Finite Group Databasebeta

← #30670888

Subgroups — #30670888

Order 16768.

900 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 159 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×#16862231C212yesmaximal, complemented
2C2×#16862221C211yesmaximal, characteristic, complemented
3C2×#16862216C212yesmaximal, complemented
4C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
5#22553247C211yesmaximal, characteristic, complemented
6#22553222C211yesmaximal, characteristic, complemented
7#22553221C211yesmaximal, characteristic, complemented
8#22553213C212yesmaximal, complemented
9#22553212C212yesmaximal, complemented
10#22553159C211yesmaximal, characteristic, complemented
11#399×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×D524C2212yes—
14C22×D524C2212yes—
15C22×D524C2211yescharacteristic, complemented
16C22×C22⋊D131C2212yes—
17C22×C4×D131C2211yescharacteristic
18C22×C4×D131C2211yescharacteristic
19#16862231—28no—
20#16862221—24no—
21#16862216—28no—
22C2×Dic131⋊C4C2212yescomplemented
23C2×C22⋊Dic131C2211yescharacteristic
24C2×(C2×D131)⋊C4C2211yescharacteristic
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2211yescharacteristic
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2212yescomplemented
29C2×(C2×D131)⋊C4C2212yescomplemented
30C2×(C2×D131)⋊C4C2212yescomplemented
31C22⋊C4×D131—24no—
32C2×C4×Dic131C2212yescomplemented
33C131⋊#95C2211yescharacteristic, complemented
34C2×C4⋊C4×C131C2212yescomplemented
35C2×C4⋊C4×C131C2211yescharacteristic, complemented
36C2×C22⋊C4×C131C2212yes—
37C2×C22⋊C4×C131C2211yescharacteristic
38#95×C131C2211yescharacteristic, complemented
39C23×D131—22no—
40C23×D131—22no—
41C23×D131—24no—
42C23×D131—22no—
43C23×D131—42no—
44C23×D131C2312yes—
45C23×D131C2311yescharacteristic
46C23×D131C2311yescharacteristic
47C2×D524—24no—
48C2×D524—24no—
49C2×D524—24no—
50C2×D524—24no—
51C2×D524—24no—
52C2×D524—22no—
53C2×D524—42no—
54C2×D524—42no—
55C2×C22⋊D131—24no—
56C2×C22⋊D131—24no—
57C2×C22⋊D131—42no—
58C22×Dic131C2311yescharacteristic
59C22×Dic131C2311yescharacteristic
60C22×Dic131C2312yes—
61C2×C4×D131—24no—
62C2×C4×D131—22no—
63C2×C4×D131—42no—
64C2×C4×D131D414yescomplemented
65Dic131⋊C4—24no—
66C22⋊Dic131—22no—
67(C2×D131)⋊C4—24no—
68(C2×D131)⋊C4—24no—
69(C2×D131)⋊C4—24no—
70(C2×D131)⋊C4—22no—
71(C2×D131)⋊C4—24no—
72(C2×D131)⋊C4—22no—
73(C2×D131)⋊C4D418yescomplemented
74C4×Dic131—42no—
75C24×C131C2311yescharacteristic
76C22×C4×C131C2311yescharacteristic
77C22×C4×C131C2312yes—
78C22×C4×C131C2311yescharacteristic
79C22×C4×C131C2312yes—
80C4⋊C4×C131—22no—
81C4⋊C4×C131—24no—
82C22⋊C4×C131—24no—
83C22⋊C4×C131—22no—
84C22×D131—21no—
85C22×D131—21no—
86C22×D131—22no—
87C22×D131—28no—
88C22×D131—21no—
89C22×D131—22no—
90C22×D131—21no—
91C22×D131—22no—
92C22×D131—22no—
93C22×D131—22no—
94C22×D131—22no—
95C22×D131—22no—
96C22×D131—22no—
97C22×D131—22no—
98C22×D131—22no—
99C22×D131—48no—
100C22×D131—48no—
101C22×D131—44no—
102C22×D131—44no—
