Finite Group Databasebeta

← #30669286

Subgroups — #30669286

Order 16768.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D131C211yesmaximal, characteristic
2C2×#16862228C212yesmaximal, complemented
3C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
4#22553228C212yesmaximal, complemented
5#22553227C212yesmaximal, complemented
6#22553188C212yesmaximal, complemented
7#22553187C211yesmaximal, characteristic, complemented
8#22553184C211yesmaximal, characteristic, complemented
9#22553168C211yesmaximal, characteristic, complemented
10#22553167C211yesmaximal, characteristic, complemented
11#379×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×C22⋊D131—24no—
14C22×C22⋊D131—28no—
15C22×C22⋊D131C2212yes—
16C22×C22⋊D131C2212yes—
17C23×Dic131C2211yescharacteristic
18C22×C4×D131C2211yescharacteristic, complemented
19C22×C4×D131C2211yescharacteristic, complemented
20#16862228—28no—
21C2×C4⋊Dic131C2212yescomplemented
22C2×Dic131⋊C4C2212yescomplemented
23C2×C22⋊Dic131C2211yescharacteristic
24C2×C22⋊Dic131C2211yescharacteristic
25C2×C22⋊Dic131C2212yescomplemented
26C2×C22⋊Dic131C2212yescomplemented
27C2×(C2×D131)⋊C4C2212yescomplemented
28C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D131)⋊C4C2212yescomplemented
31C22⋊C4×D131—24no—
32C131⋊#95C2212yescomplemented
33C131⋊#95C2212yescomplemented
34C25×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2211yescharacteristic
37C2×C22⋊C4×C131C2212yescomplemented
38C2×C22⋊C4×C131C2212yescomplemented
39C23×D131—24no—
40C23×D131—22no—
41C23×D131—22no—
42C23×D131—42no—
43C23×D131D414yes—
44C23×D131C2311yescharacteristic
45C23×D131C2311yescharacteristic
46C2×C22⋊D131—232no—
47C2×C22⋊D131—24no—
48C2×C22⋊D131—24no—
49C2×C22⋊D131—24no—
50C2×C22⋊D131—24no—
51C2×C22⋊D131—24no—
52C2×C22⋊D131D418yes—
53C22×Dic131—24no—
54C22×Dic131—22no—
55C22×Dic131C2311yescharacteristic
56C22×Dic131C2311yescharacteristic
57C22×Dic131C2312yes—
58C22×Dic131C2312yes—
59C2×C4×D131—24no—
60C2×C4×D131—22no—
61C2×C4×D131—22no—
62C2×C4×D131—42no—
63C4⋊Dic131—24no—
64Dic131⋊C4—24no—
65C22⋊Dic131—24no—
66C22⋊Dic131—22no—
67C22⋊Dic131—22no—
68C22⋊Dic131—42no—
69(C2×D131)⋊C4—24no—
70(C2×D131)⋊C4—22no—
71(C2×D131)⋊C4—22no—
72(C2×D131)⋊C4—42no—
73C24×C131—24no—
74C24×C131—22no—
75C24×C131—22no—
76C24×C131—22no—
77C24×C131—42no—
78C24×C131C2312yes—
79C24×C131C2311yescharacteristic
80C22×C4×C131C2311yescharacteristic
81C22×C4×C131C2311yescharacteristic
82C22×C4×C131C2312yes—
83C22⋊C4×C131—24no—
84C22⋊C4×C131—22no—
85C22⋊C4×C131—22no—
86C22⋊C4×C131—42no—
87C22×D131—21no—
88C22×D131—21no—
89C22×D131—22no—
90C22×D131—28no—
91C22×D131—22no—
92C22×D131—21no—
93C22×D131—21no—
94C22×D131—22no—
95C22×D131—28no—
96C22×D131—44no—
97C22×D131—48no—
98C22×D131—48no—
99C22×D131C2×D412yes—
100C22×D131C2×D414yes—
101C22×D131C2×D412yes—
102C22⋊D131—216no—
103C22⋊D131—216no—
104C2×Dic131—28no—
105C2×Dic131—22no—
106C2×Dic131—22no—
107C2×Dic131—22no—
108C2×Dic131—21no—
109C2×Dic131—21no—
110C2×Dic131—22no—
111C2×Dic131—22no—
