Finite Group Databasebeta

← #31789237

Subgroups — #31789237

Order 17536.

774 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D137C211yesmaximal, characteristic
2C2×#17984096C211yesmaximal, characteristic, complemented
3C2×#17984094C211yesmaximal, characteristic, complemented
4#22577370C211yesmaximal, characteristic, complemented
5#22577368C212yesmaximal, complemented
6#22577368C212yesmaximal, complemented
7#22577366C211yesmaximal, characteristic, complemented
8#22577365C212yesmaximal, complemented
9C2×#17984000C211yesmaximal, characteristic, complemented
10C22⋊C8×D137C212yesmaximal, complemented
11#408×C137C211yesmaximal, characteristic, complemented
12C2×#14555511C411yescharacteristic
13C2×#14555510C2212yes—
14C2×#14555510C2211yescharacteristic
15C2×#14555509C411yescharacteristic
16C2×#14555509C412yes—
17#41×D137—24no—
18#41×D137—24no—
19C2×Q8×D137C411yescharacteristic
20C2×D4×D137C411yescharacteristic
21C2×D4×D137C412yes—
22C22×C4×D137C2212yes—
23C22×C4×D137C2211yescharacteristic
24#17984096—24no—
25#17984094—24no—
26C2×#14555494C2211yescharacteristic, complemented
27C2×#14555493C2212yescomplemented
28C2×#14555493C2212yescomplemented
29C22×#11764040C2211yescharacteristic, complemented
30C2×C8×D137C2212yescomplemented
31C2×C8×D137C2212yescomplemented
32C22⋊#11764040C2212yescomplemented
33C22⋊#11764040C2212yescomplemented
34#17984000C2211yescharacteristic
35#17984000C2212yes—
36#17984000C2211yescharacteristic
37#17984000C2211yescharacteristic, complemented
38#17984000C2211yescharacteristic, complemented
39#17984000C2212yescomplemented
40C2×#41×C137C2211yescharacteristic
41C2×M16×C137C2211yescharacteristic, complemented
42C22×C8×C137C2211yescharacteristic, complemented
43C22⋊C8×C137C2212yescomplemented
44C22⋊C8×C137C2212yescomplemented
45C23×D137C2×C412yes—
46C23×D137C2×C411yescharacteristic
47#14555511—24no—
48#14555510—24no—
49#14555510—24no—
50#14555510D414yes—
51#14555510D414yes—
52#14555509—24no—
53#14555509—28no—
54C2×Dic274C2×C412yes—
55C2×Dic274C2×C411yescharacteristic
56Q8×D137—24no—
57C2×D548C2×C412yes—
58C2×D548C2×C411yescharacteristic
59C2×C22⋊D137C2×C412yes—
60C2×C22⋊D137C2×C412yes—
61C2×C22⋊D137C2×C412yes—
62D4×D137—24no—
63D4×D137—28no—
64C22×Dic137C2×C412yes—
65C22×Dic137C2×C411yescharacteristic
66C2×C4×D137—24no—
67C2×C4×D137—24no—
68C2×C4×D137—22no—
69C2×C4×D137—22no—
70C2×C4×D137C2×C412yes—
71C2×C4×D137C2×C411yescharacteristic
72C2×C4×D137C2×C412yes—
73C2×C4×D137C2×C411yescharacteristic
74C2×C4×D137C2311yescharacteristic
75C2×C4×D137C2311yescharacteristic
76C2×C4×D137C2312yes—
77#14555494—22no—
78#14555493—24no—
79#14555493—24no—
80C2×#11764040—22no—
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2312yes—
83C2×#11764040C2311yescharacteristic
84C8×D137—24no—
85C8×D137—24no—
86#41×C137—22no—
87#41×C137—22no—
88C2×Q8×C137C2×C411yescharacteristic
89C2×D4×C137C2×C412yes—
90C2×D4×C137C2×C411yescharacteristic
91C22×C4×C137C2312yes—
92C22×C4×C137C2311yescharacteristic
93M16×C137—22no—
94C2×C8×C137—22no—
95C2×C8×C137C2311yescharacteristic
96C2×C8×C137C2312yes—
97C2×C8×C137C2311yescharacteristic
98C22×D137—22no—
99C22×D137—24no—
100C22×D137—24no—
101C22×D137—22no—
102C22×D137—42no—
103C22×D137—41no—
