Finite Group Databasebeta

← #30669346

Subgroups — #30669346

Order 16768.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862229C211yesmaximal, characteristic, complemented
2C2×#16862215C211yesmaximal, characteristic
3C2×#16862205C211yesmaximal, characteristic, complemented
4#22553235C212yesmaximal, complemented
5#22553228C212yesmaximal, complemented
6#22553212C212yesmaximal, complemented
7#22553206C212yesmaximal, complemented
8#22553191C211yesmaximal, characteristic, complemented
9#22553185C211yesmaximal, characteristic, complemented
10#22553168C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13C22×C22⋊D131C2212yes—
14#16862229—24no—
15#16862215—24no—
16#16862205—24no—
17C2×C4⋊Dic131C2212yescomplemented
18C2×Dic131⋊C4C2211yescharacteristic, complemented
19C2×Dic131⋊C4C2211yescharacteristic, complemented
20C2×Dic131⋊C4C2212yescomplemented
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C22⋊Dic131C2212yes—
24C2×C22⋊Dic131C2212yes—
25C2×C22⋊Dic131C2212yes—
26C2×C22⋊Dic131C2211yescharacteristic
27C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
28C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D131)⋊C4C2212yescomplemented
30C2×C4×Dic131C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C131⋊#95C2212yescomplemented
33C25×C131C2211yescharacteristic
34C23×C4×C131C2211yescharacteristic
35C23×C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131—24no—
37C2×C22⋊C4×C131C2212yes—
38C2×C22⋊C4×C131C2212yes—
39C23×D131C2311yescharacteristic
40C2×C22⋊D131—24no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131—22no—
43C2×C22⋊D131—22no—
44C2×C22⋊D131—24no—
45C2×C22⋊D131—42no—
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2312yes—
49C22×Dic131C2311yescharacteristic
50C22×Dic131C2312yes—
51C4⋊Dic131—24no—
52Dic131⋊C4—22no—
53Dic131⋊C4—22no—
54Dic131⋊C4—42no—
55C22⋊Dic131—24no—
56C22⋊Dic131—22no—
57C22⋊Dic131—22no—
58C22⋊Dic131—24no—
59C22⋊Dic131—42no—
60C22⋊Dic131D414yes—
61(C2×D131)⋊C4—22no—
62(C2×D131)⋊C4—22no—
63(C2×D131)⋊C4—42no—
64C4×Dic131—24no—
65C24×C131—22no—
66C24×C131—24no—
67C24×C131—22no—
68C24×C131—22no—
69C24×C131—42no—
70C24×C131C2312yes—
71C24×C131C2311yescharacteristic
72C22×C4×C131—22no—
73C22×C4×C131—22no—
74C22×C4×C131—22no—
75C22×C4×C131—42no—
76C22×C4×C131D414yescomplemented
77C22×C4×C131C2312yes—
78C22×C4×C131C2311yescharacteristic
79C22×C4×C131C2311yescharacteristic
80C22⋊C4×C131—24no—
81C22⋊C4×C131—24no—
82C22×D131—21no—
83C22×D131—21no—
84C22×D131—21no—
85C22×D131—22no—
86C22×D131—22no—
87C22⋊D131—48no—
88C22⋊D131—44no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—21no—
93C2×Dic131—21no—
94C2×Dic131—21no—
95C2×Dic131—21no—
96C2×Dic131—22no—
97C2×Dic131—22no—
98C2×Dic131—21no—
99C2×Dic131—22no—
100C2×Dic131—22no—
101C2×Dic131C2×D414yes—
102C23×C131—22no—
103C23×C131—22no—
104C23×C131—28no—
105C23×C131—22no—
106C23×C131—22no—
107C23×C131—22no—
108C23×C131—22no—
109C23×C131—22no—
110C23×C131—22no—
111C23×C131—48no—
112C23×C131—48no—
113C23×C131—44no—
114C23×C131—44no—
115C23×C131C2×D412yes—
116C23×C131C2×D412yes—
117C23×C131#4112yes—
118C23×C131#4114yes—
119C23×C131C2411yescharacteristic
120C2×C4×C131—28no—
121C2×C4×C131—21no—
122C2×C4×C131—21no—
123C2×C4×C131—21no—
124C2×C4×C131—21no—
125C2×C4×C131—22no—
126C2×C4×C131—22no—
127C2×C4×C131—22no—
128C2×C4×C131—44no—
129C2×C4×C131C2×D412yes—
130C2×C4×C131C2×D412yes—
131C22×C131—22no—
132C22×C131—22no—
133C22×C131—24no—
134C22×C131—24no—
135C22×C131—24no—
136C22×C131—24no—
137C22×C131—24no—
138C22×C131—416no—
139C22×C131—42no—
140C22×C131—42no—
141C22×C131—42no—
142C22×C131—42no—
143C22×C131#12114yes—
144C22×C131C22×D411yescharacteristic
145C22×C131C22×D411yescharacteristic
146C22×C131C2×#4111yescharacteristic
147C22×C131C2×#4112yes—
148C22×C131H(5,𝔽2)12yes—
149C2×D131—41no—
150C2×D131—41no—
151C2×D131—41no—
152C2×D131—42no—
153C2×D131—42no—
154C4×C131—22no—
155C4×C131—22no—
156C4×C131—42no—
157Dic131—24no—
158Dic131—41no—
159Dic131—41no—
160Dic131—41no—
161Dic131—42no—
162C2×C131—24no—
163C2×C131—44no—
164C2×C131C2×#12111yescharacteristic