112C2×Dic131—22no—
113C2×Dic131—22no—
114C2×Dic131C2×D414yes—
115C4×D131—44no—
116C4×D131—44no—
117C23×C131—22no—
118C23×C131—22no—
119C23×C131—28no—
120C23×C131—22no—
121C23×C131—22no—
122C23×C131—22no—
123C23×C131—22no—
124C23×C131—22no—
125C23×C131—22no—
126C23×C131—48no—
127C23×C131—48no—
128C23×C131—44no—
129C23×C131—44no—
130C23×C131C2×D414yes—
131C23×C131C2×D412yes—
132C23×C131#4112yes—
133C23×C131#4112yes—
134C23×C131C2411yescharacteristic
135C2×C4×C131—22no—
136C2×C4×C131—21no—
137C2×C4×C131—21no—
138C2×C4×C131—22no—
139C2×C4×C131—22no—
140C2×C4×C131—22no—
141C2×C4×C131—22no—
142C22×C131—22no—
143C22×C131—22no—
144C22×C131—24no—
145C22×C131—24no—
146C22×C131—24no—
147C22×C131—24no—
148C22×C131—24no—
149C22×C131—416no—
150C22×C131—42no—
151C22×C131—42no—
152C22×C131—42no—
153C22×C131—42no—
154C22×C131#12314yescomplemented
155C22×C131C22×D411yescharacteristic
156C22×C131C22×D412yes—
157C22×C131C2×#4111yescharacteristic
158C22×C131C2×#4111yescharacteristic
159C22×C131H(5,𝔽2)12yes—
160C2×D131—22no—
161C2×D131—22no—
162C2×D131—24no—
163C2×D131—24no—
164C2×D131—22no—
165C2×D131—22no—
166C2×D131—22no—
167C2×D131—416no—
168C2×D131—42no—
169C2×D131—42no—
170C2×D131#12014yes—
171C4×C131—41no—
172C4×C131—41no—
173C4×C131—42no—
174Dic131—24no—
175Dic131—41no—
176Dic131—41no—
177Dic131—42no—
178C2×C131—24no—
179C2×C131—44no—
180C2×C131C2×#12011yescharacteristic
181C2×C131C2×#12311yescharacteristic
182C2×C131#53411yescharacteristic
183C2×C131#54612yes—
184C2×C131#54612yes—
185D131—24no—
186D131—42no—
187C131#240511yescharacteristic, complemented, Sylow
188#2405—1311nomaximal, Sylow
189C23×D4—1311no—
190C2×#123—1312no—
191C22×C22⋊C4—1311no—
192#394—1312no—
193#394—1312no—
194#386—1312no—
195#385—1311no—
196#385—1311no—
197#379—1311no—
198#379—1311no—
199#379D13111yescharacteristic, complemented
200C25—1311no—
201C25C2×D13111yescharacteristic
202C22×D4—2624no—
203C22×D4—2628no—
204C22×D4—1312no—
205C22×D4—1312no—
206C23×C4—1311no—
207C23×C4—1311no—
208C23×C4—1311no—
209#123—2628no—
210C2×C4⋊C4—1312no—
211C2×C4⋊C4—1312no—
212C2×C22⋊C4—2624no—
213C2×C22⋊C4—1311no—
214C2×C22⋊C4—1312no—
215C2×C22⋊C4—1311no—
216C2×C22⋊C4—1311no—
217C2×C22⋊C4—1311no—
218C2×C22⋊C4—1312no—
219C2×C22⋊C4—1312no—
220C2×C22⋊C4—1312no—
221C2×C22⋊C4C2×D13111yescharacteristic
222C2×C22⋊C4C2×D13111yescharacteristic
223C2×C22⋊C4C2×D13112yescomplemented
224C2×C22⋊C4C2×D13112yescomplemented
225#95—1312no—
226#95—1312no—
227C24—24no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24—2624no—
233C24—2622no—
234C24—2622no—
235C24—5242no—
236C24—1311no—
237C24—1314no—
238C24—1311no—
239C24C22×D13112yes—
240C24C22×D13111yescharacteristic
241C2×D4—26232no—
242C2×D4—2624no—
243C2×D4—2624no—
244C2×D4—2624no—
245C2×D4—2624no—
246C2×D4—2624no—
247C2×D4—1318no—
248C22×C4—2622no—
249C22×C4—2622no—
250C22×C4—2624no—
251C22×C4—2622no—
252C22×C4—2624no—
253C22×C4—5242no—
254C22×C4—1311no—
255C22×C4—1312no—
256C22×C4—1311no—
257C22×C4—1312no—
258C22×C4C22×D13111yescharacteristic
