Finite Group Databasebeta

← #30677812

Subgroups — #30677812

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#22554006C211yesmaximal, characteristic, complemented
2#22553950C211yesmaximal, characteristic, complemented
3#22553668C211yesmaximal, characteristic, complemented
4#22553646C211yesmaximal, characteristic, complemented
5#22553636C211yesmaximal, characteristic, complemented
6#22553610C211yesmaximal, characteristic, complemented
7#22553548C211yesmaximal, characteristic, complemented
8#22553544C211yesmaximal, characteristic, complemented
9#22553530C211yesmaximal, characteristic, complemented
10#22553457C211yesmaximal, characteristic, complemented
11#22553447C211yesmaximal, characteristic, complemented
12#22553392C211yesmaximal, characteristic, complemented
13C8×C22⋊D131C211yesmaximal, characteristic, complemented
14#102×D131C211yesmaximal, characteristic, complemented
15#416×C131C211yesmaximal, characteristic, complemented
16C2×D4×D131C2211yescharacteristic
17C2×#14523500C2211yescharacteristic, complemented
18C2×D1048C2211yescharacteristic
19C2×#14523498C2211yescharacteristic
20C2×C8×D131C2211yescharacteristic
21#16862240—21no—
22#16862237C2211yescharacteristic, complemented
23#16862235C2211yescharacteristic
24#16862229—21no—
25#16862228—21no—
26#16862222C2211yescharacteristic
27#16862221C2211yescharacteristic
28#16862220C2211yescharacteristic
29#16862220C2211yescharacteristic, complemented
30#16862219—21no—
31#16862218—21no—
32#16862214—21no—
33C4×D524—21no—
34D4×Dic131C2211yescharacteristic
35#16862208C2211yescharacteristic, complemented
36C4×C22⋊D131C2211yescharacteristic
37C4⋊C4×D131C2211yescharacteristic
38C2×(C2×D131)⋊C4—21no—
39#16862180C2211yescharacteristic, complemented
40#16862179C2211yescharacteristic, complemented
41#16862177C2211yescharacteristic, complemented
42Dic131⋊C8C2211yescharacteristic, complemented
43#16862170C2211yescharacteristic, complemented
44#16862168C2211yescharacteristic
45#16862168C2211yescharacteristic, complemented
46#16862167C2211yescharacteristic
47#16862166C2211yescharacteristic
48#16862166C2211yescharacteristic
49C22⋊(C131⋊C8)C2211yescharacteristic, complemented
50(C2×D131)⋊C8C2211yescharacteristic
51C8×Dic131C2211yescharacteristic, complemented
52C2×#41×C131C2211yescharacteristic
53C22×C8×C131C2211yescharacteristic, complemented
54#117×C131C2211yescharacteristic, complemented
55#103×C131C2211yescharacteristic, complemented
56#103×C131C2211yescharacteristic, complemented
57#102×C131C2211yescharacteristic
58#102×C131C2211yescharacteristic
59C23×D131—21no—
60C2×D524—21no—
61C2×D524C2311yescharacteristic
62C2×D524C2311yescharacteristic
63C2×C22⋊D131—21no—
64C2×C22⋊D131—21no—
65C2×C22⋊D131—21no—
66C2×C22⋊D131D411yescharacteristic
67D4×D131—22no—
68D4×D131—41no—
69C22×Dic131—21no—
70C2×C4×D131—21no—
71C2×C4×D131—21no—
72C2×C4×D131C2311yescharacteristic
73#14523500—41no—
74D1048—41no—
75#14523498—22no—
76C2×C131⋊C8C2311yescharacteristic
77C2×C131⋊C8C2311yescharacteristic
78C8×D131—41no—
79C4⋊Dic131—21no—
80C4⋊Dic131C2311yescharacteristic
81C4⋊Dic131C2311yescharacteristic
82Dic131⋊C4—21no—
83Dic131⋊C4—21no—
84Dic131⋊C4—21no—
85Dic131⋊C4D411yescharacteristic
86C22⋊Dic131—21no—
87C22⋊Dic131D411yescharacteristic
88(C2×D131)⋊C4—21no—
89(C2×D131)⋊C4—21no—
90(C2×D131)⋊C4—21no—
91(C2×D131)⋊C4—21no—
92(C2×D131)⋊C4—21no—
93(C2×D131)⋊C4—21no—