165C2×C131#52511yescharacteristic
166C2×C131#53511yescharacteristic
167C2×C131#54612yes—
168C2×C131#55112yes—
169D131—81no—
170C131#241111yescharacteristic, complemented, Sylow
171#2411—1311nomaximal, Sylow
172C2×#123—1311no—
173C2×#120—1311no—
174C2×#117—1311no—
175C22×C22⋊C4D13111yescharacteristic, complemented
176#396—1312no—
177#394—1312no—
178#390—1312no—
179#388—1312no—
180#386—1311no—
181#385—1311no—
182#379—1311no—
183C25C2×D13111yescharacteristic
184C22×D4—1311no—
185C22×D4—1312no—
186C23×C4C2×D13111yescharacteristic
187C23×C4C2×D13111yescharacteristic
188#123—2624no—
189#120—2624no—
190#117—2624no—
191C2×C4⋊C4—1312no—
192C2×C4⋊C4—1311no—
193C2×C4⋊C4—1311no—
194C2×C4⋊C4—1312no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—1311no—
197C2×C22⋊C4—1312no—
198C2×C22⋊C4—1311no—
199C2×C22⋊C4—1312no—
200C2×C22⋊C4—1311no—
201C2×C22⋊C4—1311no—
202C2×C22⋊C4—1312no—
203C2×C22⋊C4—1312no—
204C2×C22⋊C4—1311no—
205C2×C22⋊C4C2×D13112yes—
206C2×C22⋊C4C2×D13112yes—
207C2×C42—1312no—
208#95—1312no—
209#95—1312no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—1311no—
216C24C22×D13112yes—
217C24C22×D13111yescharacteristic
218C2×D4—2622no—
219C2×D4—2622no—
220C2×D4—2622no—
221C2×D4—2624no—
222C2×D4—2624no—
223C2×D4—5242no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—1311no—
229C22×C4—1311no—
230C22×C4—1312no—
231C22×C4—1311no—
232C22×C4—1312no—
233C22×C4C22⋊D13114yescomplemented
234C22×C4C22×D13112yes—
235C22×C4C22×D13111yescharacteristic
236C22×C4C22×D13111yescharacteristic
237C4⋊C4—2624no—
238C4⋊C4—2622no—
239C4⋊C4—2622no—
240C4⋊C4—5242no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—2622no—
244C22⋊C4—2622no—
245C22⋊C4—2624no—
246C22⋊C4—2624no—
247C22⋊C4—2622no—
248C22⋊C4—2622no—
249C22⋊C4—5242no—
250C22⋊C4—5242no—
251C22⋊C4—1314no—
252C42—2624no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—2621no—
267C23—2621no—
268C23—2621no—
269C23—2622no—
270C23—2622no—
271C23D4×D13112yes—
272C23C2×C22⋊D13112yes—
273C23#1452351212yes—
274C23#1452351314yes—
275C23C23×D13111yescharacteristic
276D4—5248no—
277D4—5244no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—2622no—
288C2×C4—2622no—
289C2×C4—2622no—
290C2×C4—2621no—
291C2×C4—2621no—
292C2×C4—2621no—
293C2×C4—2621no—
294C2×C4—2621no—
295C2×C4—2622no—
296C2×C4—2622no—
297C2×C4—2622no—
298C2×C4—2622no—
299C2×C4—1314no—
300C2×C4C2×C22⋊D13112yes—
301C2×C4C2×C22⋊D13112yes—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—5241no—
315C22—5242no—
316C22—5242no—
317C22—5241no—
318C22—5241no—
319C22#1686221814yes—
320C22C2×D4×D13111yescharacteristic
321C22C22×C22⋊D13111yescharacteristic
322C22C2×#1452351211yescharacteristic
323C22C2×#1452351312yes—
324C22#1686229112yes—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—2624no—
329C4—5241no—
330C4—5242no—
331C4—5241no—
332C4—5241no—
333C2—24no—
334C2—44no—
335C2—10481no—
336C2C2×#1686221811yescharacteristic
337C2#2255382311yescharacteristic
338C2#2255387911yescharacteristic
339C2#2255395212yes—
340C2#2255398812yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C22⋊C4×C131C2
prime radicalC2×#16862215C2
hypercenterC22×C22⋊C4D131
hypocenterC131#2411
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16862229C2semidirect
3. C2×#16862205C2semidirect
4. #22553235C2semidirect
5. #22553228C2semidirect
6. #22553212C2semidirect
7. #22553206C2semidirect
8. #22553191C2semidirect
9. #22553185C2semidirect
10. #22553168C2semidirect
11. C22×C22⋊C4×C131C2semidirect
17. C2×C4⋊Dic131C22semidirect
18. C2×Dic131⋊C4C22semidirect
19. C2×Dic131⋊C4C22semidirect
20. C2×Dic131⋊C4C22semidirect
27. C2×(C2×D131)⋊C4C22semidirect
28. C2×(C2×D131)⋊C4C22semidirect
29. C2×(C2×D131)⋊C4C22semidirect
30. C2×C4×Dic131C22semidirect
31. C131⋊#95C22semidirect
32. C131⋊#95C22semidirect
76. C22×C4×C131D4semidirect
170. C131#2411semidirect
175. C22×C22⋊C4D131semidirect
233. C22×C4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622292
C2×#168622152
C2×#168622052
#225532352
#225532282
#225532122
#225532062
#225531912
#225531852
#225531682
C22×C22⋊C4×C1312
#2411131