Finite Group Databasebeta

← #30677813

Subgroups — #30677813

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#22554008C211yesmaximal, characteristic, complemented
2#22553958C211yesmaximal, characteristic, complemented
3#22553667C211yesmaximal, characteristic, complemented
4#22553648C211yesmaximal, characteristic, complemented
5#22553637C211yesmaximal, characteristic, complemented
6#22553613C211yesmaximal, characteristic, complemented
7#22553548C211yesmaximal, characteristic, complemented
8#22553541C211yesmaximal, characteristic, complemented
9#22553528C211yesmaximal, characteristic, complemented
10#22553459C211yesmaximal, characteristic, complemented
11#22553448C211yesmaximal, characteristic, complemented
12C4×D1048C211yesmaximal, characteristic, complemented
13#22553309C211yesmaximal, characteristic, complemented
14C2×#16862168C211yesmaximal, characteristic, complemented
15C8×D4×C131C211yesmaximal, characteristic, complemented
16C2×#14523513C2211yescharacteristic
17C22×D524C2211yescharacteristic
18C2×#14523502C2211yescharacteristic, complemented
19C2×D1048C2211yescharacteristic
20C2×D1048C2211yescharacteristic
21#16862241—21no—
22#16862238C2211yescharacteristic, complemented
23#16862234C2211yescharacteristic
24#16862231—21no—
25#16862228—21no—
26#16862225C2211yescharacteristic
27#16862220C2211yescharacteristic
28#16862220C2211yescharacteristic
29#16862220C2211yescharacteristic, complemented
30#16862219—21no—
31#16862218—21no—
32#16862214—21no—
33C4×D524C2211yescharacteristic
34C4×D524C2211yescharacteristic, complemented
35(C22⋊D131)⋊C4—21no—
36#16862205C2211yescharacteristic, complemented
37C2×C4⋊Dic131C2211yescharacteristic
38C22⋊C4×D131—21no—
39#16862180C2211yescharacteristic, complemented
40#16862180C2211yescharacteristic, complemented
41#16862178C2211yescharacteristic, complemented
42#16862171C2211yescharacteristic, complemented
43#16862171C2211yescharacteristic, complemented
44#16862171C2211yescharacteristic, complemented
45#16862168C2211yescharacteristic
46#16862168C2211yescharacteristic
47#16862168C2211yescharacteristic
48#16862168C2211yescharacteristic
49#16862168C2211yescharacteristic
50#16862168C2211yescharacteristic
51#16862168C2211yescharacteristic, complemented
52C22×C8×C131C2211yescharacteristic
53C22×C8×C131C2211yescharacteristic, complemented
54C4×D4×C131C2211yescharacteristic
55C4⋊C8×C131C2211yescharacteristic, complemented
56C22⋊C8×C131C2211yescharacteristic
57C22⋊C8×C131C2211yescharacteristic, complemented
58C4×C8×C131C2211yescharacteristic, complemented
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×D524C2311yescharacteristic
69C2×C22⋊D131—21no—
70C2×C22⋊D131—21no—
71C2×C22⋊D131—21no—
72C2×C22⋊D131—21no—
73C22×Dic131—21no—
74C2×C4×D131—21no—
75C2×C4×D131—21no—
76C2×C4×D131—21no—
77#14523502—41no—
78D1048—22no—
79D1048—41no—
80C4⋊Dic131—21no—
81C4⋊Dic131C2311yescharacteristic
82C4⋊Dic131C2311yescharacteristic
83C4⋊Dic131C2311yescharacteristic
84C4⋊Dic131C2311yescharacteristic
85Dic131⋊C4—21no—
86Dic131⋊C4—21no—
87Dic131⋊C4—21no—
88C22⋊Dic131—21no—
89C22⋊Dic131—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—
97C4×Dic131—21no—
98C2×D4×C131D411yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C22×C4×C131C2311yescharacteristic
101C2×C8×C131—41no—
102C2×C8×C131—41no—
103C2×C8×C131C2311yescharacteristic
104C2×C8×C131C2311yescharacteristic
105C2×C8×C131C2311yescharacteristic
106C2×C8×C131C2311yescharacteristic
107C4⋊C4×C131D411yescharacteristic
108C22⋊C4×C131D411yescharacteristic
109C22⋊C4×C131D411yescharacteristic
110C42×C131C2311yescharacteristic
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—21no—
114C22×D131—42no—
115C22×D131—42no—
116C22×D131—81no—
117Dic262—21no—
118Dic262—21no—
119D524—21no—
120D524—21no—
121D524—21no—
122D524—21no—
123D524—21no—
124D524—42no—
125D524#4112yes—
126C22⋊D131—42no—
127C22⋊D131—42no—
128C22⋊D131—81no—
129C22⋊D131—81no—
130C2×Dic131—21no—
131C2×Dic131—21no—
132C2×Dic131—21no—
133C2×Dic131—21no—
134C2×Dic131—21no—
135C2×Dic131—81no—
136C4×D131—42no—
137C4×D131—81no—
138C4×D131—81no—
139C23×C131C2×D411yescharacteristic
140C23×C131C2×D411yescharacteristic
141D4×C131—22no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131C2×D411yescharacteristic
147C2×C4×C131C2×D411yescharacteristic
148C2×C4×C131C2×D411yescharacteristic
149C2×C4×C131C2×D411yescharacteristic
150C2×C4×C131C2411yescharacteristic
151C8×C131—21no—
152C8×C131—21no—
153C8×C131—21no—
