Finite Group Databasebeta

← #31792304

Subgroups — #31792304

Order 17536.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984121C211yesmaximal, characteristic, complemented
2C22×#14555499C211yesmaximal, characteristic, complemented
3D4×Dic274C211yesmaximal, characteristic, complemented
4#22578049C211yesmaximal, characteristic
5#22577797C211yesmaximal, characteristic, complemented
6#22577649C211yesmaximal, characteristic, complemented
7#22577644C211yesmaximal, characteristic, complemented
8#22577626C211yesmaximal, characteristic, complemented
9#22577609C211yesmaximal, characteristic, complemented
10#22577567C211yesmaximal, characteristic, complemented
11#22577561C211yesmaximal, characteristic, complemented
12#22577556C211yesmaximal, characteristic, complemented
13#22577544C211yesmaximal, characteristic, complemented
14#22577505C211yesmaximal, characteristic, complemented
15#435×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×Dic274C2211yescharacteristic
18C22×Dic274C2211yescharacteristic, complemented
19C22×D548C2211yescharacteristic
20#17984121—24no—
21C2×Dic548C2211yescharacteristic, complemented
22C2×Dic548C2211yescharacteristic, complemented
23C2×Dic548C2211yescharacteristic, complemented
24C2×#14555499—24no—
25C2×#14555499C2211yescharacteristic
26C2×#14555499C2211yescharacteristic
27C2×#14555499C2211yescharacteristic
28C2×#14555499C2211yescharacteristic
29C2×#14555499C2211yescharacteristic
30C2×#14555499C2211yescharacteristic
31#17984088C2211yescharacteristic, complemented
32#17984087—21no—
33#17984076C2211yescharacteristic
34#17984073—21no—
35#17984067C2211yescharacteristic
36#17984067C2211yescharacteristic
37#17984067C2211yescharacteristic, complemented
38#17984065—21no—
39#17984065—21no—
40#17984062C2211yescharacteristic
41#17984061—21no—
42#17984056—21no—
43C4×Dic274C2211yescharacteristic, complemented
44C4×D548C2211yescharacteristic
45D4×Dic137—21no—
46C22⋊C4×D137—21no—
47#17984022C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984013C2211yescharacteristic, complemented
50#17984013C2211yescharacteristic, complemented
51#17984013C2211yescharacteristic, complemented
52#17984010C2211yescharacteristic
53#17984010C2211yescharacteristic
54C2×M16×C137C2211yescharacteristic, complemented
55C22×C8×C137C2211yescharacteristic
56C4×D4×C137C2211yescharacteristic
57C4⋊C8×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C22⋊C8×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555510—22no—
63#14555510—41no—
64C2×Dic274—22no—
65C2×Dic274—22no—
66C2×Dic274—21no—
67C2×Dic274—21no—
68C2×Dic274—41no—
69C2×Dic274—41no—
70C2×Dic274C2311yescharacteristic
71C2×Dic274C2311yescharacteristic
72C2×Dic274C2311yescharacteristic
73C2×Dic274C2311yescharacteristic
74C2×D548—22no—
75C2×D548—21no—
76C2×D548—41no—
77C2×D548C2311yescharacteristic
78C2×D548C2311yescharacteristic
79C2×C22⋊D137—21no—
80C2×C22⋊D137—21no—
81C22×Dic137—21no—
82C22×Dic137—21no—
83C2×C4×D137—21no—
84C2×C4×D137—21no—
85Dic548—22no—
86Dic548—22no—
87Dic548—41no—
88#14555499—22no—
89#14555499—22no—
90#14555499—22no—
91#14555499—22no—
92#14555499—41no—
93#14555499D414yescomplemented
94C4⋊Dic137—21no—
95C4⋊Dic137—21no—
96C4⋊Dic137C2311yescharacteristic
97C4⋊Dic137C2311yescharacteristic
98#14555489—21no—
99#14555489—21no—
100#14555489—21no—
101#14555489—21no—
102C22⋊Dic137—21no—
103C22⋊Dic137—21no—
104C22⋊Dic137—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109C4×Dic137—21no—
110C2×D4×C137D411yescharacteristic
111C22×C4×C137C2311yescharacteristic
112C22×C4×C137C2311yescharacteristic
113M16×C137—22no—
114C2×C8×C137—22no—
115C2×C8×C137C2311yescharacteristic
116C2×C8×C137C2311yescharacteristic
