Finite Group Databasebeta

← #31792693

Subgroups — #31792693

Order 17536.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578099C211yesmaximal, characteristic, complemented
2#22578062C211yesmaximal, characteristic, complemented
3#22577758C211yesmaximal, characteristic, complemented
4#22577745C211yesmaximal, characteristic, complemented
5#22577722C211yesmaximal, characteristic, complemented
6#22577711C211yesmaximal, characteristic, complemented
7#22577639C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577619C211yesmaximal, characteristic, complemented
10#22577550C211yesmaximal, characteristic, complemented
11#22577544C211yesmaximal, characteristic, complemented
12C4×#14555499C211yesmaximal, characteristic, complemented
13#22577400C211yesmaximal, characteristic, complemented
14C2×#17984013C211yesmaximal, characteristic, complemented
15C8×D4×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×Dic274C2211yescharacteristic
18C2×#14555499C2211yescharacteristic
19C2×#14555499C2211yescharacteristic
20C2×D1096C2211yescharacteristic
21#17984083—21no—
22#17984080C2211yescharacteristic
23#17984076C2211yescharacteristic
24#17984074—21no—
25#17984070—21no—
26#17984067C2211yescharacteristic
27#17984066—21no—
28#17984062C2211yescharacteristic
29#17984062C2211yescharacteristic
30#17984062C2211yescharacteristic
31#17984060—21no—
32#17984060—21no—
33C4×Dic274C2211yescharacteristic
34C4×D548C2211yescharacteristic
35#17984048—21no—
36#17984048—21no—
37#17984047C2211yescharacteristic
38C2×C4⋊Dic137C2211yescharacteristic
39#17984022C2211yescharacteristic
40#17984020C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984013C2211yescharacteristic
46#17984013C2211yescharacteristic
47#17984013C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic
52C22×C8×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C4×D4×C137C2211yescharacteristic
55C4⋊C8×C137C2211yescharacteristic
56C22⋊C8×C137C2211yescharacteristic
57C22⋊C8×C137C2211yescharacteristic
58C4×C8×C137C2211yescharacteristic
59#14555510—22no—
60#14555510—41no—
61C2×Dic274—22no—
62C2×Dic274—21no—
63C2×Dic274—41no—
64C2×Dic274C2311yescharacteristic
65C2×Dic274C2311yescharacteristic
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C22×Dic137—21no—
73C22×Dic137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76#14555499—22no—
77#14555499—41no—
78D1096—41no—
79C4⋊Dic137—21no—
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96C4×Dic137—21no—
97C4×Dic137—21no—
98C2×D4×C137D411yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C2×C8×C137—41no—
102C2×C8×C137—41no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137D411yescharacteristic
108C22⋊C4×C137D411yescharacteristic
109C22⋊C4×C137D411yescharacteristic
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112C22×D137—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117Dic274#4112yes—
118D548—21no—
119D548—21no—
120D548—21no—
121D548—21no—
122C22⋊D137—42no—
123C22⋊D137—42no—
124C22⋊D137—42no—
125C22⋊D137—81no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—21no—
131C2×Dic137—21no—
132C2×Dic137—42no—
133C2×Dic137—81no—
134C4×D137—42no—
135C4×D137—81no—
136C23×C137C2×D411yescharacteristic
137C23×C137C2×D411yescharacteristic
138D4×C137—22no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137C2×D411yescharacteristic
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C8×C137—21no—
149C8×C137—21no—
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137#13512yes—
158C22×C137C22×D411yescharacteristic
159C2×D137—41no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C4×C137—21no—
166C4×C137—21no—
167C4×C137—21no—
168C4×C137—21no—
169C4×C137C2×#4111yescharacteristic
170C4×C137H(5,𝔽2)11yescharacteristic
171Dic137—22no—
172Dic137—41no—
173Dic137—41no—
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177C2×C137—22no—
178C2×C137—41no—
179C2×C137#54611yescharacteristic
180C2×C137C2×#13511yescharacteristic
181C2×C137#57711yescharacteristic
182D137—81no—
183D137—81no—
184C137#330111yescharacteristic, complemented, Sylow
