Finite Group Databasebeta

← #468×D271 #46845948

Subgroups — #468×D271 #46845948

Order 34688.

544 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D271C211yesmaximal, characteristic, complemented
2C2×M16×D271C211yesmaximal, characteristic, complemented
3C22⋊Q8×D271C211yesmaximal, characteristic, complemented
4#121×D271C211yesmaximal, characteristic
5#31774146C211yesmaximal, characteristic, complemented
6#31774145C211yesmaximal, characteristic, complemented
7#31774144C211yesmaximal, characteristic, complemented
8#31774143C211yesmaximal, characteristic, complemented
9#31774142C211yesmaximal, characteristic, complemented
10#31774141C211yesmaximal, characteristic, complemented
11#31774140C211yesmaximal, characteristic, complemented
12#107×D271C211yesmaximal, characteristic, complemented
13#103×D271C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#468×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17C2×D4×D271—21no—
18C2×D4×D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20C2×#16864643C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic, complemented
22C2×#16864641C2211yescharacteristic, complemented
23SD16×D271—24no—
24C2×#16864638C2211yescharacteristic, complemented
25C2×#16864637C2211yescharacteristic, complemented
26C2×#16864637C2211yescharacteristic, complemented
27M16×D271—24no—
28C2×C8×D271C2211yescharacteristic
29C2×C8×D271C2211yescharacteristic
30#22576140C2211yescharacteristic, complemented
31#22576139C2211yescharacteristic, complemented
32#22576138—21no—
33#22576137—21no—
34#22576136C2211yescharacteristic, complemented
35#22576135C2211yescharacteristic
36#22576134C2211yescharacteristic
37#22576133—21no—
38#22576132—21no—
39#22576131C2211yescharacteristic
40C4⋊C4×D271—21no—
41C4⋊C4×D271C2211yescharacteristic
42C4⋊C4×D271C2211yescharacteristic
43C22⋊C4×D271—21no—
44C22⋊C4×D271—21no—
45#22576094C2211yescharacteristic, complemented
46#22576093C2211yescharacteristic, complemented
47#22576093C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576084C2211yescharacteristic, complemented
50#22576083C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic, complemented
52#22576081C2211yescharacteristic, complemented
53#22576080C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×M16×C271C2211yescharacteristic, complemented
56C22⋊Q8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#107×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271D411yescharacteristic
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65Q8×D271—22no—
66Q8×D271—22no—
67C2×D1084—21no—
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72D4×D271—22no—
73D4×D271—22no—
74D4×D271—44no—
75C22×Dic271D411yescharacteristic
76C2×C4×D271—22no—
77C2×C4×D271—22no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271D411yescharacteristic
82C2×C4×D271D411yescharacteristic
83C2×C4×D271C2311yescharacteristic
84C2×C4×D271C2311yescharacteristic
85#16864643—22no—
86#16864642—22no—
87#16864641—22no—
88#16864638—22no—
89#16864637—22no—
90#16864637—22no—
91C2×C271⋊C8C2311yescharacteristic
92C2×C271⋊C8C2311yescharacteristic
93C8×D271—22no—
94C8×D271D414yescomplemented
95C4⋊Dic271—21no—
96C4⋊Dic271C2311yescharacteristic
97C4⋊Dic271C2311yescharacteristic
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100Dic271⋊C4—21no—
101Dic271⋊C4—21no—
102C22⋊Dic271—21no—
103C22⋊Dic271—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271—21no—
110C2×D4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112SD16×C271—22no—
113M16×C271—22no—
114C2×C8×C271C2311yescharacteristic
115C2×C8×C271C2311yescharacteristic
116C4⋊C4×C271—21no—
117C4⋊C4×C271C2311yescharacteristic
118C4⋊C4×C271C2311yescharacteristic
