Finite Group Databasebeta

← #31792316

Subgroups — #31792316

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×#17984107C211yesmaximal, characteristic, complemented
3#22578057C211yesmaximal, characteristic, complemented
4#22578048C211yesmaximal, characteristic, complemented
5#22577797C211yesmaximal, characteristic, complemented
6#22577662C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577618C211yesmaximal, characteristic, complemented
9#22577608C211yesmaximal, characteristic, complemented
10#22577566C211yesmaximal, characteristic, complemented
11#22577566C211yesmaximal, characteristic, complemented
12#22577555C211yesmaximal, characteristic, complemented
13#22577549C211yesmaximal, characteristic, complemented
14#22577476C211yesmaximal, characteristic, complemented
15#442×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×#14555509C2211yescharacteristic
18C2×#14555509C2211yescharacteristic, complemented
19C2×D4×D137C2211yescharacteristic
20#17984112—24no—
21#17984107—24no—
22C2×Dic548C2211yescharacteristic, complemented
23C2×Dic548C2211yescharacteristic, complemented
24C2×#14555499C2211yescharacteristic
25C2×#14555499C2211yescharacteristic
26C2×#14555498C2211yescharacteristic
27C2×#14555498C2211yescharacteristic
28C2×#14555498C2211yescharacteristic, complemented
29C2×#14555495C2211yescharacteristic
30C2×#14555493C2211yescharacteristic
31C2×C8×D137C2211yescharacteristic, complemented
32#17984088C2211yescharacteristic, complemented
33#17984076C2211yescharacteristic
34#17984072—21no—
35#17984072—21no—
36#17984072—21no—
37#17984068C2211yescharacteristic
38#17984068C2211yescharacteristic
39#17984068C2211yescharacteristic, complemented
40#17984063C2211yescharacteristic
41#17984062—21no—
42#17984062—21no—
43#17984058—21no—
44C4×D548C2211yescharacteristic
45C2×C4⋊Dic137—21no—
46C2×#14555487—21no—
47C4⋊#11764040C2211yescharacteristic, complemented
48#17984011C2211yescharacteristic, complemented
49#17984011C2211yescharacteristic, complemented
50#17984011C2211yescharacteristic, complemented
51#17984008C2211yescharacteristic
52#17984000C2211yescharacteristic
53#17984000C2211yescharacteristic, complemented
54C2×D8×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic
56C4×D4×C137C2211yescharacteristic, complemented
57#106×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic
59#102×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555509—22no—
63#14555509—22no—
64#14555509—22no—
65#14555509—41no—
66#14555509—41no—
67#14555509—41no—
68C2×Dic274C2311yescharacteristic
69C2×Dic274C2311yescharacteristic
70C2×Dic274C2311yescharacteristic
71C2×D548—21no—
72C2×D548D411yescharacteristic
73C2×D548C2311yescharacteristic
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76C2×C22⋊D137—21no—
77C2×C22⋊D137—21no—
78D4×D137—22no—
79D4×D137—41no—
80C22×Dic137—21no—
81C22×Dic137—21no—
82C22×Dic137—21no—
83C2×C4×D137C2311yescharacteristic
84C2×C4×D137C2311yescharacteristic
85Dic548—22no—
86Dic548—41no—
87#14555499—22no—
88#14555499—41no—
89#14555498—22no—
90#14555498—22no—
91#14555498—22no—
92#14555495—22no—
93#14555493—22no—
94C2×#11764040C2311yescharacteristic
95C2×#11764040C2311yescharacteristic
96C8×D137—22no—
97C4⋊Dic137—21no—
98C4⋊Dic137—21no—
99C4⋊Dic137—21no—
100C4⋊Dic137—21no—
101C4⋊Dic137—21no—