104C22×D137C22×C411yescharacteristic
105C22×D137C22×C412yes—
106C22×D137C22×C411yescharacteristic
107Dic274—24no—
108Dic274C22⋊C414yes—
109D548—24no—
110D548C22⋊C414yes—
111C22⋊D137—24no—
112C22⋊D137—24no—
113C22⋊D137C22⋊C418yes—
114C2×Dic137—22no—
115C2×Dic137—24no—
116C2×Dic137C22×C411yescharacteristic
117C2×Dic137C22×C411yescharacteristic
118C2×Dic137C22×C412yes—
119C4×D137—24no—
120C4×D137—22no—
121C4×D137—22no—
122C4×D137—21no—
123C4×D137—21no—
124C4×D137—21no—
125C4×D137—21no—
126C4×D137—24no—
127C4×D137C2×D414yes—
128C4×D137C2×D414yes—
129#11764040—21no—
130#11764040—22no—
131#11764040—21no—
132C23×C137C22×C412yes—
133C23×C137C22×C411yescharacteristic
134Q8×C137—22no—
135D4×C137—24no—
136D4×C137—22no—
137C2×C4×C137—22no—
138C2×C4×C137—22no—
139C2×C4×C137C22×C412yes—
140C2×C4×C137C22×C411yescharacteristic
141C2×C4×C137C2×D412yes—
142C2×C4×C137C2×D412yes—
143C2×C4×C137C2411yescharacteristic
144C8×C137—21no—
145C8×C137—21no—
146C8×C137—22no—
147C22×C137—21no—
148C22×C137—21no—
149C22×C137—22no—
150C22×C137—22no—
151C22×C137—22no—
152C22×C137C2×C22⋊C412yes—
153C22×C137C23×C411yescharacteristic
154C2×D137—21no—
155C2×D137—22no—
156C2×D137—21no—
157C2×D137—22no—
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—21no—
161C2×D137—48no—
162C2×D137—44no—
163C2×D137C2×C22⋊C414yes—
164C2×D137#13112yescomplemented
165C4×C137—22no—
166C4×C137C2×C22⋊C412yes—
167C4×C137C22×D411yescharacteristic
168C4×C137C22×D411yescharacteristic
169Dic137—21no—
170Dic137—21no—
171Dic137C2×C22⋊C414yes—
172C2×C137—22no—
173C2×C137—42no—
174C2×C137C22×C22⋊C411yescharacteristic
175C2×C137C2×#13111yescharacteristic
176C2×C137#56811yescharacteristic
177D137—22no—
178D137—24no—
179D137—41no—
180C137#288011yescharacteristic, complemented, Sylow
181#2880—1371nomaximal, Sylow
182C22×#41—1371no—
183C2×#131—1371no—
184C2×#131—1371no—
185#408—1371no—
186#408—1372no—
187#408—1372no—
188#408D13711yescharacteristic, complemented
189#407—1371no—
190#407—1372no—
191C2×C22⋊C8—1371no—
192C2×C22⋊C8—1372no—
193C2×#41—2744no—
194C2×#41—2744no—
195C2×#41—1371no—
196C2×#41—1371no—
197C2×#41—1372no—
198C2×#41—1372no—
199C2×#41—1371no—
200C2×#41C2×D13711yescharacteristic
201C22×Q8—1371no—
202C22×D4—1371no—
203C22×D4—1372no—
204C23×C4—1372no—
205C23×C4—1371no—
206#131—2744no—
207#131—2744no—
208C2×M16—1371no—
209C2×M16—1372no—
210C2×M16—1372no—
211C2×M16C2×D13711yescharacteristic, complemented
212C22×C8—1371no—
213C22×C8—1372no—
214C22×C8—1372no—
215C22×C8C2×D13711yescharacteristic, complemented
216C22⋊C8—1372no—
217C22⋊C8—1372no—
218C22⋊C8—1371no—
219C22⋊C8—1372no—
220C22⋊C8—1371no—
221C22⋊C8—1371no—
222C22⋊C8—1371no—
223C22⋊C8—1372no—
224C22⋊C8C2×D13712yescomplemented
225C22⋊C8C2×D13712yescomplemented
226C24—1372no—
227C24—1371no—
228#41—22no—
229#41—22no—
230#41—2744no—
231#41—2744no—
232#41—2744no—
233#41—2748no—
234#41—2744no—
235#41—1374no—
236#41—1374no—
237C2×Q8—2744no—
238C2×Q8—1372no—
239C2×Q8—1371no—
240C2×Q8C4×D13711yescharacteristic
241C2×D4—2748no—
242C2×D4—2744no—
243C2×D4—1372no—
244C2×D4—1372no—
245C2×D4—1372no—
246C2×D4—1372no—
247C2×D4—1371no—
248C2×D4C4×D13712yes—
249C2×D4C4×D13711yescharacteristic
250C22×C4—2744no—
251C22×C4—2744no—
