Finite Group Databasebeta

← #46849066

Subgroups — #46849066

Order 34688.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774606C211yesmaximal, characteristic, complemented
2#31774567C211yesmaximal, characteristic, complemented
3#31774261C211yesmaximal, characteristic, complemented
4#31774260C211yesmaximal, characteristic, complemented
5#31774242C211yesmaximal, characteristic, complemented
6#31774198C211yesmaximal, characteristic, complemented
7#31774195C211yesmaximal, characteristic, complemented
8#31774185C211yesmaximal, characteristic, complemented
9#31774090C211yesmaximal, characteristic, complemented
10#31774090C211yesmaximal, characteristic, complemented
11#31774026C211yesmaximal, characteristic, complemented
12#31774009C211yesmaximal, characteristic, complemented
13#31774007C211yesmaximal, characteristic, complemented
14D8×Dic271C211yesmaximal, characteristic, complemented
15#459×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864639C2211yescharacteristic
18#22576159C2211yescharacteristic
19#22576156C2211yescharacteristic
20#22576156C2211yescharacteristic
21#22576152C2211yescharacteristic
22#22576151C2211yescharacteristic
23#22576139—21no—
24#22576137—21no—
25#22576137—21no—
26#22576132—21no—
27#22576132—21no—
28#22576131—21no—
29C4×Dic542C2211yescharacteristic
30C4×D1084C2211yescharacteristic
31D4×Dic271C2211yescharacteristic
32D4×Dic271C2211yescharacteristic
33D4×Dic271C2211yescharacteristic
34D4×Dic271C2211yescharacteristic
35C2×Dic271⋊C4—21no—
36#22576094C2211yescharacteristic
37#22576092C2211yescharacteristic
38#22576091C2211yescharacteristic
39Dic271⋊C8C2211yescharacteristic
40Dic271⋊C8C2211yescharacteristic
41#22576082C2211yescharacteristic
42#22576082C2211yescharacteristic
43#22576081C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576071C2211yescharacteristic
50#22576070C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×D8×C271C2211yescharacteristic
53C2×D8×C271C2211yescharacteristic
54#127×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic
56C4⋊C8×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×Dic542D411yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C22×Dic271—21no—
68C22×Dic271—21no—
69C22×Dic271—21no—
70C22×Dic271—21no—
71C2×C4×D271—21no—
72#16864639—22no—
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C4⋊Dic271—21no—
76C4⋊Dic271D411yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4D411yescharacteristic
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90(C2×D271)⋊C4—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93C2×D4×C271—21no—
94C2×D4×C271—21no—
95C2×D4×C271C2311yescharacteristic
96C2×D4×C271C2311yescharacteristic
97C2×D4×C271C2311yescharacteristic
98C22×C4×C271—21no—
99D8×C271—22no—
100D8×C271—22no—
101C2×C8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C22⋊C4×C271—21no—
105C42×C271C2311yescharacteristic
106C22×D271—21no—
107Dic542—22no—
108Dic542—81no—
109D1084—21no—
110D1084—21no—
111C22⋊D271—42no—
112C22⋊D271—42no—
113C22⋊D271—81no—
114C22⋊D271—81no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—42no—
119C2×Dic271—42no—
120C2×Dic271—42no—
121C2×Dic271—81no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C4×D271—42no—
127C271⋊C8—21no—
128C271⋊C8—21no—
129C23×C271—21no—
130C23×C271—21no—
131C23×C271—21no—
132D4×C271—21no—
133D4×C271—21no—
134D4×C271—21no—
135D4×C271—21no—
136D4×C271—21no—
137D4×C271—42no—
138D4×C271—81no—
139D4×C271#4112yescomplemented
140C2×C4×C271—21no—
141C2×C4×C271—81no—
142C2×C4×C271C2×D411yescharacteristic
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2411yescharacteristic
145C8×C271—21no—
146C8×C271—21no—
147C22×C271—22no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271C22×D411yescharacteristic
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C4×C271—21no—
161C4×C271—21no—
162C4×C271—41no—
163C4×C271C2×#4111yescharacteristic
164C4×C2712−1+411yescharacteristic
165Dic271—21no—
166Dic271—21no—
167Dic271—21no—
168Dic271—41no—
169Dic271—41no—
170Dic271—41no—
171Dic271AGL(1,ℤ/8)12yes—
172C2×C271—42no—
173C2×C271—81no—
174C2×C271—81no—