102C4⋊Dic137D411yescharacteristic
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487D411yescharacteristic
110#14555487D411yescharacteristic
111C2×D4×C137C2311yescharacteristic
112C2×D4×C137C2311yescharacteristic
113C22×C4×C137—21no—
114C22×C4×C137—21no—
115D8×C137D414yescomplemented
116C2×C8×C137C2311yescharacteristic
117C2×C8×C137C2311yescharacteristic
118C4⋊C4×C137C2311yescharacteristic
119C4⋊C4×C137C2311yescharacteristic
120C22⋊C4×C137—21no—
121C22⋊C4×C137—21no—
122C42×C137C2311yescharacteristic
123C22×D137—21no—
124C22×D137—42no—
125C22×D137—42no—
126C22×D137—81no—
127C22×D137C2×D411yescharacteristic
128C22×D137C2×D411yescharacteristic
129Dic274—21no—
130Dic274—21no—
131Dic274—21no—
132Dic274—21no—
133Dic274—21no—
134Dic274—21no—
135D548—21no—
136D548—21no—
137D548—41no—
138D548—81no—
139C22⋊D137—42no—
140C22⋊D137—42no—
141C22⋊D137—42no—
142C22⋊D137—42no—
143C2×Dic137—21no—
144C2×Dic137—21no—
145C2×Dic137—21no—
146C2×Dic137—42no—
147C2×Dic137—42no—
148C2×Dic137—42no—
149C2×Dic137C2×D411yescharacteristic
150C2×Dic137C2×D411yescharacteristic
151C4×D137—21no—
152C4×D137—21no—
153C4×D137—21no—
154C4×D137—21no—
155#11764040—21no—
156#11764040—21no—
157C23×C137—21no—
158C23×C137—21no—
159D4×C137—21no—
160D4×C137—21no—
161D4×C137C2×D412yes—
162D4×C137C2×D412yes—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—42no—
166C2×C4×C137—81no—
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2×D411yescharacteristic
169C2×C4×C137C2411yescharacteristic
170C8×C137—21no—
171C8×C137C2×D412yes—
172C22×C137—22no—
173C22×C137—22no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137C22×D411yescharacteristic
179C2×D137—21no—
180C2×D137—21no—
181C2×D137—21no—
182C2×D137—21no—
183C2×D137—21no—
184C2×D137—21no—
185C2×D137—41no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—84no—
189C4×C137—21no—
190C4×C137—21no—
191C4×C137—41no—
192C4×C137—41no—
193C4×C137C22×D411yescharacteristic
194C4×C137H(5,𝔽2)11yescharacteristic
195Dic137—21no—
196Dic137—21no—
197Dic137—41no—
198Dic137—41no—
199Dic137—41no—
200C2×C137—42no—
201C2×C137—42no—
202C2×C137D4211yescharacteristic
203C2×C137#57511yescharacteristic
204C2×C137#57711yescharacteristic
205D137—41no—
206D137—41no—
207D137—81no—
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#449—1371no—
223#442D13711yescharacteristic, complemented
224#435—1371no—
225C2×#41—1371no—
226C2×#41—1371no—
227C2×#41—1371no—
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×SD16—1371no—
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#121—1371no—
253#120—2741no—
254C4×D4—1371no—
255C4×D4C2×D13711yescharacteristic
256C4×D4C2×D13711yescharacteristic, complemented
257C2×C4⋊C4—2741no—
258C2×C22⋊C4—2741no—
259#106C2×D13711yescharacteristic, complemented
260C4⋊C8—1371no—
261#103—1371no—
262#103—1371no—
263#103—1371no—
264#102—1371no—
265#102C2×D13711yescharacteristic
266#102C2×D13711yescharacteristic, complemented
267C22⋊C8—1371no—
268C22⋊C8—1371no—
269#97C2×D13711yescharacteristic, complemented
270C24—2741no—
271#41—2742no—
272#41—2742no—
273#41—2742no—
274#41—5481no—
275#41—5481no—
276#41—5481no—
277C2×Q8—1371no—
278C2×Q8—1371no—
279C2×Q8—1371no—
280C2×D4—2741no—
281C2×D4—2742no—
282C2×D4—2741no—
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—21no—
