Finite Group Databasebeta

← #30669342

Subgroups — #30669342

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×#16862227C211yesmaximal, characteristic, complemented
2C2×#16862202C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic131C211yesmaximal, characteristic, complemented
4#22553235C212yesmaximal, complemented
5#22553226C212yesmaximal, complemented
6#22553215C212yesmaximal, complemented
7#22553200C212yesmaximal, complemented
8#22553192C211yesmaximal, characteristic, complemented
9#22553187C211yesmaximal, characteristic, complemented
10#22553168C211yesmaximal, characteristic, complemented
11C2×#120×C131C211yesmaximal, characteristic
12C23×Dic131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14#16862227—24no—
15#16862202—24no—
16C2×C4⋊Dic131C2212yescomplemented
17C2×Dic131⋊C4C2211yescharacteristic, complemented
18C2×Dic131⋊C4C2212yescomplemented
19C2×Dic131⋊C4C2211yescharacteristic, complemented
20C2×C22⋊Dic131—24no—
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C22⋊Dic131C2212yes—
24C2×C22⋊Dic131C2212yes—
25C2×C22⋊Dic131C2212yes—
26C2×C22⋊Dic131C2212yes—
27C2×C22⋊Dic131C2211yescharacteristic, complemented
28C2×C22⋊Dic131C2211yescharacteristic, complemented
29C2×C22⋊Dic131C2212yescomplemented
30C2×C4×Dic131C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C131⋊#95C2212yescomplemented
33C25×C131C2211yescharacteristic
34C22×D4×C131C2211yescharacteristic
35C22×D4×C131C2212yes—
36#120×C131—24no—
37C2×C22⋊C4×C131C2212yes—
38C2×C22⋊C4×C131C2211yescharacteristic
39C22×Dic131—22no—
40C22×Dic131—22no—
41C22×Dic131—22no—
42C22×Dic131—42no—
43C22×Dic131D414yescomplemented
44C22×Dic131C2312yes—
45C22×Dic131C2311yescharacteristic
46C22×Dic131C2312yes—
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2311yescharacteristic
50C4⋊Dic131—42no—
51Dic131⋊C4—22no—
52Dic131⋊C4—22no—
53Dic131⋊C4—24no—
54C22⋊Dic131—22no—
55C22⋊Dic131—22no—
56C22⋊Dic131—24no—
57C22⋊Dic131—24no—
58C22⋊Dic131—22no—
59C22⋊Dic131—22no—
60C22⋊Dic131—24no—
61C22⋊Dic131—42no—
62C22⋊Dic131—42no—
63C4×Dic131—24no—
64C24×C131—22no—
65C24×C131—24no—
66C24×C131—22no—
67C24×C131—22no—
68C24×C131—42no—
69C24×C131C2311yescharacteristic
70C24×C131C2312yes—
71C24×C131C2311yescharacteristic
72C2×D4×C131—22no—
73C2×D4×C131—22no—
74C2×D4×C131—24no—
75C2×D4×C131—22no—
76C2×D4×C131—24no—
77C2×D4×C131—42no—
78C22×C4×C131C2311yescharacteristic
79C22×C4×C131C2312yes—
80C22⋊C4×C131—24no—
81C22⋊C4×C131D414yes—
82C2×Dic131—28no—
83C2×Dic131—21no—
84C2×Dic131—21no—
85C2×Dic131—21no—
86C2×Dic131—21no—
87C2×Dic131—22no—
88C2×Dic131—21no—
89C2×Dic131—21no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131—22no—
97C2×Dic131—44no—
98C2×Dic131C2×D412yes—
99C2×Dic131C2×D412yes—
100C23×C131—22no—
101C23×C131—22no—
102C23×C131—28no—
103C23×C131—22no—
104C23×C131—22no—
105C23×C131—22no—
106C23×C131—22no—
107C23×C131—22no—
108C23×C131—22no—
109C23×C131—21no—
110C23×C131—21no—
111C23×C131—21no—
112C23×C131—22no—
113C23×C131—22no—
114C23×C131—48no—
115C23×C131—48no—
116C23×C131—44no—
117C23×C131—44no—
118C23×C131C2×D412yes—
119C23×C131C2×D412yes—
120C23×C131#4114yes—
121C23×C131#4112yes—
122C23×C131C2411yescharacteristic
123D4×C131—48no—
124D4×C131—44no—
125C2×C4×C131—22no—
126C2×C4×C131—22no—
127C2×C4×C131—21no—
128C2×C4×C131—21no—
129C2×C4×C131—21no—
130C2×C4×C131C2×D414yes—
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—41no—
144C22×C131—41no—
145C22×C131—41no—
146C22×C131—42no—
147C22×C131—42no—
148C22×C131#12114yes—
149C22×C131C22×D411yescharacteristic
150C22×C131C22×D411yescharacteristic
151C22×C131C2×#4111yescharacteristic
152C22×C131C2×#4112yes—
153C22×C131H(5,𝔽2)12yes—
154C4×C131—24no—
155C4×C131—41no—
156Dic131—22no—
157Dic131—22no—
158Dic131—41no—
159Dic131—41no—
160Dic131—42no—
161Dic131—42no—
162C2×C131—24no—
163C2×C131—44no—
164C2×C131—81no—
165C2×C131C2×#12111yescharacteristic
166C2×C131#52511yescharacteristic
