Finite Group Databasebeta

← #46847855

Subgroups — #46847855

Order 34688.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576186C211yesmaximal, characteristic, complemented
2C22×#16864642C211yesmaximal, characteristic, complemented
3#31774556C211yesmaximal, characteristic, complemented
4#31774284C211yesmaximal, characteristic, complemented
5#31774141C211yesmaximal, characteristic, complemented
6#31774131C211yesmaximal, characteristic, complemented
7#31774112C211yesmaximal, characteristic, complemented
8#31774094C211yesmaximal, characteristic, complemented
9#31774048C211yesmaximal, characteristic, complemented
10#31774040C211yesmaximal, characteristic, complemented
11#31774035C211yesmaximal, characteristic, complemented
12#31774031C211yesmaximal, characteristic, complemented
13#31773997C211yesmaximal, characteristic, complemented
14#31773969C211yesmaximal, characteristic, complemented
15D42×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic, complemented
17C22×Dic542C2211yescharacteristic, complemented
18#22576186—24no—
19C2×#16864642—24no—
20C2×#16864642C2211yescharacteristic
21C2×#16864642C2211yescharacteristic
22C2×#16864642C2211yescharacteristic, complemented
23C2×#16864642C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic, complemented
25C2×#16864642C2211yescharacteristic, complemented
26C2×#16864639C2211yescharacteristic
27C2×#16864639C2211yescharacteristic, complemented
28C2×#16864639C2211yescharacteristic, complemented
29C2×#16864638C2211yescharacteristic, complemented
30C22×C271⋊C8C2211yescharacteristic, complemented
31#22576148C2211yescharacteristic, complemented
32#22576146—21no—
33#22576139C2211yescharacteristic, complemented
34#22576138—21no—
35#22576134C2211yescharacteristic, complemented
36#22576132—21no—
37C4×Dic542C2211yescharacteristic, complemented
38(C22⋊D271)⋊C4—21no—
39#22576093C2211yescharacteristic, complemented
40C4⋊(C271⋊C8)C2211yescharacteristic, complemented
41#22576083C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576081C2211yescharacteristic
44#22576081C2211yescharacteristic, complemented
45#22576081C2211yescharacteristic, complemented
46#22576080C2211yescharacteristic, complemented
47C22⋊(C271⋊C8)C2211yescharacteristic, complemented
48C22⋊(C271⋊C8)C2211yescharacteristic, complemented
49#22576071C2211yescharacteristic, complemented
50C22×D4×C271—21no—
51C22×D4×C271C2211yescharacteristic
52C22×D4×C271C2211yescharacteristic
53#127×C271C2211yescharacteristic
54#121×C271—21no—
55#121×C271C2211yescharacteristic
56#121×C271C2211yescharacteristic
57#120×C271—21no—
58#120×C271—21no—
59C4×D4×C271C2211yescharacteristic
60C4×D4×C271C2211yescharacteristic
61#16864654—22no—
62#16864654—41no—
63C2×Dic542—22no—
64C2×Dic542—21no—
65C2×Dic542—41no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C22×Dic271—21no—
72C2×C4×D271—21no—
73#16864642—22no—
74#16864642—22no—
75#16864642—22no—
76#16864642—22no—
77#16864642—41no—
78#16864642D414yescomplemented
79#16864639—22no—
80#16864639—22no—
81#16864639—41no—
82#16864638—22no—
83C2×C271⋊C8—22no—
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95C4×Dic271—21no—
96C24×C271—21no—
97C24×C271—21no—
98C2×D4×C271—22no—
99C2×D4×C271—22no—
100C2×D4×C271—21no—
101C2×D4×C271—21no—
102C2×D4×C271—21no—
103C2×D4×C271—21no—
104C2×D4×C271—21no—
105C2×D4×C271—21no—
106C2×D4×C271—41no—
107C2×D4×C271—41no—
108C2×D4×C271—44no—
109C2×D4×C271D411yescharacteristic
110C2×D4×C271C2311yescharacteristic
111C2×D4×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271—21no—
114C22×C4×C271C2311yescharacteristic
