Finite Group Databasebeta

← #30676140

Subgroups — #30676140

Order 16768.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22553929C211yesmaximal, characteristic, complemented
2#22553786C211yesmaximal, characteristic, complemented
3#22553653C211yesmaximal, characteristic, complemented
4#22553601C211yesmaximal, characteristic, complemented
5#22553591C211yesmaximal, characteristic, complemented
6#22553526C211yesmaximal, characteristic, complemented
7#22553526C211yesmaximal, characteristic, complemented
8#22553465C211yesmaximal, characteristic, complemented
9#22553445C211yesmaximal, characteristic
10#22553421C211yesmaximal, characteristic, complemented
11#22553411C211yesmaximal, characteristic, complemented
12#22553399C211yesmaximal, characteristic, complemented
13#22553392C211yesmaximal, characteristic, complemented
14#22553370C211yesmaximal, characteristic, complemented
15#482×C131C211yesmaximal, characteristic, complemented
16C22×D524C2211yescharacteristic
17C2×#14523502C2211yescharacteristic
18C2×#14523502C2211yescharacteristic
19C2×#14523498C2211yescharacteristic
20C2×#14523498C2211yescharacteristic
21#16862245C2211yescharacteristic, complemented
22#16862239—21no—
23#16862233—21no—
24#16862233C2211yescharacteristic, complemented
25#16862227—21no—
26#16862225C2211yescharacteristic
27#16862223—21no—
28#16862211C2211yescharacteristic, complemented
29C4×D524—21no—
30C4×D524—21no—
31D4×Dic131C2211yescharacteristic, complemented
32#16862208C2211yescharacteristic
33#16862208C2211yescharacteristic
34#16862208C2211yescharacteristic
35#16862208C2211yescharacteristic
36#16862205C2211yescharacteristic
37C22⋊C4×D131—21no—
38C22⋊C4×D131—21no—
39#16862179C2211yescharacteristic, complemented
40#16862177C2211yescharacteristic, complemented
41Dic131⋊C8C2211yescharacteristic, complemented
42Dic131⋊C8C2211yescharacteristic, complemented
43#16862169C2211yescharacteristic, complemented
44#16862169C2211yescharacteristic, complemented
45#16862168C2211yescharacteristic
46#16862168C2211yescharacteristic
47#16862166C2211yescharacteristic
48#16862166C2211yescharacteristic
49C22⋊(C131⋊C8)C2211yescharacteristic
50#16862156C2211yescharacteristic, complemented
51C8×Dic131C2211yescharacteristic, complemented
52#121×C131C2211yescharacteristic
53#117×C131C2211yescharacteristic
54#107×C131C2211yescharacteristic, complemented
55#106×C131C2211yescharacteristic, complemented
56#102×C131C2211yescharacteristic, complemented
57#102×C131C2211yescharacteristic, complemented
58C22⋊C8×C131C2211yescharacteristic
59C23×D131—21no—
60C2×Dic262—21no—
61C2×Dic262C2311yescharacteristic
62C2×D524—22no—
63C2×D524—22no—
64C2×D524—21no—
65C2×D524C2311yescharacteristic
66C2×D524C2311yescharacteristic
67C22×Dic131—21no—
68C2×C4×D131—21no—
69C2×C4×D131—21no—
70C2×C4×D131—21no—
71C2×C4×D131—21no—
72#14523502—22no—
73#14523502—22no—
74#14523498—22no—
75#14523498—22no—
76C2×C131⋊C8C2311yescharacteristic
77C2×C131⋊C8C2311yescharacteristic
78C4⋊Dic131D411yescharacteristic
79C4⋊Dic131D411yescharacteristic
80C4⋊Dic131C2311yescharacteristic
81Dic131⋊C4—21no—
82Dic131⋊C4—21no—
83Dic131⋊C4—21no—
84C22⋊Dic131—21no—
85C22⋊Dic131—21no—
86C22⋊Dic131—21no—
87C22⋊Dic131—21no—
88C22⋊Dic131D411yescharacteristic
89C22⋊Dic131D411yescharacteristic
90(C2×D131)⋊C4—21no—
91(C2×D131)⋊C4—21no—
92(C2×D131)⋊C4—21no—
