Finite Group Databasebeta

← #31792678

Subgroups — #31792678

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578093C211yesmaximal, characteristic, complemented
2#22578062C211yesmaximal, characteristic
3#22577758C211yesmaximal, characteristic, complemented
4#22577745C211yesmaximal, characteristic, complemented
5#22577717C211yesmaximal, characteristic, complemented
6#22577711C211yesmaximal, characteristic, complemented
7#22577644C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577626C211yesmaximal, characteristic, complemented
10#22577567C211yesmaximal, characteristic, complemented
11#22577561C211yesmaximal, characteristic, complemented
12C4×Dic548C211yesmaximal, characteristic, complemented
13#22577400C211yesmaximal, characteristic, complemented
14C2×#17984013C211yesmaximal, characteristic, complemented
15C8×D4×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×Dic274C2211yescharacteristic
18C2×Dic548C2211yescharacteristic
19C2×Dic548C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21#17984081—21no—
22#17984080C2211yescharacteristic
23#17984078—21no—
24#17984076C2211yescharacteristic
25#17984074—21no—
26#17984069—21no—
27#17984067C2211yescharacteristic
28#17984067C2211yescharacteristic
29#17984067C2211yescharacteristic
30#17984066—21no—
31#17984062C2211yescharacteristic
32#17984060—21no—
33C4×Dic274C2211yescharacteristic
34C4×Dic274C2211yescharacteristic
35#17984048—21no—
36#17984047C2211yescharacteristic
37#17984044—21no—
38C2×C4⋊Dic137C2211yescharacteristic
39#17984022C2211yescharacteristic
40#17984022C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984013C2211yescharacteristic
46#17984013C2211yescharacteristic
47#17984013C2211yescharacteristic
48#17984013C2211yescharacteristic
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×Dic274C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C22×Dic137—21no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73Dic548—22no—
74Dic548—41no—
75#14555499—41no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95C4×Dic137—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—
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—21no—
117Dic274—42no—
118Dic274#4112yes—
119D548—21no—
120D548—21no—
121C22⋊D137—42no—
122C22⋊D137—42no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—42no—
131C2×Dic137—42no—
132C4×D137—42no—
133C23×C137C2×D411yescharacteristic
134C23×C137C2×D411yescharacteristic
135D4×C137—22no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2×D411yescharacteristic
143C2×C4×C137C2×D411yescharacteristic
144C2×C4×C137C2411yescharacteristic
145C8×C137—21no—
146C8×C137—21no—
147C8×C137—21no—
148C8×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137#13512yes—
155C22×C137C22×D411yescharacteristic
156C2×D137—41no—
157C2×D137—41no—
158C2×D137—41no—
159C4×C137—21no—
160C4×C137—21no—
161C4×C137—21no—
162C4×C137—21no—
163C4×C137C2×#4111yescharacteristic
164C4×C137H(5,𝔽2)11yescharacteristic
165Dic137—22no—
166Dic137—41no—
167Dic137—41no—
168Dic137—41no—
169Dic137—41no—
170Dic137—41no—
171Dic137—41no—
172C2×C137—22no—
173C2×C137—41no—
174C2×C137#54611yescharacteristic
175C2×C137C2×#13511yescharacteristic
176C2×C137#57811yescharacteristic
177D137—81no—
178C137#330011yescharacteristic, complemented, Sylow
179#3300—1371nomaximal, Sylow
180#552—1371no—
181#548—1371no—
182#488—1371no—
183#485—1371no—
184#480—1371no—
185#479—1371no—
186#467—1371no—
187#465—1371no—
188#464—1371no—
189#452—1371no—
190#451—1371no—
191C4×Q16—1371no—
192C8×D4D13711yescharacteristic, complemented
193#416—1371no—
194C2×#103—1371no—
195C2×#41—1371no—
196C22×Q8—1371no—