119C22⋊C4×C271—21no—
120C22⋊C4×C271—21no—
121C22×D271—22no—
122C22×D271—22no—
123C22×D271—22no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271C2×D411yescharacteristic
129C22×D271C2×D411yescharacteristic
130Dic542—21no—
131Dic542—21no—
132Dic542—42no—
133D1084—21no—
134D1084—21no—
135D1084—42no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C22⋊D271—42no—
139C2×Dic271—22no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271C2×D411yescharacteristic
144C2×Dic271C2×D411yescharacteristic
145C4×D271—22no—
146C4×D271—21no—
147C4×D271—21no—
148C4×D271—22no—
149C4×D271—42no—
150C4×D271—42no—
151C4×D271—42no—
152C4×D271C2×D412yes—
153C4×D271#4112yes—
154C271⋊C8—21no—
155C271⋊C8C2×D412yescomplemented
156C23×C271—21no—
157C23×C271C2×D411yescharacteristic
158Q8×C271—21no—
159Q8×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—42no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2411yescharacteristic
170C8×C271—21no—
171C8×C271C2×D412yescomplemented
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271C22×D411yescharacteristic
179C2×D271—21no—
180C2×D271—21no—
181C2×D271—21no—
182C2×D271—44no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—84no—
187C2×D271#12112yes—
188C2×D271AGL(1,ℤ/8)12yes—
189C2×D271#13712yes—
190C4×C271—21no—
191C4×C271—41no—
192C4×C271—41no—
193C4×C271—41no—
194C4×C271C22×D411yescharacteristic
195C4×C271C2×#4111yescharacteristic
196Dic271—21no—
197Dic271—41no—
198Dic271—41no—
199Dic271—41no—
200Dic271#12112yes—
201C2×C271—41no—
202C2×C271—81no—
203C2×C271C2×#12111yescharacteristic
204C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C271C2×#13711yescharacteristic
206D271—41no—
207D271—81no—
208D271#46814yesdirect product factor
209C271C2×#46811yescharacteristic, complemented, Sylow
210C2×#468—2711nomaximal, Sylow
211C22×SD16—2711no—
212C22×M16—2711no—
213C2×C22⋊Q8—2711no—
214C2×#121—2711no—
215#468—2711no—
216#468—2711no—
217#468—2711no—
218#468—2711no—
219#468—2711no—
220#468—2711no—
221#468—2711no—
222#468D27111yescharacteristic, direct product factor
223C2×#107—2711no—
224C2×#103—2711no—
225C2×#102—2711no—
226C22×Q8—2711no—
227C22×D4—2711no—
228C22×D4—5421no—
229C23×C4—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16—5424no—
234C2×SD16C2×D27111yescharacteristic, complemented
235C2×M16—2711no—
236C2×M16—2711no—
237C2×M16—2711no—
238C2×M16—5424no—
239C2×M16C2×D27111yescharacteristic, complemented
240C22×C8—2711no—
241C22×C8—2711no—
242C22⋊Q8—2711no—
243C22⋊Q8—2711no—
244C22⋊Q8—2711no—
245C22⋊Q8—5421no—
246C22⋊Q8—5421no—
247C22⋊Q8C2×D27111yescharacteristic, complemented
248#121—2711no—
249#121—2711no—
250#121—2711no—
251#121—5421no—
252#121—5421no—
253#121C2×D27111yescharacteristic
254C2×C4⋊C4—2711no—
255C2×C4⋊C4—2711no—
256C2×C4⋊C4—5421no—
257C2×C22⋊C4—5421no—
258C2×C22⋊C4—5421no—
259#107—2711no—
260#107—2711no—
261#107—2711no—
262#107C2×D27111yescharacteristic, complemented
263#103—2711no—
264#103—2711no—
265#103—2711no—
266#103C2×D27111yescharacteristic, complemented
267#102—2711no—
268#102—2711no—
269#102—2711no—
270#102C2×D27111yescharacteristic, complemented
271C24—2711no—
272C24—5421no—
273C2×Q8—2711no—
274C2×Q8—5421no—
275C2×Q8—5422no—
276C2×Q8—5422no—
277C2×Q8C22×D27111yescharacteristic
278C2×D4—21no—
279C2×D4—2711no—
280C2×D4—5421no—
281C2×D4—5422no—
282C2×D4—5422no—
283C2×D4—5421no—
284C2×D4—5421no—
285C2×D4—5421no—
286C2×D4—10844no—
287C2×D4C22×D27111yescharacteristic
288C22×C4—2711no—
289C22×C4—2711no—
290C22×C4—2711no—
291C22×C4—2711no—
292C22×C4—2711no—
293C22×C4—5422no—
294C22×C4—5422no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4—5421no—
298C22×C4C22×D27111yescharacteristic
299SD16—22no—