93(C2×D131)⋊C4—21no—
94C4×Dic131—21no—
95C4×Dic131C2311yescharacteristic
96C4×Dic131C2311yescharacteristic
97C2×D4×C131—21no—
98C2×D4×C131C2311yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C2×C8×C131C2311yescharacteristic
101C2×C8×C131C2311yescharacteristic
102C4⋊C4×C131C2311yescharacteristic
103C4⋊C4×C131C2311yescharacteristic
104C4⋊C4×C131C2311yescharacteristic
105C22⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C42×C131—21no—
108C22×D131—21no—
109C22×D131—21no—
110C22×D131—24no—
111C22×D131—81no—
112C22×D131—81no—
113Dic262—21no—
114Dic262—21no—
115Dic262—81no—
116D524—24no—
117D524—21no—
118D524—21no—
119D524#4112yes—
120D524#4112yes—
121C2×Dic131—21no—
122C2×Dic131—21no—
123C2×Dic131—81no—
124C2×Dic131C2×D411yescharacteristic
125C2×Dic131C2×D411yescharacteristic
126C2×Dic131C2×D411yescharacteristic
127C2×Dic131C2×D411yescharacteristic
128C4×D131—81no—
129C4×D131—81no—
130C4×D131—81no—
131C4×D131—81no—
132C131⋊C8—21no—
133C131⋊C8—21no—
134C23×C131—21no—
135C23×C131C2×D411yescharacteristic
136D4×C131—21no—
137D4×C131—21no—
138D4×C131—81no—
139C2×C4×C131—21no—
140C2×C4×C131—21no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131C2×D411yescharacteristic
145C2×C4×C131C2411yescharacteristic
146C8×C131—21no—
147C8×C131—21no—
148C22×C131—21no—
149C22×C131—21no—
150C22×C131—21no—
151C22×C131—41no—
152C22×C131—41no—
153C22×C131—41no—
154C22×C131C22×D411yescharacteristic
155C2×D131—22no—
156C2×D131—22no—
157C2×D131—41no—
158C2×D131—41no—
159C2×D131—41no—
160C2×D131—41no—
161C2×D131—84no—
162C4×C131—21no—
163C4×C131—41no—
164C4×C131—41no—
165C4×C131—41no—
166C4×C131C2×#4111yescharacteristic
167C4×C131C2×#4111yescharacteristic
168Dic131—21no—
169Dic131—21no—
170Dic131—21no—
171Dic131—21no—
172Dic131—41no—
173Dic131—41no—
174C2×C131—41no—
175C2×C131—81no—
176C2×C131#52511yescharacteristic
177C2×C131#57611yescharacteristic
178C2×C131#57711yescharacteristic
179D131—42no—
180D131—42no—
181C131#311811yescharacteristic, complemented, Sylow
182#3118—1311nomaximal, Sylow
183#542—1311no—
184#518—1311no—
185#485—1311no—
186#482D13111yescharacteristic, complemented
187#476—1311no—
188#474—1311no—
189#463—1311no—
190#463—1311no—
191#450—1311no—
192#447—1311no—
193#442—1311no—
194#440—1311no—
195C4×SD16—1311no—
196C4×D8—1311no—
197#433—1311no—
198C22×D4—1311no—
199C2×SD16—1311no—
200C2×SD16—1311no—
201C2×D8—1311no—
202C2×D8—1311no—
203C4⋊Q8—1311no—
204#126—2621no—
205#124—2621no—
206#124—1311no—
207#123—2621no—
208C22⋊Q8—2621no—
209C22⋊Q8—1311no—
210#121C2×D13111yescharacteristic
211C4×Q8—1311no—
212C4×D4—2621no—
213C4×D4—2621no—
214C4×D4—1311no—
215C4×D4—1311no—
216C4×D4—1311no—
217C4×D4—1311no—
218C4×D4—1311no—
219#117—1311no—
220#117C2×D13111yescharacteristic
221C2×C22⋊C4—2621no—
222C2×C22⋊C4—2621no—
223#107—1311no—
224#107C2×D13111yescharacteristic, complemented
225#106—1311no—
226#106C2×D13111yescharacteristic, complemented
227C4⋊C8—1311no—
228C4⋊C8—1311no—
229#103—1311no—
230#103—1311no—
231#102—1311no—
232#102—1311no—
233#102—1311no—
234#102—1311no—
235#102C2×D13111yescharacteristic, complemented
236#102C2×D13111yescharacteristic, complemented
237C22⋊C8—1311no—
238C22⋊C8C2×D13111yescharacteristic
239#97—1311no—
240C4×C8—1311no—
241C24—2621no—
242C2×Q8—2621no—
243C2×Q8—1311no—
244C2×D4—21no—
245C2×D4—2622no—
246C2×D4—2622no—
247C2×D4—2621no—
