Finite Group Databasebeta

← #30677809

Subgroups — #30677809

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#22554010C211yesmaximal, characteristic, complemented
2#22553953C211yesmaximal, characteristic, complemented
3#22553668C211yesmaximal, characteristic, complemented
4#22553639C211yesmaximal, characteristic, complemented
5#22553615C211yesmaximal, characteristic, complemented
6#22553549C211yesmaximal, characteristic, complemented
7#22553543C211yesmaximal, characteristic, complemented
8#22553530C211yesmaximal, characteristic, complemented
9#22553460C211yesmaximal, characteristic, complemented
10#22553448C211yesmaximal, characteristic, complemented
11#22553394C211yesmaximal, characteristic, complemented
12(C22⋊D131)⋊C8C211yesmaximal, characteristic, complemented
13#22553307C211yesmaximal, characteristic, complemented
14C2×#16862166C211yesmaximal, characteristic, complemented
15#484×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C22×D524C2211yescharacteristic
18C2×#14523500C2211yescharacteristic, complemented
19C2×D1048C2211yescharacteristic
20C2×D1048C2211yescharacteristic
21C22×C131⋊C8C2211yescharacteristic
22C2×C8×D131C2211yescharacteristic, complemented
23#16862241—21no—
24#16862237C2211yescharacteristic, complemented
25#16862235C2211yescharacteristic
26#16862231—21no—
27#16862231—21no—
28#16862221—21no—
29#16862221C2211yescharacteristic
30#16862221C2211yescharacteristic, complemented
31#16862220—21no—
32#16862220C2211yescharacteristic
33#16862216—21no—
34#16862208C2211yescharacteristic
35#16862208C2211yescharacteristic, complemented
36C4×C22⋊D131—21no—
37C4×C22⋊D131C2211yescharacteristic
38#16862204C2211yescharacteristic, complemented
39C22⋊C4×D131—21no—
40#16862177C2211yescharacteristic, complemented
41Dic131⋊C8C2211yescharacteristic, complemented
42#16862170C2211yescharacteristic, complemented
43#16862168C2211yescharacteristic
44#16862168C2211yescharacteristic
45#16862166C2211yescharacteristic
46#16862166C2211yescharacteristic
47#16862166C2211yescharacteristic
48#16862166C2211yescharacteristic
49#16862166C2211yescharacteristic, complemented
50(C2×D131)⋊C8C2211yescharacteristic, complemented
51C8×Dic131C2211yescharacteristic, complemented
52C22⋊Q8×C131C2211yescharacteristic
53C2×C4⋊C4×C131C2211yescharacteristic
54#107×C131C2211yescharacteristic, complemented
55#107×C131C2211yescharacteristic, complemented
56#103×C131C2211yescharacteristic, complemented
57#103×C131C2211yescharacteristic, complemented
58C22⋊C8×C131C2211yescharacteristic
59C23×D131—21no—
60#14523514—22no—
61#14523514—41no—
62C2×D524—22no—
63C2×D524—21no—
64C2×D524—21no—
65C2×D524—21no—
66C2×D524—21no—
67C2×D524—41no—
68C2×D524C2311yescharacteristic
69C2×D524C2311yescharacteristic
70C2×D524C2311yescharacteristic
71C2×C22⋊D131—21no—
72C2×C22⋊D131D411yescharacteristic
73C2×C4×D131—21no—
74C2×C4×D131—21no—
75C2×C4×D131—21no—
76C2×C4×D131C2311yescharacteristic
77#14523500—41no—
78D1048—22no—
79D1048—41no—
80C2×C131⋊C8—41no—
81C2×C131⋊C8C2311yescharacteristic
82C2×C131⋊C8C2311yescharacteristic
83C8×D131—41no—
84C4⋊Dic131—21no—
85C4⋊Dic131C2311yescharacteristic
86Dic131⋊C4—21no—
87Dic131⋊C4D411yescharacteristic
88C22⋊Dic131D411yescharacteristic
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)⋊C4—21no—
96(C2×D131)⋊C4—21no—
97(C2×D131)⋊C4D411yescharacteristic
98C4×Dic131—21no—
99C4×Dic131C2311yescharacteristic
100C2×Q8×C131C2311yescharacteristic
101C22×C4×C131—21no—
102C22×C4×C131C2311yescharacteristic
103C2×C8×C131C2311yescharacteristic
104C2×C8×C131C2311yescharacteristic
105C4⋊C4×C131—21no—
106C4⋊C4×C131—21no—
107C4⋊C4×C131C2311yescharacteristic
108C4⋊C4×C131C2311yescharacteristic
109C4⋊C4×C131C2311yescharacteristic
110C22⋊C4×C131—21no—
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—21no—
114C22×D131—42no—
115C22×D131—42no—
116C22×D131—81no—
117C22×D131C2×D411yescharacteristic
118D524—21no—
119D524—21no—
120D524—21no—
121D524—21no—
122D524—21no—
123D524—42no—
124D524—42no—
125D524—81no—
126D524—81no—
127D524#4112yes—
128C22⋊D131—22no—
129C22⋊D131—42no—
130C2×Dic131—21no—
131C2×Dic131C2×D411yescharacteristic
132C2×Dic131C2×D411yescharacteristic
133C2×Dic131C2×D411yescharacteristic
134C4×D131—21no—
135C4×D131—21no—
136C4×D131—42no—
137C4×D131—81no—
138C4×D131—81no—
139C131⋊C8—21no—
140C131⋊C8—21no—
141C23×C131C2×D411yescharacteristic
142Q8×C131—21no—
