Finite Group Databasebeta

← #46847958

Subgroups — #46847958

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×#22576184C211yesmaximal, characteristic, complemented
2#31774546C211yesmaximal, characteristic, complemented
3D4×C22⋊D271C211yesmaximal, characteristic, complemented
4#31774268C211yesmaximal, characteristic, complemented
5#31774143C211yesmaximal, characteristic, complemented
6#31774140C211yesmaximal, characteristic, complemented
7#31774100C211yesmaximal, characteristic, complemented
8#31774090C211yesmaximal, characteristic, complemented
9#31774048C211yesmaximal, characteristic, complemented
10#31774046C211yesmaximal, characteristic, complemented
11#31774042C211yesmaximal, characteristic, complemented
12#31774039C211yesmaximal, characteristic, complemented
13D8×Dic271C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic, complemented
17C22×C22⋊D271—21no—
18C2×D4×D271C2211yescharacteristic
19#22576184—24no—
20C2×#16864643C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic
22C2×#16864642C2211yescharacteristic, complemented
23C2×#16864641C2211yescharacteristic, complemented
24C2×#16864639C2211yescharacteristic, complemented
25C2×#16864637C2211yescharacteristic, complemented
26#22576158C2211yescharacteristic, complemented
27#22576156C2211yescharacteristic
28#22576143—21no—
29#22576141—21no—
30#22576139C2211yescharacteristic, complemented
31#22576136C2211yescharacteristic, complemented
32#22576134C2211yescharacteristic
33#22576132—21no—
34#22576132—21no—
35#22576131C2211yescharacteristic
36#22576129—21no—
37#22576128—21no—
38D4×Dic271C2211yescharacteristic
39D4×Dic271C2211yescharacteristic, complemented
40C4×C22⋊D271C2211yescharacteristic
41C2×Dic271⋊C4—21no—
42#22576094C2211yescharacteristic, complemented
43Dic271⋊C8C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576084C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic, complemented
49#22576081C2211yescharacteristic, complemented
50C22⋊(C271⋊C8)C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52C8×Dic271C2211yescharacteristic, complemented
53C2×#41×C271C2211yescharacteristic, complemented
54C22×D4×C271C2211yescharacteristic
55AGL(1,ℤ/8)×C271—24no—
56C2×SD16×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic, complemented
58C2×D8×C271C2211yescharacteristic
59C2×D8×C271C2211yescharacteristic, complemented
60C2×M16×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864653—22no—
63#16864653—41no—
64C2×Dic542C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—44no—
73C2×C22⋊D271D411yescharacteristic
74D4×D271—22no—
75D4×D271—41no—
76C22×Dic271—21no—
77C22×Dic271—21no—
78C22×Dic271—21no—
79C2×C4×D271C2311yescharacteristic
80#16864643—22no—
81#16864642—22no—
82#16864642—41no—
83#16864641—41no—
84#16864639—22no—
85#16864637—22no—
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89C4⋊Dic271C2311yescharacteristic
90Dic271⋊C4—21no—
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93Dic271⋊C4—21no—
94Dic271⋊C4D411yescharacteristic
95C22⋊Dic271—21no—
96C22⋊Dic271—21no—
97C22⋊Dic271—21no—
98C22⋊Dic271—21no—
99C22⋊Dic271D411yescharacteristic
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4D411yescharacteristic
103C4×Dic271C2311yescharacteristic
104C24×C271—21no—
105#41×C271—22no—
106#41×C271—41no—
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271—22no—
109C2×D4×C271—21no—
110C2×D4×C271—21no—
111C2×D4×C271—41no—
112C2×D4×C271C2311yescharacteristic
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115C22×C4×C271—21no—
116C22×C4×C271C2311yescharacteristic
117SD16×C271—22no—
118SD16×C271—22no—
119D8×C271—22no—
120D8×C271D414yescomplemented
121M16×C271—22no—
122C2×C8×C271C2311yescharacteristic
123C2×C8×C271C2311yescharacteristic