248C2×D4—1311no—
249C2×D4—1311no—
250C2×D4C22×D13111yescharacteristic
251C22×C4—2621no—
252C22×C4—2621no—
253C22×C4—2621no—
254C22×C4—2621no—
255C22×C4—2621no—
256C22×C4C22×D13111yescharacteristic
257SD16—2622no—
258SD16—2622no—
259D8—2622no—
260D8—2622no—
261C2×C8—1311no—
262C2×C8—1311no—
263C2×C8C22×D13111yescharacteristic
264C2×C8C22×D13111yescharacteristic
265C4⋊C4—2621no—
266C4⋊C4—2621no—
267C4⋊C4—2621no—
268C4⋊C4—1311no—
269C4⋊C4—1311no—
270C4⋊C4—1311no—
271C4⋊C4C22×D13111yescharacteristic
272C4⋊C4C22×D13111yescharacteristic
273C4⋊C4C22×D13111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—2621no—
277C22⋊C4—2621no—
278C22⋊C4—2621no—
279C22⋊C4—2621no—
280C22⋊C4—2621no—
281C22⋊C4—2621no—
282C22⋊C4—2621no—
283C22⋊C4—2621no—
284C22⋊C4—1311no—
285C22⋊C4—1311no—
286C42—21no—
287C42—2621no—
288C42—1311no—
289C42—1311no—
290C23—21no—
291C23—2624no—
292C23—2621no—
293C23—2621no—
294C23—10481no—
295C23—10481no—
296C23D4×D13111yescharacteristic
297Q8—2621no—
298Q8—2621no—
299Q8—10481no—
300D4—21no—
301D4—21no—
302D4—2624no—
303D4—2621no—
304D4—2621no—
305D4—81no—
306D4—1312no—
307D4—1312no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—2621no—
314C2×C4—2621no—
315C2×C4—10481no—
316C2×C4—10481no—
317C2×C4—10481no—
318C2×C4—10481no—
319C2×C4—10481no—
320C2×C4—1311no—
321C2×C4—1311no—
322C2×C4—1311no—
323C2×C4—1311no—
324C2×C4D4×D13111yescharacteristic
325C2×C4C23×D13111yescharacteristic
326C8—21no—
327C8—21no—
328C8—2621no—
329C8—2621no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—2622no—
337C22—2622no—
338C22—5241no—
339C22—5241no—
340C22—5241no—
341C22—5241no—
342C22—10484no—
343C22C2×D4×D13111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—2621no—
349C4—2621no—
350C4—2621no—
351C4—2621no—
352C4—5241no—
353C4—5241no—
354C4#41×D13111yescharacteristic
355C4#41×D13111yescharacteristic
356C2—41no—
357C2—81no—
358C2—5242no—
359C2—5242no—
360C2#2255382011yescharacteristic
361C2#2255411611yescharacteristic
362C2#2255411911yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#482×C131C2
socleC22×C131C22×D4
prime radical#22553445C2
hypercenter#482D131
hypocenterC131#3118
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #22553929C2semidirect
2. #22553786C2semidirect
3. #22553653C2semidirect
4. #22553601C2semidirect
5. #22553591C2semidirect
6. #22553526C2semidirect
7. #22553526C2semidirect
8. #22553465C2semidirect
10. #22553421C2semidirect
11. #22553411C2semidirect
12. #22553399C2semidirect
13. #22553392C2semidirect
14. #22553370C2semidirect
15. #482×C131C2semidirect
21. #16862245C22semidirect
24. #16862233C22semidirect
28. #16862211C22semidirect
31. D4×Dic131C22semidirect
39. #16862179C22semidirect
40. #16862177C22semidirect
41. Dic131⋊C8C22semidirect
42. Dic131⋊C8C22semidirect
43. #16862169C22semidirect
44. #16862169C22semidirect
50. #16862156C22semidirect
51. C8×Dic131C22semidirect
54. #107×C131C22semidirect
55. #106×C131C22semidirect
56. #102×C131C22semidirect
57. #102×C131C22semidirect
181. C131#3118semidirect
186. #482D131semidirect
224. #107C2×D131semidirect
226. #106C2×D131semidirect
235. #102C2×D131semidirect
236. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225539292
#225537862
#225536532
#225536012
#225535912
#225535262
#225534652
#225534452
#225534212
#225534112
#225533992
#225533922
#225533702
#482×C1312
#3118131