Finite Group Databasebeta

← #46846605

Subgroups — #46846605

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#31774501C211yesmaximal, characteristic, complemented
2#31774369C211yesmaximal, characteristic, complemented
3#31774220C211yesmaximal, characteristic, complemented
4#31774219C211yesmaximal, characteristic, complemented
5#31774198C211yesmaximal, characteristic, complemented
6#31774197C211yesmaximal, characteristic, complemented
7#31774089C211yesmaximal, characteristic, complemented
8#31774029C211yesmaximal, characteristic, complemented
9#31774029C211yesmaximal, characteristic, complemented
10#31774009C211yesmaximal, characteristic, complemented
11#31774006C211yesmaximal, characteristic, complemented
12D8×Dic271C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14#31773957C211yesmaximal, characteristic, complemented
15#459×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19#22576152C2211yescharacteristic, complemented
20#22576141—21no—
21#22576141—21no—
22#22576132—21no—
23#22576132—21no—
24#22576131C2211yescharacteristic
25#22576131C2211yescharacteristic
26C4×D1084C2211yescharacteristic
27D4×Dic271C2211yescharacteristic
28D4×Dic271C2211yescharacteristic, complemented
29D4×Dic271C2211yescharacteristic, complemented
30#22576122C2211yescharacteristic
31C4×C22⋊D271—21no—
32(C22⋊D271)⋊C4—21no—
33#22576118C2211yescharacteristic
34#22576117C2211yescharacteristic
35C2×(C2×D271)⋊C4—21no—
36C22⋊C4×D271—21no—
37#22576094C2211yescharacteristic, complemented
38#22576093C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41C4⋊(C271⋊C8)C2211yescharacteristic, complemented
42#22576082C2211yescharacteristic
43#22576082C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic, complemented
48(C2×D271)⋊C8C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C2×D8×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54#127×C271C2211yescharacteristic, complemented
55C4×D4×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64D4×D271—22no—
65D4×D271—22no—
66C22×Dic271—21no—
67C22×Dic271—21no—
68C22×Dic271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271C2311yescharacteristic
72#16864639—22no—
73#16864639—22no—
74C2×C271⋊C8C2311yescharacteristic
75C2×C271⋊C8C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4D411yescharacteristic
82Dic271⋊C4D411yescharacteristic
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4D411yescharacteristic
92(C2×D271)⋊C4D411yescharacteristic
93C4×Dic271—21no—
94C4×Dic271C2311yescharacteristic
95C2×D4×C271—21no—
96C2×D4×C271—21no—
97C2×D4×C271C2311yescharacteristic
98C2×D4×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C22×C4×C271—21no—
101D8×C271—22no—
102D8×C271—22no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109C22×D271—24no—
110C22×D271—81no—
111C22×D271—81no—
112C22×D271C2×D411yescharacteristic
113D1084—21no—
114D1084#4112yes—
115C22⋊D271—24no—
116C22⋊D271—81no—
117C22⋊D271—81no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—81no—
122C2×Dic271—81no—
123C2×Dic271—81no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—21no—
128C4×D271—21no—
129C4×D271—81no—
130C4×D271—81no—
131C271⋊C8—21no—
132C271⋊C8—21no—
133C23×C271—21no—
134C23×C271—21no—
135C23×C271—21no—
136D4×C271—21no—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—81no—
142D4×C271—81no—
143D4×C271#4112yes—
144C2×C4×C271—21no—
145C2×C4×C271—81no—
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2411yescharacteristic
149C8×C271—21no—
150C8×C271—21no—
151C22×C271—22no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271C22×D411yescharacteristic
161C2×D271—21no—
162C2×D271—22no—
163C2×D271—21no—
164C2×D271—21no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—84no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271C2×#4111yescharacteristic
172C4×C271C2×#4111yescharacteristic
173Dic271—21no—
174Dic271—21no—
175Dic271—21no—
176Dic271—41no—
177Dic271—41no—
178Dic271—41no—
179C2×C271—42no—
180C2×C271—81no—
181C2×C271—81no—
182C2×C271#52511yescharacteristic
