Finite Group Databasebeta

← #46846549

Subgroups — #46846549

Order 34688.

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#31774520C211yesmaximal, characteristic, complemented
2#31774372C211yesmaximal, characteristic, complemented
3#31774235C211yesmaximal, characteristic, complemented
4#31774222C211yesmaximal, characteristic, complemented
5#31774185C211yesmaximal, characteristic, complemented
6#31774175C211yesmaximal, characteristic, complemented
7#31774114C211yesmaximal, characteristic, complemented
8#31774114C211yesmaximal, characteristic, complemented
9#31774031C211yesmaximal, characteristic
10#31774006C211yesmaximal, characteristic, complemented
11#31773998C211yesmaximal, characteristic, complemented
12SD16×Dic271C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#450×C271C211yesmaximal, characteristic, complemented
16C2×#16864639C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18#22576159C2211yescharacteristic, complemented
19#22576155—21no—
20#22576149C2211yescharacteristic, complemented
21#22576146—21no—
22#22576142—21no—
23#22576140—21no—
24#22576134C2211yescharacteristic
25Q8×Dic271C2211yescharacteristic, complemented
26D4×Dic271—21no—
27D4×Dic271—21no—
28D4×Dic271C2211yescharacteristic
29D4×Dic271C2211yescharacteristic
30D4×Dic271C2211yescharacteristic
31D4×Dic271C2211yescharacteristic
32#22576122C2211yescharacteristic, complemented
33#22576119C2211yescharacteristic
34#22576119C2211yescharacteristic
35C2×C22⋊Dic271—21no—
36C2×C22⋊Dic271—21no—
37#22576094C2211yescharacteristic, complemented
38#22576093C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576084C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22×D4×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C22×Dic271—21no—
63C22×Dic271—21no—
64C22×Dic271—21no—
65C22×Dic271—21no—
66C2×C4×D271—21no—
67#16864639—22no—
68#16864639—22no—
69C2×C271⋊C8C2311yescharacteristic
70C2×C271⋊C8C2311yescharacteristic
71C4⋊Dic271—21no—
72C4⋊Dic271D411yescharacteristic
73C4⋊Dic271D411yescharacteristic
74C4⋊Dic271C2311yescharacteristic
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78C22⋊Dic271—21no—
79C22⋊Dic271—21no—
80C22⋊Dic271—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271—21no—
83C22⋊Dic271D411yescharacteristic
84C22⋊Dic271D411yescharacteristic
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93C24×C271—21no—
94C2×Q8×C271C2311yescharacteristic
95C2×D4×C271—22no—
96C2×D4×C271—22no—
97C2×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101SD16×C271—22no—
102SD16×C271—22no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271—21no—
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22×D271—21no—
109Dic542—81no—
110D1084—21no—
111D1084—21no—
112C22⋊D271—81no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—81no—
117C2×Dic271—81no—
118C2×Dic271—81no—
119C2×Dic271—81no—
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C4×D271—81no—
125C271⋊C8—21no—
126C271⋊C8—21no—
127C23×C271—24no—
128C23×C271—21no—
129C23×C271—21no—
130C23×C271—81no—
131C23×C271—81no—
132C23×C271C2×D411yescharacteristic
133Q8×C271—21no—
134Q8×C271—21no—
135D4×C271—24no—
136D4×C271—21no—
137D4×C271—21no—
138D4×C271#4112yes—
139D4×C271#4112yes—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271—21no—
148C22×C271—21no—
149C22×C271—22no—
150C22×C271—22no—
151C22×C271—21no—
152C22×C271—21no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—84no—
158C22×C271C22×D411yescharacteristic
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C4×C271—21no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271C2×#4111yescharacteristic
166C4×C271C2×#4111yescharacteristic
167Dic271—21no—
168Dic271—21no—
169Dic271—21no—
170Dic271—21no—
171Dic271—41no—
172Dic271—41no—
173Dic271—41no—
174C2×C271—41no—
175C2×C271—42no—
176C2×C271—42no—
177C2×C271#52511yescharacteristic
178C2×C271#57611yescharacteristic
179C2×C271#57711yescharacteristic
180D271—81no—
181C271#311811yescharacteristic, complemented, Sylow
182#3118—2711nomaximal, Sylow
183#542—2711no—
184#518—2711no—
185#485—2711no—
186#482—2711no—
187#476—2711no—
