Finite Group Databasebeta

← #30677817

Subgroups — #30677817

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#22554014C211yesmaximal, characteristic, complemented
2#22553953C211yesmaximal, characteristic, complemented
3#22553668C211yesmaximal, characteristic, complemented
4#22553645C211yesmaximal, characteristic, complemented
5#22553638C211yesmaximal, characteristic, complemented
6#22553611C211yesmaximal, characteristic, complemented
7#22553547C211yesmaximal, characteristic, complemented
8#22553529C211yesmaximal, characteristic, complemented
9#22553456C211yesmaximal, characteristic, complemented
10#22553445C211yesmaximal, characteristic, complemented
11#22553392C211yesmaximal, characteristic, complemented
12C8×C22⋊D131C211yesmaximal, characteristic, complemented
13#22553307C211yesmaximal, characteristic, complemented
14C2×#16862168C211yesmaximal, characteristic, complemented
15#465×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C22×D524C2211yescharacteristic
18C2×#14523498C2211yescharacteristic
19C2×#14523498C2211yescharacteristic
20C2×C8×D131C2211yescharacteristic, complemented
21#16862239—21no—
22#16862237C2211yescharacteristic, complemented
23#16862235C2211yescharacteristic
24#16862231—21no—
25#16862230—21no—
26#16862221—21no—
27#16862221C2211yescharacteristic
28#16862218—21no—
29#16862217C2211yescharacteristic, complemented
30#16862216—21no—
31D4×Dic131—21no—
32D4×Dic131C2211yescharacteristic, complemented
33#16862208C2211yescharacteristic
34C4×C22⋊D131C2211yescharacteristic
35#16862204C2211yescharacteristic, complemented
36C2×C4⋊Dic131C2211yescharacteristic
37C22⋊C4×D131—21no—
38#16862179C2211yescharacteristic, complemented
39#16862179C2211yescharacteristic, complemented
40Dic131⋊C8C2211yescharacteristic, complemented
41#16862170C2211yescharacteristic, complemented
42#16862170C2211yescharacteristic, complemented
43#16862168C2211yescharacteristic
44#16862168C2211yescharacteristic
45#16862168C2211yescharacteristic
46#16862168C2211yescharacteristic
47#16862166C2211yescharacteristic
48#16862166C2211yescharacteristic
49C22⋊(C131⋊C8)C2211yescharacteristic
50(C2×D131)⋊C8C2211yescharacteristic, complemented
51C8×Dic131C2211yescharacteristic, complemented
52C2×SD16×C131C2211yescharacteristic, complemented
53C22×C8×C131C2211yescharacteristic
54C22⋊Q8×C131C2211yescharacteristic
55#121×C131C2211yescharacteristic
56#106×C131C2211yescharacteristic, complemented
57#103×C131C2211yescharacteristic, complemented
58#102×C131C2211yescharacteristic, complemented
59C23×D131—21no—
60#14523514—22no—
61#14523514—41no—
62C2×D524—22no—
63C2×D524—21no—
64C2×D524—21no—
65C2×D524—21no—
66C2×D524—41no—
67C2×D524C2311yescharacteristic
68C2×D524C2311yescharacteristic
69C2×C22⋊D131—21no—
70C2×C22⋊D131D411yescharacteristic
71C22×Dic131—21no—
72C22×Dic131—21no—
73C2×C4×D131—21no—
74C2×C4×D131—21no—
75C2×C4×D131C2311yescharacteristic
76#14523498—22no—
77#14523498—41no—
78C2×C131⋊C8C2311yescharacteristic
79C2×C131⋊C8C2311yescharacteristic
80C8×D131—41no—
81C4⋊Dic131—21no—
82C4⋊Dic131—21no—
83C4⋊Dic131C2311yescharacteristic
84C4⋊Dic131C2311yescharacteristic
85Dic131⋊C4—21no—
86Dic131⋊C4D411yescharacteristic
87C22⋊Dic131—21no—
88C22⋊Dic131—21no—
89C22⋊Dic131D411yescharacteristic
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×Dic131—21no—
97C4×Dic131C2311yescharacteristic
98C2×Q8×C131C2311yescharacteristic
99C2×D4×C131—21no—
100C2×D4×C131C2311yescharacteristic
101C22×C4×C131C2311yescharacteristic
102SD16×C131—41no—
103C2×C8×C131—41no—
104C2×C8×C131C2311yescharacteristic
105C2×C8×C131C2311yescharacteristic
106C4⋊C4×C131—21no—
107C4⋊C4×C131C2311yescharacteristic
108C4⋊C4×C131C2311yescharacteristic
109C22⋊C4×C131—21no—
110C22⋊C4×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—42no—
121D524—42no—
122D524—81no—
123D524#4112yes—
124C22⋊D131—22no—
125C22⋊D131—81no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131—81no—
129C2×Dic131—81no—
130C2×Dic131C2×D411yescharacteristic
131C2×Dic131C2×D411yescharacteristic
132C2×Dic131C2×D411yescharacteristic
133C4×D131—21no—
134C4×D131—21no—
135C4×D131—42no—
136C4×D131—81no—
137C131⋊C8—21no—
138C131⋊C8—21no—
139C23×C131—21no—
140C23×C131C2×D411yescharacteristic
141Q8×C131—21no—
142Q8×C131—21no—
143D4×C131—21no—
144D4×C131—21no—
145D4×C131—42no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131—21no—
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2411yescharacteristic
153C8×C131—21no—
154C8×C131—21no—