103C22×D131C2×D414yes—
104C22×D131C2×D412yes—
105C22×D131#4112yes—
106D524—48no—
107D524—44no—
108D524—48no—
109C22⋊D131—48no—
110C2×Dic131—22no—
111C2×Dic131—22no—
112C2×Dic131—22no—
113C2×Dic131—21no—
114C2×Dic131—22no—
115C2×Dic131C2×D412yes—
116C2×Dic131C2×D414yes—
117C4×D131—44no—
118C4×D131—44no—
119C23×C131—21no—
120C23×C131—22no—
121C23×C131—21no—
122C23×C131—22no—
123C23×C131#4112yes—
124C23×C131C2411yescharacteristic
125C2×C4×C131—22no—
126C2×C4×C131—22no—
127C2×C4×C131—22no—
128C2×C4×C131—21no—
129C2×C4×C131—22no—
130C2×C4×C131—22no—
131C2×C4×C131—22no—
132C2×C4×C131—22no—
133C2×C4×C131C2×D414yes—
134C2×C4×C131C2×D412yes—
135C22×C131—22no—
136C22×C131—24no—
137C22×C131—41no—
138C22×C131—41no—
139C22×C131—42no—
140C22×C131C22×D411yescharacteristic
141C22×C131C22×D412yes—
142C22×C131C2×#4111yescharacteristic
143C22×C131C2×#4111yescharacteristic
144C22×C131H(5,𝔽2)12yes—
145C2×D131—22no—
146C2×D131—22no—
147C2×D131—24no—
148C2×D131—24no—
149C2×D131—22no—
150C2×D131—24no—
151C2×D131—416no—
152C2×D131—41no—
153C2×D131—41no—
154C2×D131—42no—
155C2×D131—42no—
156C2×D131—42no—
157C2×D131—42no—
158C2×D131—42no—
159C2×D131—42no—
160C2×D131—42no—
161C2×D131—42no—
162C2×D131#12714yes—
163C4×C131—22no—
164C4×C131—24no—
165C4×C131—41no—
166C4×C131—42no—
167Dic131—22no—
168Dic131—24no—
169Dic131—41no—
170C2×C131—42no—
171C2×C131#52511yescharacteristic
172C2×C131C2×#12711yescharacteristic
173C2×C131#53411yescharacteristic
174C2×C131#54612yes—
175C2×C131#54612yes—
176D131—24no—
177D131—42no—
178D131—82no—
179C131#253911yescharacteristic, complemented, Sylow
180#2539—1311nomaximal, Sylow
181C2×#123—1312no—
182C2×#121—1311no—
183C2×#120—1312no—
184C22×C22⋊C4—1311no—
185#399—1311no—
186#399D13111yescharacteristic, complemented
187#392—1311no—
188#392—1311no—
189#390—1312no—
190#390—1312no—
191C4×C22⋊C4—1311no—
192C25—1311no—
193C22×D4—1312no—
194C22×D4—1312no—
195C22×D4—1312no—
196C22×D4—1311no—
197C23×C4—1311no—
198C23×C4—1311no—
199#123—2628no—
200#121—2624no—
201#120—2628no—
202C2×C4⋊C4—1312no—
203C2×C4⋊C4C2×D13111yescharacteristic, complemented
204C2×C4⋊C4C2×D13112yescomplemented
205C2×C22⋊C4—2624no—
206C2×C22⋊C4—1311no—
207C2×C22⋊C4—1311no—
208C2×C22⋊C4—1311no—
209C2×C22⋊C4—1312no—
210C2×C22⋊C4—1312no—
211C2×C22⋊C4—1312no—
212C2×C22⋊C4—1312no—
213C2×C22⋊C4—1312no—
214C2×C22⋊C4C2×D13111yescharacteristic
215C2×C22⋊C4C2×D13112yes—
216C2×C42—1312no—
217#95—1311no—
218#95C2×D13111yescharacteristic, complemented
219C24—2622no—
220C24—2622no—
221C24—2624no—
222C24—2622no—
223C24—5242no—
224C24—1312no—
225C24—1311no—
226C24—1311no—
227C24C22×D13111yescharacteristic
228C2×D4—2624no—
229C2×D4—2624no—
230C2×D4—2624no—
231C2×D4—2624no—
232C2×D4—2624no—
233C2×D4—2624no—
234C2×D4—2624no—
235C2×D4—2622no—
236C2×D4—5242no—
237C2×D4—5242no—
238C2×D4—5242no—
239C22×C4—2622no—
240C22×C4—2624no—
241C22×C4—5242no—
242C22×C4—1311no—
243C22×C4—1312no—
244C22×C4—1311no—
245C22×C4—1314no—
246C22×C4C22×D13111yescharacteristic