117C2×C8×C137C2311yescharacteristic
118C2×C8×C137C2311yescharacteristic
119C4⋊C4×C137D411yescharacteristic
120C22⋊C4×C137D411yescharacteristic
121C22⋊C4×C137D411yescharacteristic
122C42×C137C2311yescharacteristic
123C22×D137—21no—
124C22×D137—21no—
125C22×D137—42no—
126C22×D137—42no—
127C22×D137—81no—
128Dic274—21no—
129Dic274—21no—
130Dic274—21no—
131Dic274—21no—
132Dic274—21no—
133Dic274—21no—
134Dic274—21no—
135Dic274—42no—
136Dic274—42no—
137Dic274C2×D412yes—
138D548—21no—
139D548—21no—
140D548—21no—
141D548—42no—
142D548C2×D412yes—
143C22⋊D137—42no—
144C22⋊D137—81no—
145C2×Dic137—21no—
146C2×Dic137—21no—
147C2×Dic137—21no—
148C2×Dic137—21no—
149C2×Dic137—21no—
150C2×Dic137—21no—
151C2×Dic137—42no—
152C2×Dic137—42no—
153C4×D137—42no—
154C4×D137—81no—
155C23×C137C2×D411yescharacteristic
156C23×C137C2×D411yescharacteristic
157D4×C137—22no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137C2×D411yescharacteristic
163C2×C4×C137C2×D411yescharacteristic
164C2×C4×C137C2×D411yescharacteristic
165C2×C4×C137C2×D411yescharacteristic
166C2×C4×C137C2411yescharacteristic
167C8×C137—21no—
168C8×C137—21no—
169C8×C137—21no—
170C8×C137C2×D412yes—
171C22×C137—21no—
172C22×C137—21no—
173C22×C137—21no—
174C22×C137—21no—
175C22×C137—21no—
176C22×C137#13712yescomplemented
177C22×C137C22×D411yescharacteristic
178C2×D137—22no—
179C2×D137—41no—
180C2×D137—41no—
181C2×D137—41no—
182C2×D137—41no—
183C2×D137—41no—
184C2×D137—84no—
185C4×C137—21no—
186C4×C137—21no—
187C4×C137—21no—
188C4×C137—21no—
189C4×C137C22×D411yescharacteristic
190C4×C137H(5,𝔽2)11yescharacteristic
191Dic137—22no—
192Dic137—41no—
193Dic137—41no—
194Dic137—41no—
195Dic137—41no—
196Dic137—41no—
197C2×C137—22no—
198C2×C137—41no—
199C2×C137D4211yescharacteristic
200C2×C137C2×#13711yescharacteristic
201C2×C137#57711yescharacteristic
202D137—42no—
203D137—81no—
204C137#326611yescharacteristic, complemented, Sylow
205#3266—1371nomaximal, Sylow
206C2×#137—1371no—
207C22×SD16—1371no—
208D4×Q8—1371no—
209#546—1371no—
210#497—1371no—
211#468—1371no—
212#467—1371no—
213#464—1371no—
214#461—1371no—
215#452—1371no—
216#451—1371no—
217#450—1371no—
218#448—1371no—
219#440—1371no—
220#435D13711yescharacteristic, complemented
221C2×#41—1371no—
222C22×Q8—1371no—
223C22×Q8—1371no—
224C22×D4—1371no—
225#137—2744no—
226C2×Q16—1371no—
227C2×Q16—1371no—
228C2×Q16—1371no—
229C2×SD16—2744no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×SD16—1371no—
233C2×SD16—1371no—
234C2×SD16—1371no—
235C2×SD16—1371no—
236C2×M16C2×D13711yescharacteristic, complemented
237C22×C8C2×D13711yescharacteristic
238C4⋊Q8—2741no—
239C4⋊Q8—1371no—
240#124—1371no—
241#123—2741no—
242C22⋊Q8—2741no—
243C22⋊Q8—2741no—
244C22⋊Q8—1371no—
245C22⋊Q8—1371no—
246C22⋊Q8—1371no—
247#121—2741no—
248#121—1371no—
249#120—2741no—
250C4×Q8—1371no—
251C4×D4—2741no—
252C4×D4—1371no—
253C4×D4C2×D13711yescharacteristic
254C2×C22⋊C4—2741no—
255#107—1371no—
256C4⋊C8C2×D13711yescharacteristic, complemented
257#103—1371no—
258#103—1371no—
259#103—1371no—
260#103—1371no—
261#102—1371no—
262#102—1371no—
263C22⋊C8C2×D13711yescharacteristic
264C22⋊C8C2×D13711yescharacteristic, complemented
265#97C2×D13711yescharacteristic, complemented
266C24—2741no—
267#41—2742no—
268#41—5481no—
269C2×Q8—2742no—
270C2×Q8—2742no—
271C2×Q8—2741no—
272C2×Q8—2741no—
273C2×Q8—5481no—
274C2×Q8—5481no—
275C2×Q8—1371no—
276C2×Q8—1371no—
277C2×Q8—1371no—
278C2×Q8—1371no—
279C2×D4—2742no—
280C2×D4—2741no—
281C2×D4—2741no—
282C2×D4—2741no—
283C2×D4—5481no—
284C2×D4—1371no—
285C2×D4—1371no—
286C2×D4D54811yescharacteristic
287C22×C4—2741no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4C22×D13711yescharacteristic