154C8×C131—21no—
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—21no—
158C22×C131—21no—
159C22×C131—21no—
160C22×C131#13512yes—
161C22×C131C22×D411yescharacteristic
162C2×D131—22no—
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—21no—
174C4×C131—21no—
175C4×C131—21no—
176C4×C131C2×#4111yescharacteristic
177C4×C131H(5,𝔽2)11yescharacteristic
178Dic131—41no—
179Dic131—41no—
180Dic131—41no—
181Dic131—41no—
182Dic131—41no—
183C2×C131—22no—
184C2×C131—41no—
185C2×C131#54611yescharacteristic
186C2×C131C2×#13511yescharacteristic
187C2×C131#57611yescharacteristic
188D131—42no—
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×D4D13111yescharacteristic, complemented
206#416—1311no—
207C2×#102—1311no—
208C2×#41—1311no—
209C22×D4—1311no—
210C2×SD16—1311no—
211C2×D8—1311no—
212C2×D8—1311no—
213C22×C8C2×D13111yescharacteristic
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×D4C2×D13111yescharacteristic
231#117—1311no—
232C2×C4⋊C4—1311no—
233C2×C22⋊C4—2621no—
234#107—1311no—
235#107—1311no—
236#106—1311no—
237C4⋊C8C2×D13111yescharacteristic, complemented
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⋊C8C2×D13111yescharacteristic
249C22⋊C8C2×D13111yescharacteristic, complemented
250C4×C8C2×D13111yescharacteristic, complemented
251C24—2621no—
252#41—2622no—
253#41—5241no—
254C2×Q8—1311no—
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×D4D52411yescharacteristic
266C22×C4—2621no—
267C22×C4—2621no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4C22×D13111yescharacteristic
271C22×C4C22×D13111yescharacteristic
272SD16—5241no—
273D8—2622no—
274D8—5241no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D13111yescharacteristic
278C2×C8C22×D13111yescharacteristic
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⋊C4—1311no—
289C4⋊C4D52411yescharacteristic
290C22⋊C4—2621no—
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⋊C4D52411yescharacteristic
300C22⋊C4D52411yescharacteristic
301C42—2621no—
302C42C22×D13111yescharacteristic
303C23—2621no—
304C23—2621no—
305C23—2621no—
306C23—5242no—
307C23—5242no—
308C23—10481no—
309C23C2×D52411yescharacteristic
310C23C2×D52411yescharacteristic
311Q8—2621no—
312Q8—2621no—
313D4—22no—
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—2621no—
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×C4C2×D52411yescharacteristic
339C2×C4C2×D52411yescharacteristic
340C2×C4C2×D52411yescharacteristic
341C2×C4C2×D52411yescharacteristic
342C2×C4C23×D13111yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—2622no—
353C22—5241no—
354C22—5241no—
355C22—5241no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22—10484no—
362C22#1686226812yes—
363C22C22×D52411yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—5241no—
369C4—5241no—
370C4—5241no—
371C4—5241no—
372C4—5241no—
373C4#41×D13111yescharacteristic
374C4#1686229111yescharacteristic
375C2—22no—
376C2—41no—
377C2—5242no—
378C2—10481no—
379C2—10481no—
380C2#2255395711yescharacteristic
381C2C2×#1686226811yescharacteristic
382C2#2255411711yescharacteristic
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC8×D4×C131C2
socleC22×C131C22×D4
hypercenterC8×D4D131
hypocenterC131#3297
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22554008C2semidirect
2. #22553958C2semidirect
3. #22553667C2semidirect
4. #22553648C2semidirect
5. #22553637C2semidirect
6. #22553613C2semidirect
7. #22553548C2semidirect
8. #22553541C2semidirect
9. #22553528C2semidirect
10. #22553459C2semidirect
11. #22553448C2semidirect
12. C4×D1048C2semidirect
13. #22553309C2semidirect
14. C2×#16862168C2semidirect
15. C8×D4×C131C2semidirect
18. C2×#14523502C22semidirect
22. #16862238C22semidirect
29. #16862220C22semidirect
34. C4×D524C22semidirect
36. #16862205C22semidirect
39. #16862180C22semidirect
40. #16862180C22semidirect
41. #16862178C22semidirect
42. #16862171C22semidirect
43. #16862171C22semidirect
44. #16862171C22semidirect
51. #16862168C22semidirect
53. C22×C8×C131C22semidirect
55. C4⋊C8×C131C22semidirect
57. C22⋊C8×C131C22semidirect
58. C4×C8×C131C22semidirect
191. C131#3297semidirect
205. C8×D4D131semidirect
214. C22×C8C2×D131semidirect
237. C4⋊C8C2×D131semidirect
249. C22⋊C8C2×D131semidirect
250. C4×C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225540082
#225539582
#225536672
#225536482
#225536372
#225536132
#225535482
#225535412
#225535282
#225534592
#225534482
C4×D10482
#225533092
C2×#168621682
C8×D4×C1312
#3297131