252C22×C4—2742no—
253C22×C4—2742no—
254C22×C4—1371no—
255C22×C4—1372no—
256C22×C4—1372no—
257C22×C4—1371no—
258C22×C4—1372no—
259C22×C4—1371no—
260C22×C4—1371no—
261C22×C4—1371no—
262C22×C4—1372no—
263C22×C4C22×D13712yes—
264C22×C4C22×D13711yescharacteristic
265M16—22no—
266M16—2744no—
267M16—2744no—
268M16—2742no—
269C2×C8—22no—
270C2×C8—2744no—
271C2×C8—2744no—
272C2×C8—2742no—
273C2×C8—1371no—
274C2×C8—1372no—
275C2×C8—1371no—
276C2×C8C22×D13711yescharacteristic
277C2×C8C22×D13712yes—
278C2×C8C22×D13711yescharacteristic
279C23—2744no—
280C23—2742no—
281C23—2744no—
282C23—2742no—
283C23—5481no—
284C23—5482no—
285C23—1371no—
286C23—1371no—
287C23—1372no—
288C23C2×C4×D13712yes—
289C23C2×C4×D13711yescharacteristic
290Q8—22no—
291Q8—2744no—
292Q8—1374no—
293D4—24no—
294D4—22no—
295D4—2744no—
296D4—2744no—
297D4—2744no—
298D4—1378no—
299D4—1374no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—2744no—
303C2×C4—2744no—
304C2×C4—2742no—
305C2×C4—2744no—
306C2×C4—2742no—
307C2×C4—2742no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4—1372no—
315C2×C4—1374no—
316C2×C4—1374no—
317C2×C4C2×C4×D13712yes—
318C2×C4C2×C4×D13711yescharacteristic
319C2×C4D4×D13712yes—
320C2×C4D4×D13712yes—
321C2×C4C23×D13711yescharacteristic
322C8—21no—
323C8—21no—
324C8—22no—
325C8—2741no—
326C8—2742no—
327C8—2741no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—2741no—
334C22—2742no—
335C22—2741no—
336C22—2742no—
337C22—2741no—
338C22—2741no—
339C22—2741no—
340C22—5484no—
341C22—5488no—
342C22—1374no—
343C22—1372no—
344C22C22⋊C4×D13712yes—
345C22C22×C4×D13711yescharacteristic
346C4—22no—
347C4—2741no—
348C4—2741no—
349C4—1374no—
350C4C22⋊C4×D13712yes—
351C4C2×D4×D13711yescharacteristic
352C4C2×D4×D13711yescharacteristic
353C2—22no—
354C2—42no—
355C2—2742no—
356C2—2744no—
357C2—5481no—
358C2C2×C22⋊C4×D13711yescharacteristic
359C2#131×D13711yescharacteristic
360C2#2257816811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D137
commutator, socleC22×C137C23×C4
Fitting#408×C137C2
prime radicalC2×#41×D137C2
cosocleC2×C4×C137C24
hypercenter#408D137
hypocenterC137#2880
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#17984096C2semidirect
3. C2×#17984094C2semidirect
4. #22577370C2semidirect
5. #22577368C2semidirect
6. #22577368C2semidirect
7. #22577366C2semidirect
8. #22577365C2semidirect
9. C2×#17984000C2semidirect
10. C22⋊C8×D137C2semidirect
11. #408×C137C2semidirect
26. C2×#14555494C22semidirect
27. C2×#14555493C22semidirect
28. C2×#14555493C22semidirect
29. C22×#11764040C22semidirect
30. C2×C8×D137C22semidirect
31. C2×C8×D137C22semidirect
32. C22⋊#11764040C22semidirect
33. C22⋊#11764040C22semidirect
37. #17984000C22semidirect
38. #17984000C22semidirect
39. #17984000C22semidirect
41. C2×M16×C137C22semidirect
42. C22×C8×C137C22semidirect
43. C22⋊C8×C137C22semidirect
44. C22⋊C8×C137C22semidirect
164. C2×D137#131semidirect
180. C137#2880semidirect
188. #408D137semidirect
211. C2×M16C2×D137semidirect
215. C22×C8C2×D137semidirect
224. C22⋊C8C2×D137semidirect
225. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1372
C2×#179840962
C2×#179840942
#225773702
#225773682
#225773662
#225773652
C2×#179840002
C22⋊C8×D1372
#408×C1372
#2880137