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—81no—
128C22×D271C2×D411yescharacteristic
129Dic542—21no—
130Dic542—21no—
131D1084—21no—
132D1084—21no—
133C22⋊D271—22no—
134C22⋊D271—42no—
135C22⋊D271—42no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C22⋊D271—42no—
139C22⋊D271—81no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—42no—
144C2×Dic271—42no—
145C2×Dic271—42no—
146C2×Dic271C2×D411yescharacteristic
147C2×Dic271C2×D411yescharacteristic
148C2×Dic271C2×D411yescharacteristic
149C4×D271—21no—
150C4×D271—21no—
151C271⋊C8—21no—
152C271⋊C8—21no—
153C23×C271—21no—
154C23×C271—21no—
155C23×C271—21no—
156C23×C271—42no—
157C23×C271—42no—
158C23×C271—81no—
159C23×C271C2×D411yescharacteristic
160Q8×C271—21no—
161Q8×C271—21no—
162D4×C271—21no—
163D4×C271—21no—
164D4×C271—21no—
165D4×C271—21no—
166D4×C271—42no—
167D4×C271—42no—
168D4×C271C2×D412yes—
169D4×C271C2×D412yescomplemented
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271—42no—
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2411yescharacteristic
176C8×C271—21no—
177C8×C271C2×D412yescomplemented
178C22×C271—21no—
179C22×C271—22no—
180C22×C271—22no—
181C22×C271—21no—
182C22×C271—21no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271—84no—
191C22×C271C22×D411yescharacteristic
192C2×D271—21no—
193C2×D271—21no—
194C2×D271—21no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—84no—
199C4×C271—21no—
200C4×C271—41no—
201C4×C271C22×D411yescharacteristic
202C4×C271H(5,𝔽2)11yescharacteristic
203Dic271—21no—
204Dic271—21no—
205Dic271—41no—
206Dic271—41no—
207Dic271—41no—
208Dic271AGL(1,ℤ/8)12yescomplemented
209C2×C271—42no—
210C2×C271—41no—
211C2×C271—42no—
212C2×C271—81no—
213C2×C271D4211yescharacteristic
214C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C271#57711yescharacteristic
216D271—41no—
217D271—81no—
218C271#327211yescharacteristic, complemented, Sylow
219#3272—2711nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2711no—
221C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
222#547—2711no—
223D42—2711no—
224#492—2711no—
225#468—2711no—
226#468—2711no—
227#461—2711no—
228#459—2711no—
229#450—2711no—
230#450—2711no—
231#449—2711no—
232#449—2711no—
233C4×D8—2711no—
234#436—2711no—
235C2×#41—2711no—
236C2×#41C2×D27111yescharacteristic, complemented
237C22×D4—2711no—
238C22×D4—5421no—
239C22×D4C2×D27111yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—5424no—
242C2×SD16—2711no—
243C2×SD16—2711no—
244C2×SD16—2711no—
245C2×SD16—2711no—
246C2×SD16C2×D27111yescharacteristic
247C2×SD16C2×D27111yescharacteristic, complemented
248C2×D8—2711no—
249C2×D8C2×D27111yescharacteristic
250C2×D8C2×D27111yescharacteristic, complemented
251C2×M16—2711no—
252C2×M16C2×D27111yescharacteristic, complemented
253C4⋊Q8—2711no—
254#127—2711no—
255#123—5421no—
256#123—5421no—
257C22⋊Q8—2711no—
258C22⋊Q8—2711no—
259#121—2711no—
260#121—2711no—
261#121—5421no—
262#121—5421no—
263#120—5421no—
264#120—5421no—
265C4×D4—2711no—
266C4×D4—2711no—
267C4×D4—2711no—
268C2×C4⋊C4—5421no—
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—
279C4×C8—2711no—
280C24—21no—
281C24—5421no—
282#41—22no—
283#41—41no—
284#41—5422no—
285#41—10841no—
286C2×Q8—2711no—
287C2×Q8C22×D27111yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—2711no—
293C2×D4—2711no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5422no—
298C2×D4—5421no—
299C2×D4—5421no—
300C2×D4—5421no—
301C2×D4—10841no—
302C2×D4—10844no—
303C2×D4C22×D27111yescharacteristic
304C2×D4C22×D27111yescharacteristic
305C2×D4C22×D27111yescharacteristic
306C22×C4—21no—
307C22×C4—2711no—
308C22×C4—5421no—
309C22×C4—5421no—
310C22×C4—5421no—
311C22×C4C22×D27111yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—5422no—
