
238C22⋊Q8—1371no—
239C22⋊Q8C2×D13711yescharacteristic
240#121—1371no—
241#121C2×D13711yescharacteristic
242#120—2741no—
243C4×D4—2741no—
244C4×D4—2741no—
245#117—1371no—
246C2×C4⋊C4—2741no—
247C2×C4⋊C4—1371no—
248C2×C22⋊C4—2741no—
249#107—1371no—
250#107C2×D13711yescharacteristic, complemented
251#106—1371no—
252#106—1371no—
253#103—1371no—
254#103—1371no—
255#103—1371no—
256#103—1371no—
257#103—1371no—
258#103C2×D13711yescharacteristic, complemented
259#102—1371no—
260#102C2×D13711yescharacteristic, complemented
261C24—1371no—
262#41—2742no—
263#41—2742no—
264C2×Q8—2741no—
265C2×Q8—2742no—
266C2×Q8—2742no—
267C2×Q8—1371no—
268C2×Q8—1371no—
269C2×Q8C22×D13711yescharacteristic
270C2×D4—21no—
271C2×D4—2741no—
272C2×D4—2741no—
273C2×D4C22×D13711yescharacteristic
274C22×C4—2742no—
275C22×C4—2742no—
276C22×C4—2741no—
277C22×C4—2741no—
278C22×C4—2741no—
279C22×C4—1371no—
280C22×C4—1371no—
281C22×C4—1371no—
282C22×C4—1371no—
283C22×C4—1371no—
284C22×C4C22×D13711yescharacteristic
285Q16—2742no—
286Q16—2742no—
287SD16—22no—
288SD16—2742no—
289M16—22no—
290M16—2742no—
291M16—2742no—
292M16—1374no—
293C2×C8—2742no—
294C2×C8—2742no—
295C2×C8—1371no—
296C2×C8—1371no—
297C2×C8C22×D13711yescharacteristic
298C2×C8C22×D13711yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—2741no—
301C4⋊C4—2741no—
302C4⋊C4—2741no—
303C4⋊C4—2741no—
304C4⋊C4—2741no—
305C4⋊C4—2741no—
306C4⋊C4—1371no—
307C4⋊C4—1371no—
308C4⋊C4C22×D13711yescharacteristic
309C4⋊C4C22×D13711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2741no—
313C22⋊C4—2741no—
314C22⋊C4—2741no—
315C22⋊C4—2741no—
316C22⋊C4—2741no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C42—2741no—
320C23—21no—
321C23—2742no—
322C23—2742no—
323C23—5481no—
324C23—1371no—
325C23—1371no—
326C23D4×D13711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—2741no—
330Q8—2741no—
331Q8—2741no—
332Q8—2741no—
333Q8—5482no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—5482no—
338D4—5482no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2742no—
345C2×C4—2742no—
346C2×C4—2742no—
347C2×C4—2741no—
348C2×C4—2741no—
349C2×C4—2741no—
350C2×C4—2741no—
351C2×C4—2741no—
352C2×C4—2741no—
353C2×C4—5482no—
354C2×C4—5482no—
355C2×C4—5482no—
356C2×C4—1371no—
357C2×C4—1371no—
358C2×C4—1372no—
359C2×C4—1372no—
360C2×C4D4×D13711yescharacteristic
361C2×C4C23×D13711yescharacteristic
362C8—21no—
363C8—2741no—
364C8—1372no—
365C8D4×D13712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—2741no—
373C22—2741no—
374C22—2741no—
375C22—2741no—
376C22—5484no—
377C22—1372no—
378C22—1372no—
379C22C2×D4×D13711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—2741no—
385C4—5481no—
386C4—5481no—
387C4—5481no—
388C4—5481no—
389C4—1372no—
390C4C2×D4×D13711yescharacteristic
391C4#41×D13711yescharacteristic
392C2—41no—
393C2—81no—
394C2—2742no—
395C2—2742no—
396C2#121×D13711yescharacteristic
397C2#137×D13711yescharacteristic
398C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#468×C137C2
socleC22×C137C22×D4
prime radical#22577913C2
hypercenter#468D137
hypocenterC137#3056
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#17984117C2semidirect
2. C2×M16×D137C2semidirect
4. C22⋊Q8×D137C2semidirect
5. #22577661C2semidirect
6. #22577659C2semidirect
7. #22577647C2semidirect
8. #22577645C2semidirect
9. #22577643C2semidirect
10. #22577631C2semidirect
11. #22577629C2semidirect
12. #22577444C2semidirect
13. #22577410C2semidirect
14. #22577401C2semidirect
15. #468×C137C2semidirect
20. C2×Dic548C22semidirect
21. C2×#14555500C22semidirect
22. C2×#14555498C22semidirect
45. #17984021C22semidirect
46. #17984020C22semidirect
47. #17984019C22semidirect
48. #17984013C22semidirect
49. #17984013C22semidirect
50. #17984012C22semidirect
51. #17984011C22semidirect
52. #17984011C22semidirect
53. #17984009C22semidirect
54. C2×SD16×C137C22semidirect
58. #107×C137C22semidirect
59. #103×C137C22semidirect
60. #102×C137C22semidirect
86. #14555493D4semidirect
179. C2×D137#137semidirect
199. C137#3056semidirect
208. #468D137semidirect
223. C2×SD16C2×D137semidirect
250. #107C2×D137semidirect
258. #103C2×D137semidirect
260. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841172
C2×M16×D1372
#225779132
C22⋊Q8×D1372
#225776612
#225776592
#225776472
#225776452
#225776432
#225776312
#225776292
#225774442
#225774102
#225774012
#468×C1372
#3056137