Finite Group Databasebeta

← #46846601

Subgroups — #46846601

Order 34688.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774500C211yesmaximal, characteristic, complemented
2#31774372C211yesmaximal, characteristic, complemented
3#31774222C211yesmaximal, characteristic, complemented
4#31774218C211yesmaximal, characteristic, complemented
5#31774199C211yesmaximal, characteristic, complemented
6#31774195C211yesmaximal, characteristic, complemented
7#31774090C211yesmaximal, characteristic, complemented
8#31774090C211yesmaximal, characteristic, complemented
9#31774031C211yesmaximal, characteristic, complemented
10#31774009C211yesmaximal, characteristic, complemented
11#31774006C211yesmaximal, characteristic, complemented
12D8×Dic271C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#447×C271C211yesmaximal, characteristic, complemented
16C2×#16864639C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18#22576156C2211yescharacteristic, complemented
19#22576151C2211yescharacteristic, complemented
20#22576144—21no—
21#22576142—21no—
22#22576134C2211yescharacteristic
23#22576132—21no—
24#22576131—21no—
25D4×Dic271—21no—
26D4×Dic271—21no—
27D4×Dic271C2211yescharacteristic
28D4×Dic271C2211yescharacteristic
29D4×Dic271C2211yescharacteristic
30D4×Dic271C2211yescharacteristic
31D4×Dic271C2211yescharacteristic, complemented
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#22576082C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic, complemented
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic, complemented
48#22576081C2211yescharacteristic, complemented
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22×D4×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×D1084C2311yescharacteristic
60C2×C22⋊D271—21no—
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C22×Dic271—21no—
64C22×Dic271—21no—
65C22×Dic271—21no—
66C22×Dic271—21no—
67C22×Dic271—21no—
68C2×C4×D271—21no—
69#16864639—22no—
70#16864639—22no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C4⋊Dic271—21no—
74C4⋊Dic271D411yescharacteristic
75C4⋊Dic271D411yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79Dic271⋊C4—21no—
80C22⋊Dic271—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271—21no—
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271D411yescharacteristic
87C22⋊Dic271D411yescharacteristic
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93C24×C271—21no—
94C2×D4×C271—22no—
95C2×D4×C271—22no—
96C2×D4×C271—21no—
97C2×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C2×D4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102D8×C271—22no—
103D8×C271—22no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22×D271—21no—
109D1084—21no—
110D1084—21no—
111C22⋊D271—81no—
112C22⋊D271—81no—
113C22⋊D271—81no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—81no—
118C2×Dic271—81no—
119C2×Dic271—81no—
120C2×Dic271—81no—
121C2×Dic271—81no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C4×D271—81no—
127C271⋊C8—21no—
128C271⋊C8—21no—
129C23×C271—24no—
130C23×C271—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271—81no—
134C23×C271—81no—
135C23×C271C2×D411yescharacteristic
136D4×C271—24no—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—81no—
142D4×C271#4112yes—
143D4×C271#4112yes—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2411yescharacteristic
149C8×C271—21no—
150C8×C271—21no—
151C22×C271—21no—
152C22×C271—22no—
153C22×C271—22no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—84no—
164C22×C271C22×D411yescharacteristic
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—41no—
168C4×C271—21no—
169C4×C271—41no—
170C4×C271C2×#4111yescharacteristic
171C4×C271C2×#4111yescharacteristic
172Dic271—21no—
173Dic271—21no—
174Dic271—21no—
175Dic271—21no—
176Dic271—41no—
177Dic271—41no—
178Dic271—41no—
179C2×C271—41no—
180C2×C271—42no—
181C2×C271—42no—
182C2×C271—81no—
183C2×C271#52511yescharacteristic
184C2×C271#57611yescharacteristic
185C2×C271#57611yescharacteristic
186D271—81no—
187C271#312211yescharacteristic, complemented, Sylow
188#3122—2711nomaximal, Sylow
189#540—2711no—
190#518—2711no—
191#482—2711no—