185#3301—1371nomaximal, Sylow
186#553—1371no—
187#548—1371no—
188#488—1371no—
189#485—1371no—
190#481—1371no—
191#479—1371no—
192#466—1371no—
193#465—1371no—
194#463—1371no—
195#449—1371no—
196#448—1371no—
197C4×SD16—1371no—
198C8×D4D13711yescharacteristic, complemented
199#416—1371no—
200C2×#103—1371no—
201C2×#41—1371no—
202C22×Q8—1371no—
203C2×SD16—1371no—
204C2×SD16—1371no—
205C2×D8—1371no—
206C22×C8C2×D13711yescharacteristic
207C22×C8C2×D13711yescharacteristic
208#126—2741no—
209#125—1371no—
210#124—2741no—
211#124—1371no—
212#123—2741no—
213C22⋊Q8—2741no—
214C22⋊Q8—1371no—
215#121—2741no—
216#121—2741no—
217#121—1371no—
218#121—1371no—
219#121—1371no—
220C4×Q8—1371no—
221C4×D4—2741no—
222C4×D4—2741no—
223C4×D4—1371no—
224C4×D4C2×D13711yescharacteristic
225#117—1371no—
226C2×C4⋊C4—1371no—
227#107—1371no—
228#106—1371no—
229#106—1371no—
230C4⋊C8C2×D13711yescharacteristic
231#103—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#103—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241C22⋊C8C2×D13711yescharacteristic
242C22⋊C8C2×D13711yescharacteristic
243C4×C8C2×D13711yescharacteristic
244#41—2742no—
245#41—5481no—
246C2×Q8—2742no—
247C2×Q8—2741no—
248C2×Q8—5481no—
249C2×Q8—1371no—
250C2×Q8—1371no—
251C2×D4—2741no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4—1371no—
257C2×D4D54811yescharacteristic
258C22×C4—2741no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4C22×D13711yescharacteristic
263C22×C4C22×D13711yescharacteristic
264SD16—2742no—
265SD16—5481no—
266D8—5481no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8C22×D13711yescharacteristic
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C2×C8C22×D13711yescharacteristic
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4—1371no—
282C4⋊C4D54811yescharacteristic
283C22⋊C4—2741no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4D54811yescharacteristic
292C22⋊C4D54811yescharacteristic
293C42—2741no—
294C42—2741no—
295C42C22×D13711yescharacteristic
296C23—2741no—
297C23—2741no—
298C23C2×D54811yescharacteristic
299C23C2×D54811yescharacteristic
300Q8—2741no—
301Q8—2741no—
302Q8—2741no—
303Q8—5482no—
304Q8—1372no—
305D4—22no—
306D4—2741no—
307D4—2741no—
308D4—2741no—
309D4—2741no—
310D4—5482no—
311D4—5482no—
312D4—5482no—
313D4—10961no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—2741no—
319C2×C4—2741no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—5482no—
325C2×C4—5482no—
326C2×C4—10961no—
327C2×C4—10961no—
328C2×C4C2×D54811yescharacteristic
329C2×C4C2×D54811yescharacteristic
330C2×C4C2×D54811yescharacteristic
331C2×C4C2×D54811yescharacteristic
332C2×C4C23×D13711yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—5481no—
343C22—5481no—
344C22—5481no—
345C22—5481no—
346C22—5481no—
347C22—5481no—
348C22#1798411012yes—
349C22C22×D54811yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—2742no—
355C4—5481no—
356C4—5481no—
357C4—5481no—
358C4—5481no—
359C4—5481no—
360C4#41×D13711yescharacteristic
361C4#1798413311yescharacteristic
362C2—22no—
363C2—41no—
364C2—10961no—
365C2—10961no—
366C2#2257804811yescharacteristic
367C2C2×#1798411011yescharacteristic
368C2#2257821511yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC8×D4×C137C2
socleC22×C137C22×D4
hypercenterC8×D4D137
hypocenterC137#3301
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22578099C2semidirect
2. #22578062C2semidirect
3. #22577758C2semidirect
4. #22577745C2semidirect
5. #22577722C2semidirect
6. #22577711C2semidirect
7. #22577639C2semidirect
8. #22577632C2semidirect
9. #22577619C2semidirect
10. #22577550C2semidirect
11. #22577544C2semidirect
12. C4×#14555499C2semidirect
13. #22577400C2semidirect
14. C2×#17984013C2semidirect
15. C8×D4×C137C2semidirect
184. C137#3301semidirect
198. C8×D4D137semidirect

Maximal subgroups

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

subgroupindex
#225780992
#225780622
#225777582
#225777452
#225777222
#225777112
#225776392
#225776322
#225776192
#225775502
#225775442
C4×#145554992
#225774002
C2×#179840132
C8×D4×C1372
#3301137