Finite Group Databasebeta

← #30677805

Subgroups — #30677805

Order 16768.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22553958C211yesmaximal, characteristic, complemented
2#22553669C211yesmaximal, characteristic, complemented
3#22553647C211yesmaximal, characteristic, complemented
4#22553636C211yesmaximal, characteristic, complemented
5#22553609C211yesmaximal, characteristic, complemented
6#22553546C211yesmaximal, characteristic, complemented
7#22553539C211yesmaximal, characteristic, complemented
8#22553527C211yesmaximal, characteristic, complemented
9#22553455C211yesmaximal, characteristic, complemented
10#22553445C211yesmaximal, characteristic, complemented
11C4×#14523498C211yesmaximal, characteristic, complemented
12D4×C131⋊C8C211yesmaximal, characteristic, complemented
13#22553306C211yesmaximal, characteristic, complemented
14C2×#16862166C211yesmaximal, characteristic, complemented
15#553×C131C211yesmaximal, characteristic, complemented
16C2×#14523513C2211yescharacteristic
17C22×D524C2211yescharacteristic
18C2×#14523501C2211yescharacteristic, complemented
19C2×#14523498C2211yescharacteristic
20C2×#14523498C2211yescharacteristic
21C22×C131⋊C8C2211yescharacteristic
22C22×C131⋊C8C2211yescharacteristic, complemented
23#16862234C2211yescharacteristic
24#16862231—21no—
25#16862225C2211yescharacteristic
26#16862220C2211yescharacteristic
27#16862219—21no—
28#16862214—21no—
29C4×D524C2211yescharacteristic
30C22⋊C4×D131—21no—
31#16862179C2211yescharacteristic, complemented
32#16862179C2211yescharacteristic, complemented
33#16862177C2211yescharacteristic, complemented
34C4⋊(C131⋊C8)C2211yescharacteristic, complemented
35#16862169C2211yescharacteristic, complemented
36#16862169C2211yescharacteristic, complemented
37#16862169C2211yescharacteristic, complemented
38#16862167C2211yescharacteristic, complemented
39#16862166C2211yescharacteristic
40#16862166C2211yescharacteristic
41#16862166C2211yescharacteristic
42#16862166C2211yescharacteristic
43#16862166C2211yescharacteristic
44#16862166C2211yescharacteristic
45C22⋊(C131⋊C8)C2211yescharacteristic
46C22⋊(C131⋊C8)C2211yescharacteristic, complemented
47C4×C131⋊C8C2211yescharacteristic, complemented
48#126×C131—21no—
49#125×C131C2211yescharacteristic, complemented
50#123×C131—21no—
51#121×C131—21no—
52#121×C131C2211yescharacteristic
53#121×C131C2211yescharacteristic, complemented
54C4×D4×C131—21no—
55C4×D4×C131C2211yescharacteristic
56C4×D4×C131C2211yescharacteristic, complemented
57#117×C131C2211yescharacteristic, complemented
58C2×C4⋊C4×C131C2211yescharacteristic
59C23×D131—21no—
60#14523513—22no—
61#14523513—41no—
62C2×Dic262C2311yescharacteristic
63C2×D524—22no—
64C2×D524—21no—
65C2×D524—41no—
66C2×D524C2311yescharacteristic
67C2×D524C2311yescharacteristic
68C2×C22⋊D131—21no—
69C2×C22⋊D131—21no—
70C2×C4×D131—21no—
71C2×C4×D131—21no—
72#14523501—41no—
73#14523498—22no—
74#14523498—41no—
75C2×C131⋊C8—41no—
76C2×C131⋊C8—41no—
77C2×C131⋊C8C2311yescharacteristic
78C2×C131⋊C8C2311yescharacteristic
79C2×C131⋊C8C2311yescharacteristic
80C2×C131⋊C8C2311yescharacteristic
81C4⋊Dic131C2311yescharacteristic
82Dic131⋊C4—21no—
83C22⋊Dic131—21no—
84(C2×D131)⋊C4—21no—
85(C2×D131)⋊C4—21no—
86(C2×D131)⋊C4—21no—
87(C2×D131)⋊C4—21no—
88C2×D4×C131—21no—
89C2×D4×C131—21no—
90C2×D4×C131D411yescharacteristic
91C2×D4×C131C2311yescharacteristic
92C22×C4×C131—21no—
93C22×C4×C131—21no—
94C22×C4×C131C2311yescharacteristic