94(C2×D131)⋊C4—21no—
95(C2×D131)⋊C4D411yescharacteristic
96C4×Dic131C2311yescharacteristic
97#41×C131—22no—
98#41×C131—41no—
99C2×Q8×C131C2311yescharacteristic
100C2×D4×C131—21no—
101C2×D4×C131C2311yescharacteristic
102C22×C4×C131—21no—
103C22×C4×C131C2311yescharacteristic
104C2×C8×C131—41no—
105C2×C8×C131C2311yescharacteristic
106C2×C8×C131C2311yescharacteristic
107C4⋊C4×C131C2311yescharacteristic
108C4⋊C4×C131C2311yescharacteristic
109C22⋊C4×C131—21no—
110C42×C131—21no—
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—42no—
114C22×D131—42no—
115C22×D131—81no—
116C22×D131C2×D411yescharacteristic
117D524—21no—
118D524—21no—
119D524—21no—
120D524—21no—
121D524—42no—
122C22⋊D131—22no—
123C22⋊D131—42no—
124C22⋊D131—81no—
125C22⋊D131—81no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131—81no—
129C2×Dic131C2×D411yescharacteristic
130C2×Dic131C2×D411yescharacteristic
131C2×Dic131C2×D411yescharacteristic
132C4×D131—21no—
133C4×D131—21no—
134C4×D131—81no—
135C4×D131—81no—
136C131⋊C8—21no—
137C131⋊C8—21no—
138C23×C131—21no—
139C23×C131C2×D411yescharacteristic
140Q8×C131—21no—
141Q8×C131—21no—
142D4×C131—21no—
143D4×C131—42no—
144D4×C131#4112yes—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131—42no—
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2411yescharacteristic
153C8×C131—21no—
154C8×C131—21no—
155C22×C131—21no—
156C22×C131—22no—
157C22×C131—21no—
158C22×C131—21no—
159C22×C131—41no—
160C22×C131—41no—
161C22×C131C22×D411yescharacteristic
162C2×D131—21no—
163C2×D131—21no—
164C2×D131—41no—
165C2×D131—41no—
166C2×D131—41no—
167C2×D131—41no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—84no—
171C2×D131#13512yes—
172C4×C131—21no—
173C4×C131—41no—
174C4×C131—41no—
175C4×C131—41no—
176C4×C131C2×#4111yescharacteristic
177C4×C131H(5,𝔽2)11yescharacteristic
178Dic131—21no—
179Dic131—21no—
180Dic131—21no—
181Dic131—41no—
182Dic131—41no—
183C2×C131—42no—
184C2×C131—41no—
185C2×C131#54611yescharacteristic
186C2×C131C2×#13511yescharacteristic
187C2×C131#57611yescharacteristic
188D131—22no—
189D131—81no—
190D131—81no—
191C131#329711yescharacteristic, complemented, Sylow
192#3297—1311nomaximal, Sylow
193#553—1311no—
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#416D13111yescharacteristic, complemented
207C2×#102—1311no—
208C2×#41C2×D13111yescharacteristic
209C22×D4—1311no—
210C2×SD16—1311no—
211C2×D8—1311no—
212C2×D8—1311no—
213C22×C8—1311no—
214C22×C8C2×D13111yescharacteristic, complemented
215#126—2621no—
216#125—1311no—
217#124—1311no—
218#123—2621no—
219#123—2621no—
220C22⋊Q8—1311no—
221#121—2621no—
222#121—2621no—
223#121—1311no—
224#121—1311no—
225#121—1311no—
226#120—2621no—
227C4×D4—2621no—
228C4×D4—1311no—
229C4×D4—1311no—
230C4×D4—1311no—
231#117C2×D13111yescharacteristic, complemented
232C2×C4⋊C4—1311no—
233C2×C22⋊C4—2621no—
234#107—1311no—
235#107—1311no—
236#106—1311no—
237C4⋊C8—1311no—
238#103—1311no—
239#103C2×D13111yescharacteristic, complemented
240#103C2×D13111yescharacteristic, complemented
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102—1311no—
246#102C2×D13111yescharacteristic
247#102C2×D13111yescharacteristic
248C22⋊C8—1311no—
249C22⋊C8—1311no—
250C4×C8—1311no—
251C24—2621no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—21no—
256C2×D4—2621no—
257C2×D4—2622no—
258C2×D4—2621no—
259C2×D4—2621no—
260C2×D4—2621no—