192#482—2711no—
193#478—2711no—
194#478—2711no—
195#459—2711no—
196#459—2711no—
197#447—2711no—
198#447D27111yescharacteristic, complemented
199#442—2711no—
200#442—2711no—
201C4×D8—2711no—
202C4×D8—2711no—
203#433—2711no—
204C22×D4C2×D27111yescharacteristic
205C2×D8—2711no—
206C2×D8—2711no—
207C2×D8C2×D27111yescharacteristic
208C2×D8C2×D27111yescharacteristic
209#127—2711no—
210#125—2711no—
211#123—5421no—
212#123—5421no—
213#121—2711no—
214#121—5421no—
215#121—5421no—
216#121C2×D27111yescharacteristic
217C4×D4—2711no—
218C4×D4—2711no—
219C4×D4—2711no—
220C4×D4—2711no—
221C4×D4—2711no—
222C4×D4—2711no—
223C4×D4—5421no—
224C4×D4—5421no—
225#117—2711no—
226#117—2711no—
227C2×C22⋊C4—5421no—
228C2×C22⋊C4—5421no—
229#107—2711no—
230#107—2711no—
231#106—2711no—
232#106—2711no—
233C4⋊C8—2711no—
234C4⋊C8—2711no—
235#102—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102C2×D27111yescharacteristic
242#102C2×D27111yescharacteristic
243C22⋊C8—2711no—
244C22⋊C8C2×D27111yescharacteristic
245#97—2711no—
246C4×C8—2711no—
247C24—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—2711no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4C22×D27111yescharacteristic
257C2×D4C22×D27111yescharacteristic
258C2×D4C22×D27111yescharacteristic
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4—5421no—
264C22×C4—5421no—
265C22×C4C22×D27111yescharacteristic
266D8—22no—
267D8—22no—
268D8—5422no—
269D8—5422no—
270C2×C8—2711no—
271C2×C8—2711no—
272C2×C8C22×D27111yescharacteristic
273C2×C8C22×D27111yescharacteristic
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—2711no—
278C4⋊C4—2711no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4C22×D27111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—2711no—
284C22⋊C4—2711no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C42—2711no—
294C42—2711no—
295C42—5421no—
296C23—24no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—81no—
301C23—81no—
302C23—5421no—
303C23D4×D27111yescharacteristic
304D4—24no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—5421no—
311D4—5421no—
312D4—21681no—
313D4—21681no—
314D4—21681no—
315D4#1686465312yes—
316D4#1686465312yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—21681no—
328C2×C4—21681no—
329C2×C4—21681no—
330C2×C4—21681no—
331C2×C4—21681no—
332C2×C4—21681no—
333C2×C4D4×D27111yescharacteristic
334C2×C4C23×D27111yescharacteristic
335C8—21no—
336C8—21no—
337C8—5421no—
338C8—5421no—
339C22—21no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—84no—
352C22—10841no—
353C22—10841no—
354C22—10841no—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—41no—
358C4—5421no—
359C4—5421no—
360C4—5421no—
361C4—5421no—
362C4—10841no—
363C4—10841no—
364C4—10841no—
365C4C2×#1686465311yescharacteristic
366C4C2×#1686465311yescharacteristic
367C2—41no—
368C2—42no—
369C2—42no—
370C2—81no—
371C2—21681no—
372C2#3177440311yescharacteristic
373C2#3177469811yescharacteristic
374C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#447×C271C2
socleC22×C271C22×D4
hypercenter#447D271
hypocenterC271#3122
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #31774500C2semidirect
2. #31774372C2semidirect
3. #31774222C2semidirect
4. #31774218C2semidirect
5. #31774199C2semidirect
6. #31774195C2semidirect
7. #31774090C2semidirect
8. #31774090C2semidirect
9. #31774031C2semidirect
10. #31774009C2semidirect
11. #31774006C2semidirect
12. D8×Dic271C2semidirect
13. #31773977C2semidirect
14. #31773955C2semidirect
15. #447×C271C2semidirect
18. #22576156C22semidirect
19. #22576151C22semidirect
31. D4×Dic271C22semidirect
32. #22576122C22semidirect
37. #22576094C22semidirect
38. #22576093C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576082C22semidirect
44. #22576082C22semidirect
47. #22576081C22semidirect
48. #22576081C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
187. C271#3122semidirect
198. #447D271semidirect

Maximal subgroups

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

subgroupindex
#317745002
#317743722
#317742222
#317742182
#317741992
#317741952
#317740902
#317740312
#317740092
#317740062
D8×Dic2712
#317739772
#317739552
#447×C2712
#3122271