Finite Group Databasebeta

← #46846461

Subgroups — #46846461

Order 34688.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774451C211yesmaximal, characteristic, complemented
2C4×D4×D271C211yesmaximal, characteristic
3#31774234C211yesmaximal, characteristic, complemented
4#31774231C211yesmaximal, characteristic, complemented
5#31774176C211yesmaximal, characteristic, complemented
6#31774175C211yesmaximal, characteristic, complemented
7#31774101C211yesmaximal, characteristic, complemented
8#31774100C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic
10#31774046C211yesmaximal, characteristic
11#31773999C211yesmaximal, characteristic, complemented
12#31773997C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773952C211yesmaximal, characteristic, complemented
15#440×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C22×C4×D271—21no—
18C2×#16864643C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864641C2211yescharacteristic
21#22576159C2211yescharacteristic, complemented
22#22576158C2211yescharacteristic, complemented
23#22576150—21no—
24#22576140C2211yescharacteristic
25#22576138—21no—
26#22576136C2211yescharacteristic
27C4×Dic542—21no—
28C4×Dic542C2211yescharacteristic, complemented
29C4×D1084C2211yescharacteristic
30D4×Dic271C2211yescharacteristic
31#22576122C2211yescharacteristic
32C4×C22⋊D271—21no—
33(C22⋊D271)⋊C4—21no—
34#22576117—21no—
35C4⋊C4×D271C2211yescharacteristic
36C22⋊C4×D271—21no—
37C42×D271C2211yescharacteristic
38#22576094C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576093C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576083C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576071C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic
53C4×Q8×C271C2211yescharacteristic, complemented
54C4×D4×C271C2211yescharacteristic
55#107×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C2×Dic542—21no—
61C2×Dic542C2311yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64D4×D271—22no—
65D4×D271—22no—
66C22×Dic271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—44no—
71C2×C4×D271D411yescharacteristic
72C2×C4×D271D411yescharacteristic
73C2×C4×D271C2311yescharacteristic
74#16864643—22no—
75#16864642—22no—
76#16864641—22no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C4⋊Dic271—21no—
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88C22⋊Dic271—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93C4×Dic271—21no—
94C4×Dic271D411yescharacteristic
95C4×Dic271D411yescharacteristic
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271—21no—
100SD16×C271—22no—
101C2×C8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C4⋊C4×C271—21no—
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C42×C271C2311yescharacteristic
109C22×D271—24no—
110C22×D271—21no—
111C22×D271—42no—
112C22×D271—42no—
113C22×D271C2×D411yescharacteristic
114Dic542—21no—
115Dic542—21no—
116Dic542—42no—
117D1084—21no—
118D1084#4112yes—
119C22⋊D271—24no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—42no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—22no—
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—22no—
131C4×D271—42no—
132C4×D271—42no—
133C4×D271—42no—
134C271⋊C8—21no—
135C271⋊C8—21no—
136C23×C271—21no—
137Q8×C271—21no—
138Q8×C271—21no—
139D4×C271—21no—
140D4×C271#4112yes—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—42no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C8×C271—21no—
149C8×C271—21no—
150C22×C271—22no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271C22×D411yescharacteristic
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—22no—
157C2×D271—48no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271#13712yes—
161C4×C271—21no—
162C4×C271—21no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271—41no—
166C4×C271C2×#4111yescharacteristic
167C4×C271C2×#4111yescharacteristic
168Dic271—21no—
169Dic271—21no—
170Dic271—41no—
171Dic271—41no—
172Dic271—41no—
173Dic271AGL(1,ℤ/8)12yes—
174C2×C271—42no—
175C2×C271#52511yescharacteristic
176C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C271C2×#13711yescharacteristic
178D271—22no—
179D271—42no—
180C271#311011yescharacteristic, complemented, Sylow
181#3110—2711nomaximal, Sylow
182#533—2711no—
183C2×C4×D4—2711no—
184#484—2711no—
185#484—2711no—
186#474—2711no—
187#474—2711no—