115C22×C4×C271C2311yescharacteristic
116C4⋊C4×C271D411yescharacteristic
117C4⋊C4×C271C2311yescharacteristic
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22⋊C4×C271—21no—
121C22⋊C4×C271D411yescharacteristic
122C22⋊C4×C271D411yescharacteristic
123C42×C271C2311yescharacteristic
124C22×D271—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—42no—
129Dic542C2×D412yescomplemented
130D1084—21no—
131D1084—21no—
132C22⋊D271—42no—
133C22⋊D271—42no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—42no—
138C4×D271—42no—
139C271⋊C8—21no—
140C271⋊C8—21no—
141C271⋊C8—21no—
142C271⋊C8C2×D412yescomplemented
143C23×C271—21no—
144C23×C271—21no—
145C23×C271—21no—
146C23×C271—42no—
147C23×C271—42no—
148C23×C271—42no—
149C23×C271—42no—
150C23×C271—81no—
151C23×C271—81no—
152C23×C271C2×D411yescharacteristic
153C23×C271C2×D411yescharacteristic
154D4×C271—22no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—21no—
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—42no—
161D4×C271—42no—
162D4×C271—42no—
163D4×C271—42no—
164D4×C271—42no—
165D4×C271—81no—
166D4×C271C2×D412yes—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—42no—
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2×D411yescharacteristic
176C2×C4×C271C2×D411yescharacteristic
177C2×C4×C271C2411yescharacteristic
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271—22no—
182C22×C271—21no—
183C22×C271—21no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271—41no—
191C22×C271—41no—
192C22×C271—84no—
193C22×C271—84no—
194C22×C271AGL(1,ℤ/8)12yescomplemented
195C22×C271C22×D411yescharacteristic
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—41no—
199C4×C271—21no—
200C4×C271—21no—
201C4×C271—21no—
202C4×C271—21no—
203C4×C271—41no—
204C4×C271C22×D411yescharacteristic
205C4×C271H(5,𝔽2)11yescharacteristic
206Dic271—22no—
207Dic271—41no—
208Dic271—41no—
209C2×C271—22no—
210C2×C271—41no—
211C2×C271—42no—
212C2×C271—81no—
213C2×C271—81no—
214C2×C271D4211yescharacteristic
215C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C271#57711yescharacteristic
217D271—81no—
218C271#326511yescharacteristic, complemented, Sylow
219#3265—2711nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2711no—
221C22×SD16—2711no—
222#548—2711no—
223D42D27111yescharacteristic, complemented
224#496—2711no—
225#468—2711no—
226#466—2711no—
227#463—2711no—
228#460—2711no—
229#450—2711no—
230#449—2711no—
231#448—2711no—
232#447—2711no—
233#440—2711no—
234#435—2711no—
235C2×#41—2711no—
236C22×Q8—2711no—
237C22×D4—21no—
238C22×D4C2×D27111yescharacteristic
239C22×D4C2×D27111yescharacteristic
240AGL(1,ℤ/8)—5424no—
241C2×SD16—2711no—
242C2×SD16—2711no—
243C2×SD16—2711no—
244C2×SD16—2711no—
245C2×SD16—2711no—
246C2×SD16—2711no—
247C2×SD16—5424no—
248C2×D8—2711no—
249C2×D8—2711no—
250C2×D8—2711no—
251C2×M16—2711no—
252C22×C8—2711no—
253#127C2×D27111yescharacteristic
254#124—2711no—
255#124—5421no—
256C22⋊Q8—2711no—
257C22⋊Q8—5421no—
258#121—21no—
259#121—2711no—
260#121—5421no—
261#121C2×D27111yescharacteristic
262#121C2×D27111yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—2711no—
266C4×D4—5421no—
267C4×D4C2×D27111yescharacteristic
268C4×D4C2×D27111yescharacteristic
269#107—2711no—
270C4⋊C8—2711no—
271#103—2711no—
272#103—2711no—
273#102—2711no—
274#102—2711no—
275#102—2711no—
276#102—2711no—
277C22⋊C8—2711no—
278C22⋊C8—2711no—
279#97—2711no—
280C24—21no—
281C24—21no—
282#41—5422no—
283#41—10841no—
284C2×Q8—2711no—
285C2×Q8—2711no—
286C2×Q8—5422no—
287C2×Q8—5421no—
288C2×Q8—10841no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—2711no—