261C2×D4—5241no—
262C2×D4—1311no—
263C2×D4—1311no—
264C2×D4—1311no—
265C2×D4C22×D13111yescharacteristic
266C22×C4—21no—
267C22×C4—2621no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4—1311no—
271C22×C4C22×D13111yescharacteristic
272SD16—5241no—
273D8—2622no—
274D8—5241no—
275C2×C8—41no—
276C2×C8—5241no—
277C2×C8—1311no—
278C2×C8—1311no—
279C2×C8C22×D13111yescharacteristic
280C2×C8C22×D13111yescharacteristic
281C4⋊C4—2621no—
282C4⋊C4—2621no—
283C4⋊C4—2621no—
284C4⋊C4—2621no—
285C4⋊C4—1311no—
286C4⋊C4—1311no—
287C4⋊C4—1311no—
288C4⋊C4C22×D13111yescharacteristic
289C4⋊C4C22×D13111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—2621no—
292C22⋊C4—2621no—
293C22⋊C4—2621no—
294C22⋊C4—2621no—
295C22⋊C4—2621no—
296C22⋊C4—2621no—
297C22⋊C4—2621no—
298C22⋊C4—2621no—
299C22⋊C4—1311no—
300C22⋊C4—1311no—
301C42—21no—
302C42—1311no—
303C23—21no—
304C23—2621no—
305C23—2621no—
306C23—5242no—
307C23—5242no—
308C23—10481no—
309C23—1311no—
310C23D4×D13111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—2622no—
316D4—2621no—
317D4—2621no—
318D4—2621no—
319D4—2621no—
320D4—5242no—
321D4—5242no—
322D4—10481no—
323D4—10481no—
324D4#1452351312yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—2621no—
335C2×C4—10481no—
336C2×C4—10481no—
337C2×C4—10481no—
338C2×C4—1311no—
339C2×C4—1311no—
340C2×C4—1311no—
341C2×C4D4×D13111yescharacteristic
342C2×C4C23×D13111yescharacteristic
343C8—21no—
344C8—21no—
345C8—2621no—
346C8—2621no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—2621no—
354C22—2621no—
355C22—5241no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22—10484no—
362C22—1312no—
363C22C2×D4×D13111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—2621no—
369C4—2621no—
370C4—2621no—
371C4—5241no—
372C4—5241no—
373C4C2×#1452351311yescharacteristic
374C4#1686229111yescharacteristic
375C2—41no—
376C2—42no—
377C2—2622no—
378C2—10481no—
379C2—10481no—
380C2#2255395211yescharacteristic
381C2#135×D13111yescharacteristic
382C2#2255411411yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#416×C131C2
socleC22×C131C22×D4
hypercenter#416D131
hypocenterC131#3297
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22554006C2semidirect
2. #22553950C2semidirect
3. #22553668C2semidirect
4. #22553646C2semidirect
5. #22553636C2semidirect
6. #22553610C2semidirect
7. #22553548C2semidirect
8. #22553544C2semidirect
9. #22553530C2semidirect
10. #22553457C2semidirect
11. #22553447C2semidirect
12. #22553392C2semidirect
13. C8×C22⋊D131C2semidirect
14. #102×D131C2semidirect
15. #416×C131C2semidirect
17. C2×#14523500C22semidirect
22. #16862237C22semidirect
29. #16862220C22semidirect
35. #16862208C22semidirect
39. #16862180C22semidirect
40. #16862179C22semidirect
41. #16862177C22semidirect
42. Dic131⋊C8C22semidirect
43. #16862170C22semidirect
45. #16862168C22semidirect
49. C22⋊(C131⋊C8)C22semidirect
51. C8×Dic131C22semidirect
53. C22×C8×C131C22semidirect
54. #117×C131C22semidirect
55. #103×C131C22semidirect
56. #103×C131C22semidirect
191. C131#3297semidirect
206. #416D131semidirect
214. C22×C8C2×D131semidirect
231. #117C2×D131semidirect
239. #103C2×D131semidirect
240. #103C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225540062
#225539502
#225536682
#225536462
#225536362
#225536102
#225535482
#225535442
#225535302
#225534572
#225534472
#225533922
C8×C22⋊D1312
#102×D1312
#416×C1312
#3297131