292C22×C4C22×D13711yescharacteristic
293Q16—2742no—
294Q16—2742no—
295Q16—5481no—
296SD16—2742no—
297SD16—2742no—
298SD16—2742no—
299SD16—2742no—
300SD16—5481no—
301SD16—1374no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D13711yescharacteristic
305C2×C8C22×D13711yescharacteristic
306C2×C8C22×D13711yescharacteristic
307C2×C8C22×D13711yescharacteristic
308C4⋊C4—2741no—
309C4⋊C4—2741no—
310C4⋊C4—2741no—
311C4⋊C4—2741no—
312C4⋊C4—2741no—
313C4⋊C4—2741no—
314C4⋊C4—1371no—
315C4⋊C4—1371no—
316C4⋊C4D54811yescharacteristic
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4D54811yescharacteristic
325C22⋊C4D54811yescharacteristic
326C42—2741no—
327C42C22×D13711yescharacteristic
328C23—2741no—
329C23—2741no—
330C23—5482no—
331C23—5482no—
332C23—10961no—
333C23C2×D54811yescharacteristic
334C23C2×D54811yescharacteristic
335Q8—2741no—
336Q8—2741no—
337Q8—2741no—
338Q8—2741no—
339Q8—2741no—
340Q8—2741no—
341Q8—2741no—
342Q8—5482no—
343Q8—5482no—
344Q8—1372no—
345D4—22no—
346D4—2741no—
347D4—2741no—
348D4—2741no—
349D4—5482no—
350D4—5482no—
351D4—10961no—
352D4—1372no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—2741no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—2741no—
361C2×C4—2741no—
362C2×C4—2741no—
363C2×C4—5482no—
364C2×C4—5482no—
365C2×C4—5482no—
366C2×C4—10961no—
367C2×C4C2×D54811yescharacteristic
368C2×C4C2×D54811yescharacteristic
369C2×C4C2×D54811yescharacteristic
370C2×C4C2×D54811yescharacteristic
371C2×C4C23×D13711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D13712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—2742no—
382C22—5481no—
383C22—5481no—
384C22—5481no—
385C22—5481no—
386C22—5481no—
387C22—10964no—
388C22#1798412112yescomplemented
389C22C22×D54811yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—2742no—
395C4—5481no—
396C4—5481no—
397C4—5481no—
398C4—5481no—
399C4—5481no—
400C4C2×D4×D13711yescharacteristic
401C4#1798413411yescharacteristic
402C2—22no—
403C2—41no—
404C2—5482no—
405C2—10961no—
406C2D4×D54811yescharacteristic
407C2C2×#1798412111yescharacteristic
408C2#2257821511yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#435×C137C2
socleC22×C137C22×D4
prime radical#22578049C2
hypercenter#435D137
hypocenterC137#3266
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#17984121C2semidirect
2. C22×#14555499C2semidirect
3. D4×Dic274C2semidirect
5. #22577797C2semidirect
6. #22577649C2semidirect
7. #22577644C2semidirect
8. #22577626C2semidirect
9. #22577609C2semidirect
10. #22577567C2semidirect
11. #22577561C2semidirect
12. #22577556C2semidirect
13. #22577544C2semidirect
14. #22577505C2semidirect
15. #435×C137C2semidirect
18. C22×Dic274C22semidirect
21. C2×Dic548C22semidirect
22. C2×Dic548C22semidirect
23. C2×Dic548C22semidirect
31. #17984088C22semidirect
37. #17984067C22semidirect
43. C4×Dic274C22semidirect
47. #17984022C22semidirect
48. #17984013C22semidirect
49. #17984013C22semidirect
50. #17984013C22semidirect
51. #17984013C22semidirect
54. C2×M16×C137C22semidirect
57. C4⋊C8×C137C22semidirect
59. C22⋊C8×C137C22semidirect
60. #97×C137C22semidirect
93. #14555499D4semidirect
176. C22×C137#137semidirect
204. C137#3266semidirect
220. #435D137semidirect
236. C2×M16C2×D137semidirect
256. C4⋊C8C2×D137semidirect
264. C22⋊C8C2×D137semidirect
265. #97C2×D137semidirect
388. C22#17984121semidirect

Maximal subgroups

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

subgroupindex
C2×#179841212
C22×#145554992
D4×Dic2742
#225780492
#225777972
#225776492
#225776442
#225776262
#225776092
#225775672
#225775612
#225775562
#225775442
#225775052
#435×C1372
#3266137