301C2×D4—5421no—
302C2×D4—5421no—
303C2×D4C22⋊D27111yescharacteristic
304C2×D4C22×D27111yescharacteristic
305C2×D4C22×D27111yescharacteristic
306C2×D4C22×D27111yescharacteristic
307C22×C4—21no—
308C22×C4—5421no—
309C22×C4—5421no—
310C22×C4C22×D27111yescharacteristic
311C22×C4C22×D27111yescharacteristic
312SD16—2714no—
313SD16—5422no—
314SD16—5422no—
315SD16—5422no—
316SD16—5422no—
317SD16—10841no—
318D8—5422no—
319D8—5422no—
320D8—10841no—
321M16—5422no—
322C2×C8—2711no—
323C2×C8—2711no—
324C2×C8—2711no—
325C2×C8—2711no—
326C2×C8—5422no—
327C4⋊C4—2711no—
328C4⋊C4—5421no—
329C4⋊C4—5421no—
330C4⋊C4C22⋊D27111yescharacteristic
331C4⋊C4C22×D27111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4C22⋊D27111yescharacteristic
340C22⋊C4C22⋊D27111yescharacteristic
341C42—5421no—
342C42C22×D27111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—5421no—
353C23C2×C22⋊D27111yescharacteristic
354C23C2×C22⋊D27111yescharacteristic
355Q8—2712no—
356Q8—5421no—
357Q8—5421no—
358Q8—5421no—
359Q8—10842no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—5421no—
373D4—5421no—
374D4—10842no—
375D4—10842no—
376D4D4×D27112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—5421no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—10842no—
387C2×C4—10842no—
388C2×C4C2×C22⋊D27111yescharacteristic
389C2×C4C2×C22⋊D27111yescharacteristic
390C2×C4C2×C22⋊D27111yescharacteristic
391C2×C4C2×C22⋊D27111yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—2712no—
394C8—5421no—
395C8—5421no—
396C8—5421no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—10841no—
414C22—10841no—
415C22—10841no—
416C22#2257618612yescomplemented
417C22C22×C22⋊D27111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—5422no—
424C4—10841no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620511yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—21681no—
434C2D4×C22⋊D27111yescharacteristic
435C2C2×#2257618611yescharacteristic
436C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingD42×C271C2
socleC22×C271C22×D4
hypercenterD42D271
hypocenterC271#3265
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. C22×#16864642C2semidirect
3. #31774556C2semidirect
4. #31774284C2semidirect
5. #31774141C2semidirect
6. #31774131C2semidirect
7. #31774112C2semidirect
8. #31774094C2semidirect
9. #31774048C2semidirect
10. #31774040C2semidirect
11. #31774035C2semidirect
12. #31774031C2semidirect
13. #31773997C2semidirect
14. #31773969C2semidirect
15. D42×C271C2semidirect
16. C2×#16864654C22semidirect
17. C22×Dic542C22semidirect
22. C2×#16864642C22semidirect
23. C2×#16864642C22semidirect
24. C2×#16864642C22semidirect
25. C2×#16864642C22semidirect
27. C2×#16864639C22semidirect
28. C2×#16864639C22semidirect
29. C2×#16864638C22semidirect
30. C22×C271⋊C8C22semidirect
31. #22576148C22semidirect
33. #22576139C22semidirect
35. #22576134C22semidirect
37. C4×Dic542C22semidirect
39. #22576093C22semidirect
40. C4⋊(C271⋊C8)C22semidirect
41. #22576083C22semidirect
42. #22576083C22semidirect
44. #22576081C22semidirect
45. #22576081C22semidirect
46. #22576080C22semidirect
47. C22⋊(C271⋊C8)C22semidirect
48. C22⋊(C271⋊C8)C22semidirect
49. #22576071C22semidirect
78. #16864642D4semidirect
129. Dic542C2×D4semidirect
142. C271⋊C8C2×D4semidirect
194. C22×C271AGL(1,ℤ/8)semidirect
218. C271#3265semidirect
223. D42D271semidirect
416. C22#22576186semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
C22×#168646422
#317745562
#317742842
#317741412
#317741312
#317741122
#317740942
#317740482
#317740402
#317740352
#317740312
#317739972
#317739692
D42×C2712
#3265271