188#461—2711no—
189#461—2711no—
190#450—2711no—
191#450—2711no—
192#440—2711no—
193#440—2711no—
194#440—2711no—
195#440D27111yescharacteristic, complemented
196#432—2711no—
197C22×D4—2711no—
198C23×C4—5421no—
199C2×SD16—2711no—
200C2×SD16—2711no—
201C2×SD16—2711no—
202C2×SD16C2×D27111yescharacteristic
203C4⋊Q8—2711no—
204C4⋊Q8—2711no—
205#125—5421no—
206C22⋊Q8—2711no—
207C22⋊Q8—2711no—
208C22⋊Q8—5421no—
209C4×Q8—2711no—
210C4×Q8—5421no—
211C4×Q8C2×D27111yescharacteristic, complemented
212C4×D4—2711no—
213C4×D4—2711no—
214C4×D4—2711no—
215C4×D4—5421no—
216C4×D4—5421no—
217C4×D4C2×D27111yescharacteristic
218#117—5421no—
219C2×C4⋊C4—2711no—
220C2×C22⋊C4—5421no—
221C2×C42—2711no—
222#107—2711no—
223#107—2711no—
224#107—2711no—
225#107C2×D27111yescharacteristic, complemented
226C4⋊C8—2711no—
227C4⋊C8—2711no—
228#103—2711no—
229#103—2711no—
230#103—2711no—
231#103C2×D27111yescharacteristic, complemented
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102C2×D27111yescharacteristic
236C22⋊C8—2711no—
237C22⋊C8—2711no—
238#97—2711no—
239#97C2×D27111yescharacteristic, complemented
240C24—5421no—
241C2×Q8—2711no—
242C2×Q8—5421no—
243C2×Q8C22×D27111yescharacteristic
244C2×D4—2711no—
245C2×D4—5421no—
246C2×D4—5422no—
247C2×D4—5422no—
248C2×D4C22×D27111yescharacteristic
249C22×C4—21no—
250C22×C4—2711no—
251C22×C4—2711no—
252C22×C4—2711no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4—5421no—
257C22×C4—10844no—
258SD16—22no—
259SD16—5422no—
260SD16—5422no—
261SD16—5422no—
262C2×C8—2711no—
263C2×C8—2711no—
264C2×C8C22×D27111yescharacteristic
265C2×C8C22×D27111yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4C22×D27111yescharacteristic
277C4⋊C4C22×D27111yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C42—21no—
285C42—2711no—
286C42—2711no—
287C42—2711no—
288C42—5421no—
289C42C22×D27111yescharacteristic
290C23—21no—
291C23—2711no—
292C23—5424no—
293C23—5421no—
294C23—10842no—
295C23—10842no—
296Q8—21no—
297Q8—21no—
298Q8—5421no—
299Q8—5421no—
300Q8—10842no—
301D4—21no—
302D4—2712no—
303D4—5424no—
304D4—5421no—
305D4#1686465412yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—2711no—
311C2×C4—2711no—
312C2×C4—2711no—
313C2×C4—5422no—
314C2×C4—5422no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—10842no—
321C2×C4—10842no—
322C2×C4—10842no—
323C2×C4—10842no—
324C2×C4D4×D27111yescharacteristic
325C2×C4D4×D27111yescharacteristic
326C2×C4C23×D27111yescharacteristic
327C8—21no—
328C8—21no—
329C8—5421no—
330C8—5421no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—2712no—
335C22—5421no—
336C22—5421no—
337C22—5422no—
338C22—10848no—
339C22—10841no—
340C22—10841no—
341C22C2×D4×D27111yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—2712no—
348C4—5421no—
349C4—5421no—
350C4—10841no—
351C4—10841no—
352C4—10841no—
353C4#41×D27111yescharacteristic
354C4C2×#1686465411yescharacteristic
355C2—42no—
356C2—5422no—
357C2—10842no—
358C2#3177440611yescharacteristic
359C2AGL(1,ℤ/8)×D27111yescharacteristic
360C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#440×C271C2
socleC22×C271C22×D4
prime radicalC2×D4×D271C22
hypercenter#440D271
hypocenterC271#3110
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #31774451C2semidirect
3. #31774234C2semidirect
4. #31774231C2semidirect
5. #31774176C2semidirect
6. #31774175C2semidirect
7. #31774101C2semidirect
8. #31774100C2semidirect
11. #31773999C2semidirect
12. #31773997C2semidirect
13. #31773995C2semidirect
14. #31773952C2semidirect
15. #440×C271C2semidirect
21. #22576159C22semidirect
22. #22576158C22semidirect
28. C4×Dic542C22semidirect
38. #22576094C22semidirect
39. #22576093C22semidirect
40. #22576093C22semidirect
41. Dic271⋊C8C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576085C22semidirect
44. #22576084C22semidirect
45. #22576083C22semidirect
51. #22576071C22semidirect
53. C4×Q8×C271C22semidirect
55. #107×C271C22semidirect
56. #103×C271C22semidirect
58. #97×C271C22semidirect
180. C271#3110semidirect
195. #440D271semidirect
211. C4×Q8C2×D271semidirect
225. #107C2×D271semidirect
231. #103C2×D271semidirect
239. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744512
C4×D4×D2712
#317742342
#317742312
#317741762
#317741752
#317741012
#317741002
#317740492
#317740462
#317739992
#317739972
#317739952
#317739522
#440×C2712
#3110271