95C22×C4×C131C2311yescharacteristic
96C4⋊C4×C131—21no—
97C4⋊C4×C131—21no—
98C4⋊C4×C131—21no—
99C4⋊C4×C131D411yescharacteristic
100C4⋊C4×C131C2311yescharacteristic
101C4⋊C4×C131C2311yescharacteristic
102C4⋊C4×C131C2311yescharacteristic
103C22⋊C4×C131—21no—
104C22⋊C4×C131—21no—
105C22⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C22⋊C4×C131D411yescharacteristic
108C22⋊C4×C131D411yescharacteristic
109C42×C131—21no—
110C42×C131C2311yescharacteristic
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—42no—
114C22×D131—42no—
115C22×D131—81no—
116Dic262—21no—
117Dic262—21no—
118D524—21no—
119D524—21no—
120D524—21no—
121D524—42no—
122D524#4112yes—
123C22⋊D131—42no—
124C22⋊D131—81no—
125C2×Dic131—21no—
126C2×Dic131—21no—
127C4×D131—42no—
128C4×D131—81no—
129C131⋊C8—21no—
130C131⋊C8—21no—
131C131⋊C8—21no—
132C131⋊C8—21no—
133C23×C131—21no—
134C23×C131C2×D411yescharacteristic
135C23×C131C2×D411yescharacteristic
136D4×C131—22no—
137D4×C131—21no—
138D4×C131—21no—
139D4×C131—42no—
140D4×C131—81no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—81no—
149C2×C4×C131—81no—
150C2×C4×C131C2×D411yescharacteristic
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2×D411yescharacteristic
153C2×C4×C131C2×D411yescharacteristic
154C2×C4×C131C2411yescharacteristic
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—21no—
158C22×C131—21no—
159C22×C131—21no—
160C22×C131—41no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131#13512yes—
164C22×C131C22×D411yescharacteristic
165C2×D131—22no—
166C2×D131—41no—
167C2×D131—41no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—41no—
171C2×D131—84no—
172C4×C131—21no—
173C4×C131—21no—
174C4×C131—21no—
175C4×C131—21no—
176C4×C131—41no—
177C4×C131—41no—
178C4×C131—41no—
179C4×C131C2×#4111yescharacteristic
180C4×C131H(5,𝔽2)11yescharacteristic
181Dic131—41no—
182Dic131—41no—
183C2×C131—22no—
184C2×C131—41no—
185C2×C131—81no—
186C2×C131#54611yescharacteristic
187C2×C131C2×#13511yescharacteristic
188C2×C131#57611yescharacteristic
189D131—42no—
190D131—81no—
191C131#329711yescharacteristic, complemented, Sylow
192#3297—1311nomaximal, Sylow
193#553D13111yescharacteristic, complemented
194#546—1311no—
195#488—1311no—
196#484—1311no—
197#482—1311no—
198#478—1311no—
199#466—1311no—
200#465—1311no—
201#463—1311no—
202#449—1311no—
203#447—1311no—
204C4×D8—1311no—
205C8×D4—1311no—
206#416—1311no—
207C2×#102—1311no—
208C2×#41—1311no—
209C22×D4—1311no—
210C2×SD16—1311no—
211C2×D8—1311no—
212C2×D8—1311no—
213C22×C8—1311no—
214C22×C8—1311no—
215#126—21no—
216#125C2×D13111yescharacteristic, complemented
217#124—1311no—
218#123—21no—
219#123—2621no—
220C22⋊Q8—1311no—
221#121—21no—
222#121—2621no—
223#121—1311no—
224#121C2×D13111yescharacteristic
225#121C2×D13111yescharacteristic, complemented
226#120—2621no—
227C4×D4—21no—
228C4×D4—1311no—
229C4×D4C2×D13111yescharacteristic
230C4×D4C2×D13111yescharacteristic, complemented
231#117C2×D13111yescharacteristic, complemented
232C2×C4⋊C4C2×D13111yescharacteristic
233C2×C22⋊C4—2621no—
234#107—1311no—
235#107—1311no—
236#106—1311no—
237C4⋊C8—1311no—
238#103—1311no—
239#103—1311no—
240#103—1311no—
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102—1311no—
246#102—1311no—
247#102—1311no—
248C22⋊C8—1311no—
249C22⋊C8—1311no—
250C4×C8—1311no—
251C24—2621no—
252#41—2622no—
253#41—5241no—
254C2×Q8—1311no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2622no—