247C22×C4C22×D13112yes—
248C22×C4C22×D13111yescharacteristic
249C22×C4C22×D13112yes—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—2624no—
253C22⋊C4—22no—
254C22⋊C4—24no—
255C22⋊C4—2622no—
256C22⋊C4—2624no—
257C22⋊C4—2624no—
258C22⋊C4—2624no—
259C22⋊C4—2622no—
260C22⋊C4—2622no—
261C22⋊C4—2624no—
262C22⋊C4—1318no—
263C42—5242no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—2628no—
269C23—2621no—
270C23—2621no—
271C23—2622no—
272C23—2621no—
273C23—2622no—
274C23—2621no—
275C23—2622no—
276C23—2622no—
277C23—2622no—
278C23—2622no—
279C23—2622no—
280C23—2622no—
281C23—2622no—
282C23—2622no—
283C23—5248no—
284C23—5248no—
285C23—5244no—
286C23—5244no—
287C23—1312no—
288C23—1314no—
289C23—1312no—
290C23#1452351412yes—
291C23C23×D13111yescharacteristic
292D4—5248no—
293D4—5244no—
294D4—5248no—
295D4—5248no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—2622no—
305C2×C4—2622no—
306C2×C4—2622no—
307C2×C4—2621no—
308C2×C4—2622no—
309C2×C4—5244no—
310C2×C4—5244no—
311C2×C4—1312no—
312C2×C4—1314no—
313C2×C4D4×D13114yes—
314C2×C4D4×D13112yes—
315C22—24no—
316C22—22no—
317C22—41no—
318C22—42no—
319C22—41no—
320C22—2622no—
321C22—2624no—
322C22—2624no—
323C22—2622no—
324C22—2624no—
325C22—2622no—
326C22—52416no—
327C22—5241no—
328C22—5242no—
329C22—5241no—
330C22—5242no—
331C22—5242no—
332C22—5242no—
333C22—5242no—
334C22—5242no—
335C22—5242no—
336C22—5242no—
337C22—1314no—
338C22C2×D4×D13111yescharacteristic
339C22C2×D4×D13112yes—
340C22#41×D13111yescharacteristic
341C22C2×#1452351411yescharacteristic
342C22#1686229212yes—
343C4—22no—
344C4—24no—
345C4—41no—
346C4—42no—
347C4—2622no—
348C4—2624no—
349C4—5241no—
350C2—42no—
351C2—2624no—
352C2—5242no—
353C2—10482no—
354C2#2255382511yescharacteristic
355C2#127×D13111yescharacteristic
356C2#2255387311yescharacteristic
357C2#2255395312yes—
358C2#2255395412yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
hypercenter#399D131
hypocenterC131#2539
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16862231C2semidirect
2. C2×#16862221C2semidirect
3. C2×#16862216C2semidirect
4. C2×C22⋊C4×D131C2semidirect
5. #22553247C2semidirect
6. #22553222C2semidirect
7. #22553221C2semidirect
8. #22553213C2semidirect
9. #22553212C2semidirect
10. #22553159C2semidirect
11. #399×C131C2semidirect
15. C22×D524C22semidirect
22. C2×Dic131⋊C4C22semidirect
28. C2×(C2×D131)⋊C4C22semidirect
29. C2×(C2×D131)⋊C4C22semidirect
30. C2×(C2×D131)⋊C4C22semidirect
32. C2×C4×Dic131C22semidirect
33. C131⋊#95C22semidirect
34. C2×C4⋊C4×C131C22semidirect
35. C2×C4⋊C4×C131C22semidirect
38. #95×C131C22semidirect
64. C2×C4×D131D4semidirect
73. (C2×D131)⋊C4D4semidirect
179. C131#2539semidirect
186. #399D131semidirect
203. C2×C4⋊C4C2×D131semidirect
204. C2×C4⋊C4C2×D131semidirect
218. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622312
C2×#168622212
C2×#168622162
C2×C22⋊C4×D1312
#225532472
#225532222
#225532212
#225532132
#225532122
#225531592
#399×C1312
#2539131