Finite Group Databasebeta

← #46848299

Subgroups — #46848299

Order 34688.

396 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774615C211yesmaximal, characteristic, complemented
2#31774566C211yesmaximal, characteristic
3#31774242C211yesmaximal, characteristic, complemented
4#31774221C211yesmaximal, characteristic, complemented
5#31774220C211yesmaximal, characteristic, complemented
6#31774180C211yesmaximal, characteristic, complemented
7#31774126C211yesmaximal, characteristic, complemented
8#31774125C211yesmaximal, characteristic, complemented
9#31774096C211yesmaximal, characteristic, complemented
10#31774062C211yesmaximal, characteristic, complemented
11#31774059C211yesmaximal, characteristic, complemented
12#31773989C211yesmaximal, characteristic, complemented
13C8×C22⋊D271C211yesmaximal, characteristic, complemented
14#31773892C211yesmaximal, characteristic, complemented
15#416×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864644C2211yescharacteristic
18C2×#16864643C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576156—21no—
23#22576156C2211yescharacteristic
24#22576154—21no—
25#22576153—21no—
26#22576146—21no—
27#22576140C2211yescharacteristic
28#22576139C2211yescharacteristic
29#22576134—21no—
30#22576134C2211yescharacteristic
31#22576131—21no—
32#22576131C2211yescharacteristic
33Q8×Dic271C2211yescharacteristic
34C4×Dic542—21no—
35#22576125C2211yescharacteristic
36C4×C22⋊D271—21no—
37C4×C22⋊D271C2211yescharacteristic
38#22576118C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576092C2211yescharacteristic
41#22576091C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54#117×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—41no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271D411yescharacteristic
70C2×C4×D271—21no—
71C2×C4×D271C2311yescharacteristic
72#16864644—22no—
73#16864643—41no—
74#16864642—41no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C8×D271—41no—
78C4⋊Dic271—21no—
79C4⋊Dic271—21no—
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271D411yescharacteristic
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4D411yescharacteristic
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104C2×C8×C271—41no—
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C42×C271—21no—
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—42no—
116D1084—21no—
117D1084—21no—
118D1084—42no—
119C22⋊D271—22no—
120C22⋊D271—42no—
121C22⋊D271—81no—
122C22⋊D271—81no—
123C22⋊D271—81no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136C23×C271C2×D411yescharacteristic
137Q8×C271—21no—
138Q8×C271#4112yes—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—42no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—42no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C4×C271—21no—
166C4×C271—22no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271H(5,𝔽2)11yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—41no—
174Dic271—41no—
175Dic271—41no—
176Dic271#13512yes—
177C2×C271—41no—
178C2×C271—81no—
179C2×C271#54611yescharacteristic
180C2×C271C2×#13511yescharacteristic
181C2×C271#57711yescharacteristic
182D271—41no—
183D271—81no—
184C271#330411yescharacteristic, complemented, Sylow
185#3304—2711nomaximal, Sylow
186#556—2711no—
187#550—2711no—
188#486—2711no—
189#482—2711no—
190#482—2711no—
191#475—2711no—
192#465—2711no—
193#465—2711no—
194#460—2711no—
195#452—2711no—
196#452—2711no—
197C4×Q16—2711no—
198C8×D4—2711no—
199#416—2711no—
200#416D27111yescharacteristic, complemented
201C2×#41—2711no—
202C2×#41C2×D27111yescharacteristic
203C2×Q16—2711no—
204C2×SD16—2711no—
205C2×SD16—2711no—
206C22×C8—2711no—
207C22×C8C2×D27111yescharacteristic
208C4⋊Q8—2711no—
209#127—2711no—
210#127—5421no—
211#126—5421no—
212#126—5421no—
213#124—5421no—
214C22⋊Q8—2711no—
215C22⋊Q8—2711no—
216#121—2711no—
217#121—2711no—
218#121—5421no—
219#121—5421no—
220C4×Q8—2711no—
221C4×Q8—2711no—
222C4×Q8—5421no—
223C4×D4—2711no—
224C4×D4—5421no—
225#117—2711no—
226#117C2×D27111yescharacteristic
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103—2711no—
233#103C2×D27111yescharacteristic
234#103C2×D27111yescharacteristic
235#102—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102C2×D27111yescharacteristic
240#102C2×D27111yescharacteristic
241C22⋊C8—2711no—
242C22⋊C8—2711no—
243C4×C8—2711no—
244#41—22no—
245#41—41no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8—5421no—
250C2×Q8C22×D27111yescharacteristic
251C2×D4—21no—
252C2×D4—2711no—
253C2×D4—2711no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C2×D4—5421no—
258C2×D4—5421no—
259C2×D4C22×D27111yescharacteristic
260C22×C4—21no—
261C22×C4—2711no—
262C22×C4—5421no—
263C22×C4C22×D27111yescharacteristic
264Q16—5422no—
265SD16—10841no—
266SD16—10841no—
267C2×C8—41no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—10841no—
271C2×C8C22×D27111yescharacteristic
272C2×C8C22×D27111yescharacteristic
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4C22×D27111yescharacteristic
282C4⋊C4C22×D27111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2711no—
285C22⋊C4—2711no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C42—21no—
292C42—2711no—
293C42—5421no—
294C42—5421no—
295C42—5421no—
296C23—21no—
297C23—2711no—
298C23—5421no—
299C23D4×D27111yescharacteristic
300Q8—21no—
301Q8—5421no—
302Q8—5421no—
303Q8—10842no—
304Q8#1686465412yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—5422no—
309D4—5421no—
310D4—5421no—
311D4—10842no—
312D4—10842no—
313D4—21681no—
314D4—21681no—
315D4—21681no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—2711no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—10842no—
331C2×C4D4×D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5421no—
336C8—5421no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—5421no—
344C22—5421no—
345C22—5421no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—2712no—
355C4—5421no—
356C4—5421no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4C2×#1686465411yescharacteristic
361C4#2257620511yescharacteristic
362C2—41no—
363C2—81no—
364C2—10841no—
365C2—21681no—
366C2#3177453711yescharacteristic
367C2#135×D27111yescharacteristic
368C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#416×C271C2
socleC22×C271C22×D4
prime radical#31774566C2
hypercenter#416D271
hypocenterC271#3304
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #31774615C2semidirect
3. #31774242C2semidirect
4. #31774221C2semidirect
5. #31774220C2semidirect
6. #31774180C2semidirect
7. #31774126C2semidirect
8. #31774125C2semidirect
9. #31774096C2semidirect
10. #31774062C2semidirect
11. #31774059C2semidirect
12. #31773989C2semidirect
13. C8×C22⋊D271C2semidirect
14. #31773892C2semidirect
15. #416×C271C2semidirect
184. C271#3304semidirect
200. #416D271semidirect

Maximal subgroups

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

subgroupindex
#317746152
#317745662
#317742422
#317742212
#317742202
#317741802
#317741262
#317741252
#317740962
#317740622
#317740592
#317739892
C8×C22⋊D2712
#317738922
#416×C2712
#3304271