Finite Group Databasebeta

← #30669151

Subgroups — #30669151

Order 16768.

456 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862241C211yesmaximal, characteristic
2#22553211C211yesmaximal, characteristic, complemented
3#22553210C211yesmaximal, characteristic, complemented
4#22553209C211yesmaximal, characteristic, complemented
5#22553204C211yesmaximal, characteristic
6#22553203C211yesmaximal, characteristic
7#22553201C211yesmaximal, characteristic
8#22553195C211yesmaximal, characteristic, complemented
9#22553194C211yesmaximal, characteristic, complemented
10#22553186C211yesmaximal, characteristic
11#22553186C211yesmaximal, characteristic
12#22553183C211yesmaximal, characteristic
13#22553181C211yesmaximal, characteristic, complemented
14#22553179C211yesmaximal, characteristic, complemented
15#387×C131C211yesmaximal, characteristic, complemented
16C22×C4×D131C2211yescharacteristic
17#16862241C418yescomplemented
18C2×C4⋊Dic131C2211yescharacteristic
19C2×C4⋊Dic131C2211yescharacteristic
20C2×C4⋊Dic131C2211yescharacteristic
21C2×C4⋊Dic131C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×Dic131⋊C4C2211yescharacteristic
26C2×Dic131⋊C4C2211yescharacteristic
27C2×Dic131⋊C4C2211yescharacteristic
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×(C2×D131)⋊C4C2211yescharacteristic
33C2×(C2×D131)⋊C4C2211yescharacteristic
34C2×C4×Dic131C2211yescharacteristic
35C2×C4×Dic131C2211yescharacteristic
36C2×C4×Dic131C2211yescharacteristic
37C131⋊#95C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C2×C4⋊C4×C131C2211yescharacteristic
46C2×C4⋊C4×C131C2211yescharacteristic
47C2×C42×C131C2211yescharacteristic
48#95×C131C2211yescharacteristic
49#95×C131C2211yescharacteristic
50#95×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52C23×D131C2311yescharacteristic
53C22×Dic131C2311yescharacteristic
54C22×Dic131C2311yescharacteristic
55C22×Dic131C2311yescharacteristic
56C22×Dic131C2311yescharacteristic
57C22×Dic131C2311yescharacteristic
58C22×Dic131C2311yescharacteristic
59C22×Dic131C2311yescharacteristic
60C2×C4×D131—41no—
61C2×C4×D131—41no—
62C2×C4×D131—41no—
63C4⋊Dic131—41no—
64C4⋊Dic131—41no—
65C4⋊Dic131—41no—
66C4⋊Dic131C2×C414yescomplemented
67Dic131⋊C4—41no—
68Dic131⋊C4—41no—
69Dic131⋊C4—41no—
70Dic131⋊C4—41no—
71Dic131⋊C4—41no—
72Dic131⋊C4C2×C414yescomplemented
73(C2×D131)⋊C4—41no—
74(C2×D131)⋊C4—41no—
75(C2×D131)⋊C4—41no—
76(C2×D131)⋊C4C2×C414yes—
77(C2×D131)⋊C4C2×C414yes—
78(C2×D131)⋊C4C2×C414yes—
79C4×Dic131—41no—
80C4×Dic131—41no—
81C4×Dic131C2×C414yescomplemented
82C22×C4×C131C2311yescharacteristic
83C22×C4×C131C2311yescharacteristic
84C22×C4×C131C2311yescharacteristic
85C22×C4×C131C2311yescharacteristic
86C22×C4×C131C2311yescharacteristic
87C22×C4×C131C2311yescharacteristic
88C22×C4×C131C2311yescharacteristic
89C4⋊C4×C131—41no—
90C4⋊C4×C131C2×C414yescomplemented
91C42×C131—41no—
92C22×D131—21no—
93C22×D131—21no—
94C22×D131—21no—
95C22×D131—21no—
96C22×D131—21no—
97C22×D131—21no—
98C22×D131C22×C412yes—