197C2×Q16—1371no—
198C2×Q16—1371no—
199C2×SD16—1371no—
200C22×C8C2×D13711yescharacteristic
201C22×C8C2×D13711yescharacteristic
202#126—2741no—
203#125—2741no—
204#125—1371no—
205#124—2741no—
206#124—1371no—
207#123—2741no—
208C22⋊Q8—2741no—
209C22⋊Q8—1371no—
210C22⋊Q8—1371no—
211C22⋊Q8—1371no—
212#121—2741no—
213#121—1371no—
214C4×Q8—1371no—
215C4×Q8—1371no—
216C4×D4—2741no—
217C4×D4C2×D13711yescharacteristic
218#117—2741no—
219#117—1371no—
220C2×C4⋊C4—1371no—
221#107—1371no—
222#107—1371no—
223#106—1371no—
224C4⋊C8C2×D13711yescharacteristic
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231#103—1371no—
232#102—1371no—
233#102—1371no—
234#102—1371no—
235C22⋊C8C2×D13711yescharacteristic
236C22⋊C8C2×D13711yescharacteristic
237C4×C8C2×D13711yescharacteristic
238#41—2742no—
239#41—5481no—
240C2×Q8—2742no—
241C2×Q8—2741no—
242C2×Q8—5481no—
243C2×Q8—1371no—
244C2×Q8—1371no—
245C2×Q8—1371no—
246C2×D4—2741no—
247C2×D4—2741no—
248C2×D4—1371no—
249C2×D4D54811yescharacteristic
250C22×C4—2741no—
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4C22×D13711yescharacteristic
254C22×C4C22×D13711yescharacteristic
255Q16—2742no—
256Q16—5481no—
257SD16—5481no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D13711yescharacteristic
261C2×C8C22×D13711yescharacteristic
262C2×C8C22×D13711yescharacteristic
263C2×C8C22×D13711yescharacteristic
264C4⋊C4—2741no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—1371no—
273C4⋊C4—1371no—
274C4⋊C4—1371no—
275C4⋊C4—1371no—
276C4⋊C4D54811yescharacteristic
277C22⋊C4—2741no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4D54811yescharacteristic
285C22⋊C4D54811yescharacteristic
286C42—2741no—
287C42—2741no—
288C42—2741no—
289C42C22×D13711yescharacteristic
290C23—2741no—
291C23C2×D54811yescharacteristic
292C23C2×D54811yescharacteristic
293Q8—2741no—
294Q8—2741no—
295Q8—2741no—
296Q8—2741no—
297Q8—2741no—
298Q8—5482no—
299Q8—1372no—
300D4—22no—
301D4—2741no—
302D4—2741no—
303D4—5482no—
304D4—5482no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—5482no—
317C2×C4—5482no—
318C2×C4—5482no—
319C2×C4C2×D54811yescharacteristic
320C2×C4C2×D54811yescharacteristic
321C2×C4C2×D54811yescharacteristic
322C2×C4C2×D54811yescharacteristic
323C2×C4C23×D13711yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—5481no—
334C22—5481no—
335C22—5481no—
336C22#1798411012yes—
337C22C22×D54811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—2742no—
343C4—5481no—
344C4—5481no—
345C4—5481no—
346C4—5481no—
347C4—5481no—
348C4—5481no—
349C4#41×D13711yescharacteristic
350C4#1798413311yescharacteristic
351C2—22no—
352C2—41no—
353C2—10961no—
354C2#2257804811yescharacteristic
355C2C2×#1798411011yescharacteristic
356C2#2257822011yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC8×D4×C137C2
socleC22×C137C22×D4
prime radical#22578062C2
hypercenterC8×D4D137
hypocenterC137#3300
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. #22578093C2semidirect
3. #22577758C2semidirect
4. #22577745C2semidirect
5. #22577717C2semidirect
6. #22577711C2semidirect
7. #22577644C2semidirect
8. #22577632C2semidirect
9. #22577626C2semidirect
10. #22577567C2semidirect
11. #22577561C2semidirect
12. C4×Dic548C2semidirect
13. #22577400C2semidirect
14. C2×#17984013C2semidirect
15. C8×D4×C137C2semidirect
178. C137#3300semidirect
192. C8×D4D137semidirect

Maximal subgroups

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

subgroupindex
#225780932
#225780622
#225777582
#225777452
#225777172
#225777112
#225776442
#225776322
#225776262
#225775672
#225775612
C4×Dic5482
#225774002
C2×#179840132
C8×D4×C1372
#3300137