183C2×C271#57611yescharacteristic
184C2×C271#57611yescharacteristic
185D271—41no—
186D271—42no—
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#459D27111yescharacteristic, complemented
197#447—2711no—
198#447—2711no—
199#442—2711no—
200#442—2711no—
201C4×D8—2711no—
202C4×D8—2711no—
203#433—2711no—
204C22×D4—2711no—
205C2×D8—2711no—
206C2×D8—2711no—
207C2×D8C2×D27111yescharacteristic
208C2×D8C2×D27111yescharacteristic
209#127C2×D27111yescharacteristic, complemented
210#125—2711no—
211#123—5421no—
212#123—5421no—
213#121—2711no—
214#121—2711no—
215#121—5421no—
216#121—5421no—
217C4×D4—2711no—
218C4×D4—2711no—
219C4×D4—2711no—
220C4×D4—2711no—
221C4×D4—2711no—
222C4×D4—5421no—
223C4×D4—5421no—
224C4×D4C2×D27111yescharacteristic
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⋊C8C2×D27111yescharacteristic, complemented
235#102—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102C2×D27111yescharacteristic, complemented
242#102C2×D27111yescharacteristic, complemented
243C22⋊C8—2711no—
244C22⋊C8—2711no—
245#97—2711no—
246C4×C8—2711no—
247C24—5421no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—2711no—
251C2×D4—5421no—
252C2×D4—5422no—
253C2×D4—5422no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4C22×D27111yescharacteristic
257C2×D4C22×D27111yescharacteristic
258C2×D4C22×D27111yescharacteristic
259C22×C4—21no—
260C22×C4—2711no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4—5421no—
264C22×C4—5421no—
265C22×C4—5421no—
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—5421no—
295C42C22×D27111yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—2711no—
300C23—5424no—
301C23—5421no—
302C23—21681no—
303C23—21681no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—81no—
311D4—2712no—
312D4—5424no—
313D4—5421no—
314D4—21681no—
315D4—21681no—
316D4#1686465312yes—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—2711no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—5421no—
323C2×C4—5421no—
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×C4D4×D27111yescharacteristic
333C2×C4D4×D27111yescharacteristic
334C2×C4C23×D27111yescharacteristic
335C8—21no—
336C8—21no—
337C8—5421no—
338C8—5421no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—5421no—
349C22—5422no—
350C22—5421no—
351C22—5421no—
352C22—10841no—
353C22—10841no—
354C22—21684no—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—5421no—
360C4—5421no—
361C4—5421no—
362C4—10841no—
363C4—10841no—
364C4—10841no—
365C4#41×D27111yescharacteristic
366C4C2×#1686465311yescharacteristic
367C2—42no—
368C2—81no—
369C2—81no—
370C2—10841no—
371C2—10842no—
372C2#3177440711yescharacteristic
373C2#3177469811yescharacteristic
374C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#459×C271C2
socleC22×C271C22×D4
hypercenter#459D271
hypocenterC271#3122
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #31774501C2semidirect
2. #31774369C2semidirect
3. #31774220C2semidirect
4. #31774219C2semidirect
5. #31774198C2semidirect
6. #31774197C2semidirect
7. #31774089C2semidirect
8. #31774029C2semidirect
9. #31774029C2semidirect
10. #31774009C2semidirect
11. #31774006C2semidirect
12. D8×Dic271C2semidirect
13. #31773977C2semidirect
14. #31773957C2semidirect
15. #459×C271C2semidirect
19. #22576152C22semidirect
28. D4×Dic271C22semidirect
29. D4×Dic271C22semidirect
37. #22576094C22semidirect
38. #22576093C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. C4⋊(C271⋊C8)C22semidirect
46. #22576081C22semidirect
47. #22576081C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
54. #127×C271C22semidirect
56. C4⋊C8×C271C22semidirect
57. #102×C271C22semidirect
58. #102×C271C22semidirect
187. C271#3122semidirect
196. #459D271semidirect
209. #127C2×D271semidirect
234. C4⋊C8C2×D271semidirect
241. #102C2×D271semidirect
242. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745012
#317743692
#317742202
#317742192
#317741982
#317741972
#317740892
#317740292
#317740092
#317740062
D8×Dic2712
#317739772
#317739572
#459×C2712
#3122271