99C2×Dic131—21no—
100C2×Dic131—21no—
101C2×Dic131—21no—
102C2×Dic131—21no—
103C2×Dic131—21no—
104C2×Dic131—21no—
105C2×Dic131—21no—
106C2×Dic131—21no—
107C2×Dic131—21no—
108C2×Dic131—21no—
109C2×Dic131—21no—
110C2×Dic131—21no—
111C2×Dic131—21no—
112C2×Dic131—21no—
113C2×Dic131—21no—
114C2×Dic131—21no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131C22×C412yes—
118C2×Dic131C22×C412yes—
119C2×Dic131C22×C412yes—
120C4×D131—44no—
121C23×C131C2411yescharacteristic
122C2×C4×C131—21no—
123C2×C4×C131—21no—
124C2×C4×C131—21no—
125C2×C4×C131—21no—
126C2×C4×C131—21no—
127C2×C4×C131—21no—
128C2×C4×C131—21no—
129C2×C4×C131—21no—
130C2×C4×C131—21no—
131C2×C4×C131—21no—
132C2×C4×C131—21no—
133C2×C4×C131—21no—
134C2×C4×C131—21no—
135C2×C4×C131—21no—
136C2×C4×C131—21no—
137C2×C4×C131—21no—
138C2×C4×C131—21no—
139C2×C4×C131—21no—
140C2×C4×C131C22×C412yes—
141C2×C4×C131C22×C412yes—
142C2×C4×C131C22×C412yes—
143C22×C131C23×C411yescharacteristic
144C22×C131H(5,𝔽2)11yescharacteristic
145C22×C131H(5,𝔽2)11yescharacteristic
146C22×C131H(5,𝔽2)11yescharacteristic
147C22×C1312−1+411yescharacteristic
148C22×C1312−1+411yescharacteristic
149C22×C1312−1+411yescharacteristic
150C2×D131—22no—
151C2×D131—22no—
152C2×D131—22no—
153C2×D131—41no—
154C2×D131—41no—
155C2×D131—41no—
156C2×D131—41no—
157C4×C131—22no—
158C4×C131—22no—
159C4×C131—22no—
160C4×C131—41no—
161C4×C131—41no—
162C4×C131—41no—
163C4×C131—41no—
164Dic131—22no—
165Dic131—22no—
166Dic131—22no—
167Dic131—41no—
168Dic131—41no—
169Dic131—41no—
170Dic131—41no—
171C2×C131#52011yescharacteristic
172C2×C131#52011yescharacteristic
173C2×C131#52011yescharacteristic
174C2×C131#56011yescharacteristic
175C2×C131#56311yescharacteristic
176C2×C131#56311yescharacteristic
177C2×C131#56311yescharacteristic
178D131—42no—
179C131#238311yescharacteristic, complemented, Sylow
180#2383—1311nomaximal, Sylow
181C2×#126—1311no—
182#389—1311no—
183#389—1311no—
184#389—1311no—
185#388—1311no—
186#388—1311no—
187#388—1311no—
188#387—1311no—
189#387—1311no—
190#387D13111yescharacteristic, complemented
191#385—1311no—
192#385—1311no—
193#385—1311no—
194#384—1311no—
195#383—1311no—
196C23×C4—1311no—
197#126—1318no—
198C2×C4⋊C4—1311no—
199C2×C4⋊C4—1311no—
200C2×C4⋊C4—1311no—
201C2×C4⋊C4—1311no—
202C2×C4⋊C4—1311no—
203C2×C4⋊C4—1311no—
204C2×C4⋊C4—1311no—
205C2×C4⋊C4—1311no—
206C2×C4⋊C4—1311no—
207C2×C4⋊C4—1311no—
208C2×C4⋊C4C2×D13111yescharacteristic
209C2×C4⋊C4C2×D13111yescharacteristic
210C2×C22⋊C4—1311no—
211C2×C22⋊C4—1311no—
212C2×C22⋊C4—1311no—
213C2×C22⋊C4—1311no—
214C2×C22⋊C4—1311no—
215C2×C22⋊C4—1311no—
216C2×C42—1311no—
217C2×C42—1311no—
218C2×C42—1311no—
219C2×C42C2×D13111yescharacteristic
220#95—1311no—
221#95—1311no—
222#95—1311no—
223#95—1311no—
224#95—1311no—
225#95—1311no—
226#95—1311no—
227#95—1311no—
228#95C2×D13111yescharacteristic
229#95C2×D13111yescharacteristic
230#95C2×D13111yescharacteristic
231#95C2×D13111yescharacteristic
232C24—1311no—
233C22×C4—5241no—
234C22×C4—5241no—
235C22×C4—5241no—