259C22×C4C22×D13112yes—
260C22×C4C22×D13111yescharacteristic
261C4⋊C4—2624no—
262C4⋊C4—2624no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—2622no—
268C22⋊C4—2622no—
269C22⋊C4—2624no—
270C22⋊C4—2622no—
271C22⋊C4—2624no—
272C22⋊C4—2622no—
273C22⋊C4—5242no—
274C22⋊C4—5242no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—48no—
285C23—48no—
286C23—44no—
287C23—44no—
288C23—2628no—
289C23—2621no—
290C23—2621no—
291C23—2622no—
292C23—2628no—
293C23—2622no—
294C23—2621no—
295C23—2621no—
296C23—2622no—
297C23—5244no—
298C23—5248no—
299C23—5248no—
300C23—1312no—
301C23—1314no—
302C23—1312no—
303C23D4×D13114yes—
304C23D4×D13112yes—
305C23#1452351212yes—
306C23#1452351312yes—
307C23C23×D13111yescharacteristic
308D4—26216no—
309D4—26216no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—2628no—
318C2×C4—2622no—
319C2×C4—2622no—
320C2×C4—2622no—
321C2×C4—2621no—
322C2×C4—2621no—
323C2×C4—2622no—
324C2×C4—2622no—
325C2×C4—2622no—
326C2×C4—2622no—
327C2×C4—5244no—
328C2×C4—5244no—
329C2×C4—1314no—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—416no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—2622no—
343C22—2624no—
344C22—2624no—
345C22—2622no—
346C22—2622no—
347C22—2622no—
348C22—2622no—
349C22—52416no—
350C22—5242no—
351C22—5242no—
352C22—1314no—
353C22#1686222814yescomplemented
354C22C2×D4×D13111yescharacteristic
355C22C2×D4×D13112yes—
356C22C2×#1452351211yescharacteristic
357C22C2×#1452351311yescharacteristic
358C22#1686229112yes—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—2624no—
363C4—5241no—
364C4—5242no—
365C4—5241no—
366C2—24no—
367C2—44no—
368C2—2624no—
369C2—5242no—
370C2#120×D13111yescharacteristic
371C2C2×#1686222811yescharacteristic
372C2#2255387211yescharacteristic
373C2#2255394912yes—
374C2#2255395212yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#379×C131C2
prime radicalC23×C22⋊D131C2
hypercenter#379D131
hypocenterC131#2405
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#16862228C2semidirect
3. C2×C22⋊C4×D131C2semidirect
4. #22553228C2semidirect
5. #22553227C2semidirect
6. #22553188C2semidirect
7. #22553187C2semidirect
8. #22553184C2semidirect
9. #22553168C2semidirect
10. #22553167C2semidirect
11. #379×C131C2semidirect
18. C22×C4×D131C22semidirect
19. C22×C4×D131C22semidirect
21. C2×C4⋊Dic131C22semidirect
22. C2×Dic131⋊C4C22semidirect
25. C2×C22⋊Dic131C22semidirect
26. C2×C22⋊Dic131C22semidirect
27. C2×(C2×D131)⋊C4C22semidirect
28. C2×(C2×D131)⋊C4C22semidirect
29. C2×(C2×D131)⋊C4C22semidirect
30. C2×(C2×D131)⋊C4C22semidirect
32. C131⋊#95C22semidirect
33. C131⋊#95C22semidirect
37. C2×C22⋊C4×C131C22semidirect
38. C2×C22⋊C4×C131C22semidirect
154. C22×C131#123semidirect
187. C131#2405semidirect
199. #379D131semidirect
223. C2×C22⋊C4C2×D131semidirect
224. C2×C22⋊C4C2×D131semidirect
353. C22#16862228semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1312
C2×#168622282
C2×C22⋊C4×D1312
#225532282
#225532272
#225531882
#225531872
#225531842
#225531682
#225531672
#379×C1312
#2405131