300SD16—5422no—
301SD16—5422no—
302SD16—5422no—
303M16—22no—
304M16—5422no—
305M16—5422no—
306M16—5422no—
307C2×C8—2711no—
308C2×C8—2714no—
309C2×C8—2711no—
310C2×C8—5422no—
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2711no—
315C4⋊C4—2711no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C4⋊C4—5421no—
319C4⋊C4—5421no—
320C4⋊C4—5421no—
321C4⋊C4C22×D27111yescharacteristic
322C4⋊C4C22×D27111yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4—5421no—
330C22⋊C4—5421no—
331C23—21no—
332C23—2711no—
333C23—2711no—
334C23—5422no—
335C23—5422no—
336C23—5422no—
337C23—5421no—
338C23—10842no—
339C23—10842no—
340C23—10842no—
341C23D4×D27111yescharacteristic
342Q8—21no—
343Q8—21no—
344Q8—5421no—
345Q8—5421no—
346Q8—10842no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—5421no—
351D4—5421no—
352D4—10842no—
353D4—10842no—
354D4—10842no—
355D4—10842no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—2711no—
362C2×C4—2712no—
363C2×C4—2712no—
364C2×C4—2711no—
365C2×C4—5422no—
366C2×C4—5422no—
367C2×C4—5422no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—5421no—
373C2×C4—10842no—
374C2×C4—10842no—
375C2×C4—10842no—
376C2×C4D4×D27111yescharacteristic
377C2×C4C23×D27111yescharacteristic
378C8—21no—
379C8—2712no—
380C8—5421no—
381C8D4×D27112yescomplemented
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—2712no—
389C22—2712no—
390C22—2712no—
391C22—5421no—
392C22—5421no—
393C22—5421no—
394C22—10844no—
395C22—10841no—
396C22—10841no—
397C22—10841no—
398C22—21684no—
399C22C2×D4×D27111yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—2712no—
405C4—5421no—
406C4—10841no—
407C4—10841no—
408C4—10841no—
409C4C2×D4×D27111yescharacteristic
410C4#41×D27111yescharacteristic
411C2—41no—
412C2—81no—
413C2—2714no—
414C2—10841no—
415C2—21681no—
416C2#121×D27111yescharacteristic
417C2AGL(1,ℤ/8)×D27111yescharacteristic
418C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#468×C271C2
socleC22×C271C22×D4
prime radical#121×D271C2
hypercenter#468D271
hypocenterC271C2×#468
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D271C2semidirect
2. C2×M16×D271C2semidirect
3. C22⋊Q8×D271C2semidirect
5. #31774146C2semidirect
6. #31774145C2semidirect
7. #31774144C2semidirect
8. #31774143C2semidirect
9. #31774142C2semidirect
10. #31774141C2semidirect
11. #31774140C2semidirect
12. #107×D271C2semidirect
13. #103×D271C2semidirect
14. #102×D271C2semidirect
15. #468×C271C2semidirect
20. C2×#16864643C22semidirect
21. C2×#16864642C22semidirect
22. C2×#16864641C22semidirect
24. C2×#16864638C22semidirect
25. C2×#16864637C22semidirect
26. C2×#16864637C22semidirect
30. #22576140C22semidirect
31. #22576139C22semidirect
34. #22576136C22semidirect
45. #22576094C22semidirect
46. #22576093C22semidirect
47. #22576093C22semidirect
48. #22576085C22semidirect
49. #22576084C22semidirect
50. #22576083C22semidirect
51. #22576082C22semidirect
52. #22576081C22semidirect
53. #22576080C22semidirect
54. C2×SD16×C271C22semidirect
55. C2×M16×C271C22semidirect
56. C22⋊Q8×C271C22semidirect
58. #107×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
94. C8×D271D4semidirect
155. C271⋊C8C2×D4semidirect
171. C8×C271C2×D4semidirect
208. D271#468direct (N × H)
209. C271C2×#468semidirect
222. #468D271direct (N × H)
234. C2×SD16C2×D271semidirect
239. C2×M16C2×D271semidirect
247. C22⋊Q8C2×D271semidirect
262. #107C2×D271semidirect
266. #103C2×D271semidirect
270. #102C2×D271semidirect
381. C8D4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2712
C2×M16×D2712
C22⋊Q8×D2712
#121×D2712
#317741462
#317741452
#317741442
#317741432
#317741422
#317741412
#317741402
#107×D2712
#103×D2712
#102×D2712
#468×C2712
C2×#468271