236C22×C4—1311no—
237C22×C4—1311no—
238C22×C4—1311no—
239C22×C4—1311no—
240C22×C4—1311no—
241C22×C4—1311no—
242C22×C4—1311no—
243C22×C4C22×D13111yescharacteristic
244C22×C4C22×D13111yescharacteristic
245C22×C4C22×D13111yescharacteristic
246C22×C4C22×D13111yescharacteristic
247C22×C4C22×D13111yescharacteristic
248C22×C4C22×D13111yescharacteristic
249C22×C4C22×D13111yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—5241no—
252C4⋊C4—5241no—
253C4⋊C4—5241no—
254C4⋊C4—5241no—
255C4⋊C4—5241no—
256C4⋊C4—5241no—
257C4⋊C4—5241no—
258C4⋊C4—5241no—
259C4⋊C4—1314no—
260C4⋊C4—1314no—
261C4⋊C4C4×D13114yescomplemented
262C22⋊C4—5241no—
263C22⋊C4—5241no—
264C22⋊C4—5241no—
265C22⋊C4—1314no—
266C22⋊C4—1314no—
267C22⋊C4—1314no—
268C42—41no—
269C42—5241no—
270C42—5241no—
271C42—1314no—
272C23—2621no—
273C23—2621no—
274C23—2621no—
275C23—2621no—
276C23—2621no—
277C23—2621no—
278C23—1312no—
279C23C23×D13111yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—2621no—
299C2×C4—2621no—
300C2×C4—2621no—
301C2×C4—2621no—
302C2×C4—2621no—
303C2×C4—2621no—
304C2×C4—2621no—
305C2×C4—2621no—
306C2×C4—2621no—
307C2×C4—2621no—
308C2×C4—2621no—
309C2×C4—2621no—
310C2×C4—2621no—
311C2×C4—2621no—
312C2×C4—2621no—
313C2×C4—2621no—
314C2×C4—2621no—
315C2×C4—2621no—
316C2×C4—5244no—
317C2×C4—1312no—
318C2×C4—1312no—
319C2×C4—1312no—
320C2×C4C2×C4×D13112yes—
321C2×C4C2×C4×D13112yes—
322C2×C4C2×C4×D13112yes—
323C22—2622no—
324C22—2622no—
325C22—2622no—
326C22—5241no—
327C22—5241no—
328C22—5241no—
329C22—5241no—
330C22C22×C4×D13111yescharacteristic
331C22#1686229111yescharacteristic
332C22#1686229211yescharacteristic
333C22#1686229211yescharacteristic
334C22#1686229311yescharacteristic
335C22#1686229311yescharacteristic
336C22#1686229411yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—2622no—
345C4—2622no—
346C4—2622no—
347C4—5241no—
348C4—5241no—
349C4—5241no—
350C4—5241no—
351C2—5242no—
352C2#2255379111yescharacteristic
353C2#2255379211yescharacteristic
354C2#2255379411yescharacteristic
355C2#2255405211yescharacteristic
356C2#2255406311yescharacteristic
357C2#2255406411yescharacteristic
358C2#2255406611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutatorC22×C131C23×C4
Fitting#387×C131C2
socle, cosocleC23×C131C24
prime radicalC23×D131C23
hypercenter#387D131
hypocenterC131#2383
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #22553211C2semidirect
3. #22553210C2semidirect
4. #22553209C2semidirect
8. #22553195C2semidirect
9. #22553194C2semidirect
13. #22553181C2semidirect
14. #22553179C2semidirect
15. #387×C131C2semidirect
17. #16862241C4semidirect
66. C4⋊Dic131C2×C4semidirect
72. Dic131⋊C4C2×C4semidirect
81. C4×Dic131C2×C4semidirect
90. C4⋊C4×C131C2×C4semidirect
179. C131#2383semidirect
190. #387D131semidirect
261. C4⋊C4C4×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622412
#225532112
#225532102
#225532092
#225532042
#225532032
#225532012
#225531952
#225531942
#225531862
#225531832
#225531812
#225531792
#387×C1312
#2383131