293C22×C4—2741no—
294C22×C4—2741no—
295C22×C4—2741no—
296C22×C4—1371no—
297C22×C4—1371no—
298Q16—2742no—
299Q16—5481no—
300SD16—2742no—
301SD16—2742no—
302SD16—2742no—
303SD16—2742no—
304SD16—5481no—
305D8—2742no—
306D8D54814yescomplemented
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⋊C4C22×D13711yescharacteristic
320C4⋊C4C22×D13711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C22⋊C4—2741no—
326C22⋊C4—2741no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C22⋊C4—1371no—
330C22⋊C4—1371no—
331C42C22×D13711yescharacteristic
332C23—21no—
333C23—21no—
334C23—2741no—
335C23—5482no—
336C23—5482no—
337C23—10961no—
338C23—1371no—
339C23—1371no—
340Q8—2741no—
341Q8—2741no—
342Q8—2741no—
343Q8—2741no—
344Q8—2741no—
345Q8—2741no—
346D4—21no—
347D4—21no—
348D4—2741no—
349D4—2741no—
350D4—5481no—
351D4—5482no—
352D4—5482no—
353D4—5482no—
354D4—5482no—
355D4—10961no—
356D4C2×D54812yes—
357D4C2×D54812yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—2741no—
363C2×C4—2741no—
364C2×C4—2741no—
365C2×C4—2741no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—2741no—
369C2×C4—5482no—
370C2×C4—5482no—
371C2×C4—5482no—
372C2×C4—1371no—
373C2×C4—1371no—
374C2×C4D4×D13711yescharacteristic
375C2×C4D4×D13711yescharacteristic
376C2×C4C23×D13711yescharacteristic
377C8—21no—
378C8—2741no—
379C8—2741no—
380C8C2×D54812yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—2741no—
388C22—2741no—
389C22—2741no—
390C22—2741no—
391C22—2741no—
392C22—2741no—
393C22—5481no—
394C22—5481no—
395C22—5481no—
396C22—10964no—
397C22C2×D4×D13711yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—2741no—
403C4—2741no—
404C4—5481no—
405C4—5481no—
406C4—5481no—
407C4C22×D54811yescharacteristic
408C4#1798413411yescharacteristic
409C2—42no—
410C2—42no—
411C2—5481no—
412C2—5481no—
413C2—10961no—
414C2D4×D54811yescharacteristic
415C2#2257819911yescharacteristic
416C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#442×C137C2
socleC22×C137C22×D4
hypercenter#442D137
hypocenterC137#3267
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×#17984107C2semidirect
3. #22578057C2semidirect
4. #22578048C2semidirect
5. #22577797C2semidirect
6. #22577662C2semidirect
7. #22577633C2semidirect
8. #22577618C2semidirect
9. #22577608C2semidirect
10. #22577566C2semidirect
11. #22577566C2semidirect
12. #22577555C2semidirect
13. #22577549C2semidirect
14. #22577476C2semidirect
15. #442×C137C2semidirect
18. C2×#14555509C22semidirect
22. C2×Dic548C22semidirect
23. C2×Dic548C22semidirect
28. C2×#14555498C22semidirect
31. C2×C8×D137C22semidirect
32. #17984088C22semidirect
39. #17984068C22semidirect
47. C4⋊#11764040C22semidirect
48. #17984011C22semidirect
49. #17984011C22semidirect
50. #17984011C22semidirect
53. #17984000C22semidirect
56. C4×D4×C137C22semidirect
57. #106×C137C22semidirect
59. #102×C137C22semidirect
60. #97×C137C22semidirect
115. D8×C137D4semidirect
208. C137#3267semidirect
223. #442D137semidirect
256. C4×D4C2×D137semidirect
259. #106C2×D137semidirect
266. #102C2×D137semidirect
269. #97C2×D137semidirect
306. D8D548semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×#179841072
#225780572
#225780482
#225777972
#225776622
#225776332
#225776182
#225776082
#225775662
#225775552
#225775492
#225774762
#442×C1372
#3267137