167C2×C131#53511yescharacteristic
168C2×C131#54612yes—
169C2×C131#55112yes—
170C131#241111yescharacteristic, complemented, Sylow
171#2411—1311nomaximal, Sylow
172C2×#123—1311no—
173C2×#120D13111yescharacteristic
174C2×#117—1311no—
175C22×C22⋊C4—1311no—
176#396—1312no—
177#394—1312no—
178#390—1312no—
179#388—1312no—
180#386—1311no—
181#385—1311no—
182#379—1311no—
183C25C2×D13111yescharacteristic
184C22×D4C2×D13111yescharacteristic
185C22×D4C2×D13112yes—
186C23×C4—1311no—
187C23×C4—1311no—
188#123—2624no—
189#120—24no—
190#117—2624no—
191C2×C4⋊C4—1312no—
192C2×C4⋊C4—1311no—
193C2×C4⋊C4—1311no—
194C2×C4⋊C4—1312no—
195C2×C22⋊C4—2624no—
196C2×C22⋊C4—1312no—
197C2×C22⋊C4—1312no—
198C2×C22⋊C4—1311no—
199C2×C22⋊C4—1311no—
200C2×C22⋊C4—1311no—
201C2×C22⋊C4—1311no—
202C2×C22⋊C4—1312no—
203C2×C22⋊C4—1312no—
204C2×C22⋊C4—1312no—
205C2×C22⋊C4C2×D13112yes—
206C2×C22⋊C4C2×D13111yescharacteristic
207C2×C42—1312no—
208#95—1312no—
209#95—1312no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24C22×D13111yescharacteristic
216C24C22×D13112yes—
217C24C22×D13111yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C22×C4—2622no—
225C22×C4—2622no—
226C22×C4—2622no—
227C22×C4—5242no—
228C22×C4—1311no—
229C22×C4—1312no—
230C22×C4—1312no—
231C22×C4—1311no—
232C22×C4—1314no—
233C22×C4—1311no—
234C22×C4—1311no—
235C22×C4C22×D13111yescharacteristic
236C22×C4C22×D13112yes—
237C4⋊C4—2622no—
238C4⋊C4—2622no—
239C4⋊C4—2624no—
240C4⋊C4—5242no—
241C22⋊C4—24no—
242C22⋊C4—2622no—
243C22⋊C4—2622no—
244C22⋊C4—2624no—
245C22⋊C4—2624no—
246C22⋊C4—2624no—
247C22⋊C4—2622no—
248C22⋊C4—2622no—
249C22⋊C4—5242no—
250C22⋊C4—5242no—
251C22⋊C4C22⋊D13114yes—
252C42—2624no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—21no—
266C23—22no—
267C23—48no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23D4×D13112yes—
272C23C2×C22⋊D13112yes—
273C23#1452351214yes—
274C23#1452351212yes—
275C23C23×D13111yescharacteristic
276D4—48no—
277D4—44no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—2628no—
284C2×C4—2621no—
285C2×C4—2621no—
286C2×C4—2621no—
287C2×C4—2621no—
288C2×C4—2621no—
289C2×C4—2621no—
290C2×C4—2622no—
291C2×C4—2622no—
292C2×C4—2622no—
293C2×C4—2622no—
294C2×C4—2622no—
295C2×C4—2622no—
296C2×C4—2622no—
297C2×C4—2622no—
298C2×C4—5244no—
299C2×C4—1312no—
300C2×C4—1312no—
301C2×C4C2×C22⋊D13114yes—
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—41no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22#1686221714yes—
320C22C2×D4×D13111yescharacteristic
321C22C22×C22⋊D13111yescharacteristic
322C22C2×#1452351212yes—
323C22C2×#1452351211yescharacteristic
324C22#1686229112yes—
325C4—24no—
326C4—41no—
327C4—2622no—
328C4—2622no—
329C4—5242no—
330C4—5241no—
331C4—5242no—
332C4—5241no—
333C2—24no—
334C2—44no—
335C2—81no—
336C2C2×#1686221711yescharacteristic
337C2#2255381811yescharacteristic
338C2#2255387711yescharacteristic
339C2#2255395612yes—
340C2#2255399312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting, prime radicalC2×#120×C131C2
hypercenterC2×#120D131
hypocenterC131#2411
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#16862227C2semidirect
2. C2×#16862202C2semidirect
3. C22×C22⋊Dic131C2semidirect
4. #22553235C2semidirect
5. #22553226C2semidirect
6. #22553215C2semidirect
7. #22553200C2semidirect
8. #22553192C2semidirect
9. #22553187C2semidirect
10. #22553168C2semidirect
16. C2×C4⋊Dic131C22semidirect
17. C2×Dic131⋊C4C22semidirect
18. C2×Dic131⋊C4C22semidirect
19. C2×Dic131⋊C4C22semidirect
27. C2×C22⋊Dic131C22semidirect
28. C2×C22⋊Dic131C22semidirect
29. C2×C22⋊Dic131C22semidirect
30. C2×C4×Dic131C22semidirect
31. C131⋊#95C22semidirect
32. C131⋊#95C22semidirect
43. C22×Dic131D4semidirect
170. C131#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#168622272
C2×#168622022
C22×C22⋊Dic1312
#225532352
#225532262
#225532152
#225532002
#225531922
#225531872
#225531682
C2×#120×C1312
#2411131