175C2×C271#54811yescharacteristic
176C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C271#57611yescharacteristic
178D271—81no—
179C271#336611yescharacteristic, complemented, Sylow
180#3366—2711nomaximal, Sylow
181#554—2711no—
182#550—2711no—
183#490—2711no—
184#490—2711no—
185#486—2711no—
186#478—2711no—
187#478—2711no—
188#476—2711no—
189#459—2711no—
190#459—2711no—
191#459D27111yescharacteristic, complemented
192#446—2711no—
193#442—2711no—
194#442—2711no—
195C4×D8—2711no—
196C2×#41—5421no—
197C2×D8—2711no—
198C2×D8C2×D27111yescharacteristic
199C2×D8C2×D27111yescharacteristic
200C4⋊Q8—2711no—
201#127—2711no—
202#127—2711no—
203#127C2×D27111yescharacteristic
204#125—2711no—
205#125—2711no—
206C22⋊Q8—5421no—
207C22⋊Q8—5421no—
208C22⋊Q8—5421no—
209#121—5421no—
210#121—5421no—
211#121—5421no—
212C4×Q8—2711no—
213C4×D4—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—2711no—
217C4×D4—2711no—
218C4×D4C2×D27111yescharacteristic
219C2×C4⋊C4—5421no—
220#107—2711no—
221#106—2711no—
222#106—2711no—
223C4⋊C8—2711no—
224C4⋊C8—2711no—
225C4⋊C8C2×D27111yescharacteristic
226#102—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102C2×D27111yescharacteristic
235#102C2×D27111yescharacteristic
236#97—2711no—
237#97—2711no—
238C4×C8—2711no—
239#41—10844no—
240C2×Q8—2711no—
241C2×Q8—5421no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—2711no—
245C2×D4—5421no—
246C2×D4—5421no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4C22×D27111yescharacteristic
250C2×D4C22×D27111yescharacteristic
251C2×D4C22×D27111yescharacteristic
252C22×C4—21no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4—5421no—
257C22×C4—5421no—
258D8—22no—
259D8—22no—
260D8—5422no—
261C2×C8—2711no—
262C2×C8—2711no—
263C2×C8C22×D27111yescharacteristic
264C2×C8C22×D27111yescharacteristic
265C4⋊C4—2711no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—2711no—
270C4⋊C4—2711no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C42—2711no—
284C42—2711no—
285C42C22×D27111yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—5421no—
290Q8—5422no—
291Q8—21681no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—81no—
299D4—5421no—
300D4—5421no—
301D4—10842no—
302D4—10842no—
303D4—21681no—
304D4—21681no—
305D4#1686465312yescomplemented
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—2711no—
309C2×C4—2711no—
310C2×C4—2711no—
311C2×C4—2711no—
312C2×C4—5421no—
313C2×C4—5421no—
314C2×C4—5421no—
315C2×C4—10842no—
316C2×C4—10842no—
317C2×C4—10842no—
318C2×C4—10842no—
319C2×C4—21681no—
320C2×C4D4×D27111yescharacteristic
321C2×C4D4×D27111yescharacteristic
322C2×C4C23×D27111yescharacteristic
323C8—21no—
324C8—21no—
325C8—5421no—
326C8—5421no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—10841no—
337C22—10841no—
338C22—10841no—
339C22C2×D4×D27111yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—2712no—
344C4—5421no—
345C4—5421no—
346C4—5421no—
347C4—10841no—
348C4—10841no—
349C4—10841no—
350C4C2×#1686465311yescharacteristic
351C4#2257620711yescharacteristic
352C2—42no—
353C2—81no—
354C2—81no—
355C2—21681no—
356C2#3177455311yescharacteristic
357C2AGL(1,ℤ/8)×D27111yescharacteristic
358C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#459×C271C2
socleC22×C271C22×D4
hypercenter#459D271
hypocenterC271#3366
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. #31774606C2semidirect
2. #31774567C2semidirect
3. #31774261C2semidirect
4. #31774260C2semidirect
5. #31774242C2semidirect
6. #31774198C2semidirect
7. #31774195C2semidirect
8. #31774185C2semidirect
9. #31774090C2semidirect
10. #31774090C2semidirect
11. #31774026C2semidirect
12. #31774009C2semidirect
13. #31774007C2semidirect
14. D8×Dic271C2semidirect
15. #459×C271C2semidirect
139. D4×C271#41semidirect
179. C271#3366semidirect
191. #459D271semidirect
305. D4#16864653semidirect

Maximal subgroups

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

subgroupindex
#317746062
#317745672
#317742612
#317742602
#317742422
#317741982
#317741952
#317741852
#317740902
#317740262
#317740092
#317740072
D8×Dic2712
#459×C2712
#3366271