143Q8×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131—81no—
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2411yescharacteristic
153C8×C131—21no—
154C8×C131—21no—
155C22×C131—21no—
156C22×C131—21no—
157C22×C131#13512yes—
158C22×C131C22×D411yescharacteristic
159C2×D131—21no—
160C2×D131—22no—
161C2×D131—21no—
162C2×D131—21no—
163C2×D131—41no—
164C2×D131—41no—
165C2×D131—41no—
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×C131—41no—
177C4×C131C2×#4111yescharacteristic
178C4×C131H(5,𝔽2)11yescharacteristic
179Dic131—21no—
180Dic131—21no—
181Dic131—21no—
182Dic131—41no—
183C2×C131—22no—
184C2×C131#54611yescharacteristic
185C2×C131C2×#13511yescharacteristic
186C2×C131#57611yescharacteristic
187D131—41no—
188D131—42no—
189D131—81no—
190D131—81no—
191C131#329711yescharacteristic, complemented, Sylow
192#3297—1311nomaximal, Sylow
193#553—1311no—
194#546—1311no—
195#488—1311no—
196#484D13111yescharacteristic, complemented
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—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#121—1311no—
226#120—2621no—
227C4×D4—2621no—
228C4×D4—1311no—
229C4×D4—1311no—
230C4×D4—1311no—
231#117—1311no—
232C2×C4⋊C4C2×D13111yescharacteristic
233C2×C22⋊C4—2621no—
234#107C2×D13111yescharacteristic, complemented
235#107C2×D13111yescharacteristic, complemented
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#102—1311no—
247#102—1311no—
248C22⋊C8—1311no—
249C22⋊C8C2×D13111yescharacteristic
250C4×C8—1311no—
251C24—2621no—
252#41—2622no—
253#41—5241no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—2622no—
256C2×D4—2621no—
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×D4—1311no—
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—5241no—
276C2×C8—5241no—
277C2×C8—1311no—
278C2×C8—1311no—
279C2×C8C22×D13111yescharacteristic
280C2×C8C22×D13111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—2621no—
284C4⋊C4—2621no—
285C4⋊C4—1311no—
286C4⋊C4—1311no—
287C4⋊C4C22×D13111yescharacteristic
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—2621no—
302C42—1311no—
303C23—2621no—
304C23—2621no—
305C23—2621no—
306C23—5242no—
307C23—5242no—
308C23—10481no—
309C23—1311no—
310C23D4×D13111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—2622no—
314D4—2621no—
315D4—2621no—
316D4—2621no—
317D4—2621no—
318D4—2621no—
319D4—5242no—
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—21no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—81no—
335C2×C4—5242no—
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—2621no—
350C22—2621no—
351C22—2622no—
352C22—2621no—
353C22—5241no—
354C22—5241no—
355C22—5241no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22—10484no—
362C22#1686226912yes—
363C22C2×D4×D13111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—2621no—
370C4—2621no—
371C4—2621no—
372C4—5241no—
373C4#41×D13111yescharacteristic
374C4#1686229211yescharacteristic
375C2—22no—
376C2—5241no—
377C2—5242no—
378C2—10481no—
379C2—10481no—
380C2#2255395411yescharacteristic
381C2C2×#1686226911yescharacteristic
382C2#2255411611yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#484×C131C2
socleC22×C131C22×D4
hypercenter#484D131
hypocenterC131#3297
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #22554010C2semidirect
2. #22553953C2semidirect
3. #22553668C2semidirect
4. #22553639C2semidirect
5. #22553615C2semidirect
6. #22553549C2semidirect
7. #22553543C2semidirect
8. #22553530C2semidirect
9. #22553460C2semidirect
10. #22553448C2semidirect
11. #22553394C2semidirect
12. (C22⋊D131)⋊C8C2semidirect
13. #22553307C2semidirect
14. C2×#16862166C2semidirect
15. #484×C131C2semidirect
18. C2×#14523500C22semidirect
22. C2×C8×D131C22semidirect
24. #16862237C22semidirect
30. #16862221C22semidirect
35. #16862208C22semidirect
38. #16862204C22semidirect
40. #16862177C22semidirect
41. Dic131⋊C8C22semidirect
42. #16862170C22semidirect
49. #16862166C22semidirect
50. (C2×D131)⋊C8C22semidirect
51. C8×Dic131C22semidirect
54. #107×C131C22semidirect
55. #107×C131C22semidirect
56. #103×C131C22semidirect
57. #103×C131C22semidirect
191. C131#3297semidirect
196. #484D131semidirect
234. #107C2×D131semidirect
235. #107C2×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
#225540102
#225539532
#225536682
#225536392
#225536152
#225535492
#225535432
#225535302
#225534602
#225534482
#225533942
(C22⋊D131)⋊C82
#225533072
C2×#168621662
#484×C1312
#3297131