315SD16—5422no—
316SD16—10841no—
317SD16—10841no—
318D8—22no—
319D8—5422no—
320D8C22⋊D27114yescomplemented
321M16—22no—
322M16—5422no—
323C2×C8—2711no—
324C2×C8—2711no—
325C2×C8C22×D27111yescharacteristic
326C2×C8C22×D27111yescharacteristic
327C4⋊C4—2711no—
328C4⋊C4—2711no—
329C4⋊C4—2711no—
330C4⋊C4—5421no—
331C4⋊C4—5421no—
332C4⋊C4—5421no—
333C4⋊C4—5421no—
334C22⋊C4—2711no—
335C22⋊C4—2711no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4—5421no—
340C22⋊C4—5421no—
341C22⋊C4—5421no—
342C42—2711no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—2711no—
350C23—5421no—
351C23—10842no—
352C23—10842no—
353C23—21681no—
354C23D4×D27111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—5421no—
358Q8—5421no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—5422no—
366D4—5421no—
367D4—5421no—
368D4—10842no—
369D4—10842no—
370D4—10842no—
371D4—10842no—
372D4—10842no—
373D4—21681no—
374D4C2×C22⋊D27112yes—
375D4C2×C22⋊D27112yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—2711no—
381C2×C4—2711no—
382C2×C4—2711no—
383C2×C4—5421no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—5421no—
387C2×C4—5421no—
388C2×C4—10842no—
389C2×C4—10842no—
390C2×C4—10842no—
391C2×C4D4×D27111yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—21no—
394C8—5421no—
395C8—5421no—
396C8C2×C22⋊D27112yescomplemented
397C22—21no—
398C22—22no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—5421no—
411C22—5421no—
412C22—5421no—
413C22—10841no—
414C22—10841no—
415C22—10841no—
416C22—21684no—
417C22C2×D4×D27111yescharacteristic
418C4—21no—
419C4—41no—
420C4—2712no—
421C4—5421no—
422C4—5421no—
423C4—10841no—
424C4—10841no—
425C4—10841no—
426C4C22×C22⋊D27111yescharacteristic
427C4#2257620511yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—10841no—
433C2—21681no—
434C2D4×C22⋊D27111yescharacteristic
435C2AGL(1,ℤ/8)×D27111yescharacteristic
436C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×AGL(1,ℤ/8)×C271C2
socleC22×C271C22×D4
hypercenterC2×AGL(1,ℤ/8)D271
hypocenterC271#3272
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. #31774546C2semidirect
3. D4×C22⋊D271C2semidirect
4. #31774268C2semidirect
5. #31774143C2semidirect
6. #31774140C2semidirect
7. #31774100C2semidirect
8. #31774090C2semidirect
9. #31774048C2semidirect
10. #31774046C2semidirect
11. #31774042C2semidirect
12. #31774039C2semidirect
13. D8×Dic271C2semidirect
14. #31773973C2semidirect
15. C2×AGL(1,ℤ/8)×C271C2semidirect
16. C2×#16864653C22semidirect
20. C2×#16864643C22semidirect
22. C2×#16864642C22semidirect
23. C2×#16864641C22semidirect
24. C2×#16864639C22semidirect
25. C2×#16864637C22semidirect
26. #22576158C22semidirect
30. #22576139C22semidirect
31. #22576136C22semidirect
39. D4×Dic271C22semidirect
42. #22576094C22semidirect
43. Dic271⋊C8C22semidirect
44. #22576085C22semidirect
45. #22576084C22semidirect
46. #22576082C22semidirect
48. #22576081C22semidirect
49. #22576081C22semidirect
50. C22⋊(C271⋊C8)C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. C8×Dic271C22semidirect
53. C2×#41×C271C22semidirect
57. C2×SD16×C271C22semidirect
59. C2×D8×C271C22semidirect
60. C2×M16×C271C22semidirect
120. D8×C271D4semidirect
169. D4×C271C2×D4semidirect
177. C8×C271C2×D4semidirect
208. Dic271AGL(1,ℤ/8)semidirect
218. C271#3272semidirect
221. C2×AGL(1,ℤ/8)D271semidirect
236. C2×#41C2×D271semidirect
247. C2×SD16C2×D271semidirect
250. C2×D8C2×D271semidirect
252. C2×M16C2×D271semidirect
320. D8C22⋊D271semidirect
375. D4C2×C22⋊D271semidirect
396. C8C2×C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
#317745462
D4×C22⋊D2712
#317742682
#317741432
#317741402
#317741002
#317740902
#317740482
#317740462
#317740422
#317740392
D8×Dic2712
#317739732
C2×AGL(1,ℤ/8)×C2712
#3272271