258C2×D4—2621no—
259C2×D4—2621no—
260C2×D4—2621no—
261C2×D4—5241no—
262C2×D4—1311no—
263C2×D4—1311no—
264C2×D4C22⋊D13111yescharacteristic
265C2×D4C22×D13111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4C22×D13111yescharacteristic
271C22×C4C22×D13111yescharacteristic
272SD16—5241no—
273D8—2622no—
274D8—5241no—
275C2×C8—5241no—
276C2×C8—5241no—
277C2×C8—1311no—
278C2×C8—1311no—
279C2×C8—1311no—
280C2×C8—1311no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—2621no—
285C4⋊C4—1311no—
286C4⋊C4C22⋊D13111yescharacteristic
287C4⋊C4C22×D13111yescharacteristic
288C4⋊C4C22×D13111yescharacteristic
289C4⋊C4C22×D13111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—2621no—
295C22⋊C4—2621no—
296C22⋊C4—2621no—
297C22⋊C4—2621no—
298C22⋊C4—2621no—
299C22⋊C4C22⋊D13111yescharacteristic
300C22⋊C4C22⋊D13111yescharacteristic
301C42—21no—
302C42C22×D13111yescharacteristic
303C23—21no—
304C23—2621no—
305C23—2621no—
306C23—5242no—
307C23—5242no—
308C23—10481no—
309C23C2×C22⋊D13111yescharacteristic
310C23C2×C22⋊D13111yescharacteristic
311Q8—2621no—
312Q8—2621no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—2621no—
318D4—2621no—
319D4—2621no—
320D4—81no—
321D4—5242no—
322D4—5242no—
323D4—10481no—
324D4—1312no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—5242no—
337C2×C4—10481no—
338C2×C4C2×C22⋊D13111yescharacteristic
339C2×C4C2×C22⋊D13111yescharacteristic
340C2×C4C2×C22⋊D13111yescharacteristic
341C2×C4C2×C22⋊D13111yescharacteristic
342C2×C4C23×D13111yescharacteristic
343C8—2621no—
344C8—2621no—
345C8—2621no—
346C8—2621no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—2622no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22—10484no—
362C22#1686226612yes—
363C22C22×C22⋊D13111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—5241no—
372C4—5241no—
373C4#41×D13111yescharacteristic
374C4#1686229211yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—5242no—
379C2—10481no—
380C2#2255395011yescharacteristic
381C2C2×#1686226611yescharacteristic
382C2#2255411511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#553×C131C2
socleC22×C131C22×D4
hypercenter#553D131
hypocenterC131#3297
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #22553958C2semidirect
2. #22553669C2semidirect
3. #22553647C2semidirect
4. #22553636C2semidirect
5. #22553609C2semidirect
6. #22553546C2semidirect
7. #22553539C2semidirect
8. #22553527C2semidirect
9. #22553455C2semidirect
10. #22553445C2semidirect
11. C4×#14523498C2semidirect
12. D4×C131⋊C8C2semidirect
13. #22553306C2semidirect
14. C2×#16862166C2semidirect
15. #553×C131C2semidirect
18. C2×#14523501C22semidirect
22. C22×C131⋊C8C22semidirect
31. #16862179C22semidirect
32. #16862179C22semidirect
33. #16862177C22semidirect
34. C4⋊(C131⋊C8)C22semidirect
35. #16862169C22semidirect
36. #16862169C22semidirect
37. #16862169C22semidirect
38. #16862167C22semidirect
46. C22⋊(C131⋊C8)C22semidirect
47. C4×C131⋊C8C22semidirect
49. #125×C131C22semidirect
53. #121×C131C22semidirect
56. C4×D4×C131C22semidirect
57. #117×C131C22semidirect
191. C131#3297semidirect
193. #553D131semidirect
216. #125C2×D131semidirect
225. #121C2×D131semidirect
230. C4×D4C2×D131semidirect
231. #117C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225539582
#225536692
#225536472
#225536362
#225536092
#225535462
#225535392
#225535272
#225534552
#225534452
C4×#145234982
D4×C131⋊C82
#225533062
C2×#168621662
#553×C1312
#3297131