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—41no—
158C22×C131—41no—
159C22×C131—41no—
160C22×C131#13512yes—
161C22×C131C22×D411yescharacteristic
162C2×D131—21no—
163C2×D131—22no—
164C2×D131—21no—
165C2×D131—21no—
166C2×D131—41no—
167C2×D131—41no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—41no—
171C2×D131—84no—
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—22no—
184C2×C131—81no—
185C2×C131#54611yescharacteristic
186C2×C131C2×#13511yescharacteristic
187C2×C131#57611yescharacteristic
188D131—41no—
189D131—42no—
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#465D13111yescharacteristic, complemented
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×SD16C2×D13111yescharacteristic, complemented
211C2×D8—1311no—
212C2×D8—1311no—
213C22×C8—1311no—
214C22×C8C2×D13111yescharacteristic
215#126—2621no—
216#125—1311no—
217#124—1311no—
218#123—2621no—
219#123—2621no—
220C22⋊Q8C2×D13111yescharacteristic
221#121—2621no—
222#121—2621no—
223#121—1311no—
224#121—1311no—
225#121C2×D13111yescharacteristic
226#120—2621no—
227C4×D4—2621no—
228C4×D4—1311no—
229C4×D4—1311no—
230C4×D4—1311no—
231#117—1311no—
232C2×C4⋊C4—1311no—
233C2×C22⋊C4—2621no—
234#107—1311no—
235#107—1311no—
236#106C2×D13111yescharacteristic, complemented
237C4⋊C8—1311no—
238#103—1311no—
239#103—1311no—
240#103C2×D13111yescharacteristic, complemented
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102—1311no—
246#102—1311no—
247#102C2×D13111yescharacteristic, complemented
248C22⋊C8—1311no—
249C22⋊C8—1311no—
250C4×C8—1311no—
251C24—2621no—
252#41—2622no—
253#41—5241no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—21no—
256C2×D4—2622no—
257C2×D4—2621no—
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—2621no—
267C22×C4—2621no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4—1311no—
271C22×C4C22×D13111yescharacteristic
272SD16—41no—
273D8—2622no—
274D8—5241no—
275C2×C8—41no—
276C2×C8—5241no—
277C2×C8—1311no—
278C2×C8—1311no—
279C2×C8C22×D13111yescharacteristic
280C2×C8C22×D13111yescharacteristic
281C4⋊C4—21no—
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—21no—
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—2621no—
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—21no—
315D4—42no—
316D4—2622no—
317D4—2621no—
318D4—2621no—
319D4—2621no—
320D4—5242no—
321D4—5242no—
322D4—10481no—
323D4—10481no—
324D4—1312no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2621no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—5242no—
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—41no—
350C22—41no—
351C22—41no—
352C22—2621no—
353C22—2621no—
354C22—2622no—
355C22—2621no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22—10484no—
362C22#1686226712yes—
363C22C2×D4×D13111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—2621no—
369C4—2621no—
370C4—2621no—
371C4—5241no—
372C4—5241no—
373C4#41×D13111yescharacteristic
374C4#1686229111yescharacteristic
375C2—22no—
376C2—81no—
377C2—5241no—
378C2—5242no—
379C2—10481no—
380C2#2255395111yescharacteristic
381C2C2×#1686226711yescharacteristic
382C2#2255411411yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#465×C131C2
socleC22×C131C22×D4
hypercenter#465D131
hypocenterC131#3297
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22554014C2semidirect
2. #22553953C2semidirect
3. #22553668C2semidirect
4. #22553645C2semidirect
5. #22553638C2semidirect
6. #22553611C2semidirect
7. #22553547C2semidirect
8. #22553529C2semidirect
9. #22553456C2semidirect
10. #22553445C2semidirect
11. #22553392C2semidirect
12. C8×C22⋊D131C2semidirect
13. #22553307C2semidirect
14. C2×#16862168C2semidirect
15. #465×C131C2semidirect
20. C2×C8×D131C22semidirect
22. #16862237C22semidirect
29. #16862217C22semidirect
32. D4×Dic131C22semidirect
35. #16862204C22semidirect
38. #16862179C22semidirect
39. #16862179C22semidirect
40. Dic131⋊C8C22semidirect
41. #16862170C22semidirect
42. #16862170C22semidirect
50. (C2×D131)⋊C8C22semidirect
51. C8×Dic131C22semidirect
52. C2×SD16×C131C22semidirect
56. #106×C131C22semidirect
57. #103×C131C22semidirect
58. #102×C131C22semidirect
191. C131#3297semidirect
200. #465D131semidirect
210. C2×SD16C2×D131semidirect
236. #106C2×D131semidirect
240. #103C2×D131semidirect
247. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225540142
#225539532
#225536682
#225536452
#225536382
#225536112
#225535472
#225535292
#225534562
#225534452
#225533922
C8×C22⋊D1312
#225533072
C2×#168621682
#465×C1312
#3297131