188#474—2711no—
189#463—2711no—
190#463—2711no—
191#450D27111yescharacteristic, complemented
192#447—2711no—
193#442—2711no—
194#440—2711no—
195C4×SD16—2711no—
196C4×D8—2711no—
197#433—2711no—
198C22×D4C2×D27111yescharacteristic
199C2×SD16C2×D27111yescharacteristic
200C2×SD16C2×D27111yescharacteristic
201C2×D8—2711no—
202C2×D8—2711no—
203C4⋊Q8—2711no—
204#126—5421no—
205#124—2711no—
206#124—5421no—
207#123—5421no—
208C22⋊Q8—5421no—
209C22⋊Q8C2×D27111yescharacteristic
210#121—2711no—
211C4×Q8—2711no—
212C4×D4—2711no—
213C4×D4—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—2711no—
217C4×D4—5421no—
218C4×D4—5421no—
219#117—2711no—
220#117—2711no—
221C2×C22⋊C4—5421no—
222C2×C22⋊C4—5421no—
223#107—2711no—
224#107—2711no—
225#106—2711no—
226#106—2711no—
227C4⋊C8—2711no—
228C4⋊C8—2711no—
229#103—2711no—
230#103—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102C2×D27111yescharacteristic
236#102C2×D27111yescharacteristic
237C22⋊C8—2711no—
238C22⋊C8C2×D27111yescharacteristic
239#97—2711no—
240C4×C8—2711no—
241C24—21no—
242C2×Q8—5421no—
243C2×Q8C22×D27111yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—2711no—
248C2×D4—5421no—
249C2×D4C22×D27111yescharacteristic
250C2×D4C22×D27111yescharacteristic
251C22×C4—5421no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4C22×D27111yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—5422no—
260D8—5422no—
261C2×C8—2711no—
262C2×C8—2711no—
263C2×C8C22×D27111yescharacteristic
264C2×C8C22×D27111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—2711no—
270C4⋊C4—2711no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4C22×D27111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—2711no—
276C22⋊C4—2711no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—2711no—
287C42—2711no—
288C42—5421no—
289C42—5421no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—5421no—
296C23D4×D27111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—21681no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—5421no—
304D4—5421no—
305D4—21681no—
306D4#1686465312yes—
307D4#1686465312yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—2711no—
315C2×C4—2711no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—21681no—
320C2×C4—21681no—
321C2×C4—21681no—
322C2×C4—21681no—
323C2×C4—21681no—
324C2×C4D4×D27111yescharacteristic
325C2×C4C23×D27111yescharacteristic
326C8—21no—
327C8—21no—
328C8—5421no—
329C8—5421no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—10841no—
341C22—10841no—
342C22—10841no—
343C22C2×D4×D27111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—5421no—
348C4—5421no—
349C4—5421no—
350C4—5421no—
351C4—10841no—
352C4—10841no—
353C4—10841no—
354C4C2×#1686465311yescharacteristic
355C4C2×#1686465311yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—21681no—
360C2#3177440311yescharacteristic
361C2#3177470111yescharacteristic
362C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#450×C271C2
socleC22×C271C22×D4
prime radical#31774031C2
hypercenter#450D271
hypocenterC271#3118
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31774520C2semidirect
2. #31774372C2semidirect
3. #31774235C2semidirect
4. #31774222C2semidirect
5. #31774185C2semidirect
6. #31774175C2semidirect
7. #31774114C2semidirect
8. #31774114C2semidirect
10. #31774006C2semidirect
11. #31773998C2semidirect
12. SD16×Dic271C2semidirect
13. #31773977C2semidirect
14. #31773955C2semidirect
15. #450×C271C2semidirect
18. #22576159C22semidirect
20. #22576149C22semidirect
25. Q8×Dic271C22semidirect
32. #22576122C22semidirect
37. #22576094C22semidirect
38. #22576093C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576084C22semidirect
44. #22576084C22semidirect
45. #22576082C22semidirect
46. #22576082C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
181. C271#3118semidirect
191. #450D271semidirect

Maximal subgroups

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

subgroupindex
#317745202
#317743722
#317742352
#317742222
#317741852
#317741752
#317741142
#317740312
#317740062
#317739982
SD16×Dic2712
#317739772
#317739552
#450×C2712
#3118271