Finite Group Databasebeta

← #46847978

Subgroups — #46847978

Order 34688.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576187C211yesmaximal, characteristic, complemented
2#31774566C211yesmaximal, characteristic, complemented
3D4×C22⋊D271C211yesmaximal, characteristic, complemented
4#31774272C211yesmaximal, characteristic, complemented
5#31774150C211yesmaximal, characteristic, complemented
6#31774147C211yesmaximal, characteristic, complemented
7#31774095C211yesmaximal, characteristic, complemented
8#31774090C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774041C211yesmaximal, characteristic, complemented
11#31774031C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13SD16×Dic271C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic, complemented
17C22×C22⋊D271—21no—
18C2×D4×D271C2211yescharacteristic
19#22576187—24no—
20C2×#16864641C2211yescharacteristic, complemented
21C2×D2168C2211yescharacteristic, complemented
22C2×#16864639C2211yescharacteristic
23C2×#16864639C2211yescharacteristic, complemented
24C2×#16864639C2211yescharacteristic, complemented
25C2×#16864637C2211yescharacteristic, complemented
26#22576156—21no—
27#22576156C2211yescharacteristic
28#22576156C2211yescharacteristic, complemented
29#22576134—21no—
30#22576134C2211yescharacteristic
31#22576134C2211yescharacteristic, complemented
32#22576132—21no—
33#22576131—21no—
34#22576131C2211yescharacteristic
35#22576131C2211yescharacteristic, complemented
36#22576129—21no—
37#22576128—21no—
38Q8×Dic271C2211yescharacteristic, complemented
39D4×Dic271C2211yescharacteristic
40C4×C22⋊D271—21no—
41C4×C22⋊D271C2211yescharacteristic
42#22576092C2211yescharacteristic, complemented
43Dic271⋊C8C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic, 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#16864655—22no—
63#16864655—41no—
64C2×D1084—21no—
65C2×D1084C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—44no—
77C2×C22⋊D271D411yescharacteristic
78D4×D271—22no—
79D4×D271—41no—
80C22×Dic271—21no—
81C2×C4×D271—21no—
82C2×C4×D271C2311yescharacteristic
83#16864641—22no—
84D2168—22no—
85#16864639—22no—
86#16864639—41no—
87#16864639—41no—
88#16864637—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C4⋊Dic271—21no—
92C4⋊Dic271C2311yescharacteristic
93C4⋊Dic271C2311yescharacteristic
94Dic271⋊C4D411yescharacteristic
95C22⋊Dic271—21no—
96C22⋊Dic271—21no—
97C22⋊Dic271—21no—
98C22⋊Dic271D411yescharacteristic
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4D411yescharacteristic
102C4×Dic271—21no—
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×C271D414yescomplemented
119D8×C271—22no—
120D8×C271—22no—
121M16×C271—22no—
122C2×C8×C271C2311yescharacteristic
123C2×C8×C271C2311yescharacteristic
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—81no—
129C22×D271C2×D411yescharacteristic
130D1084—21no—
131D1084—21no—
132D1084—21no—
133D1084—21no—
134D1084—42no—
135C22⋊D271—22no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C22⋊D271—42no—
139C22⋊D271—42no—
140C22⋊D271—42no—
141C22⋊D271—81no—
142C22⋊D271—81no—
143C22⋊D271—81no—
144C22⋊D271—81no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—42no—
148C2×Dic271C2×D411yescharacteristic
149C2×Dic271C2×D411yescharacteristic
150C2×Dic271C2×D411yescharacteristic
151C4×D271—21no—
152C4×D271—21no—
153C4×D271—42no—
154C271⋊C8—21no—
155C271⋊C8—21no—
156C23×C271—21no—
157C23×C271—21no—
158C23×C271—21no—
159C23×C271—42no—
160C23×C271—42no—
161C23×C271—81no—
162C23×C271C2×D411yescharacteristic
163Q8×C271—21no—
164Q8×C271C2×D412yescomplemented
165D4×C271—21no—
166D4×C271—21no—
167D4×C271—21no—
168D4×C271—21no—
169D4×C271—21no—
170D4×C271—42no—
171D4×C271—42no—
172D4×C271C2×D412yes—
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—42no—
177C2×C4×C271C2×D411yescharacteristic
178C2×C4×C271C2411yescharacteristic
179C8×C271—21no—
180C8×C271C2×D412yescomplemented
181C22×C271—21no—
182C22×C271—22no—
183C22×C271—21no—
184C22×C271—21no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271—41no—
191C22×C271—41no—
192C22×C271—41no—
193C22×C271—84no—
194C22×C271C22×D411yescharacteristic
195C2×D271—21no—
196C2×D271—21no—
197C2×D271—21no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C2×D271—41no—
203C2×D271—41no—
204C2×D271—84no—
205C4×C271—21no—
206C4×C271—22no—
207C4×C271C22×D411yescharacteristic
208C4×C271H(5,𝔽2)11yescharacteristic
209Dic271—21no—
210Dic271—21no—
211Dic271—41no—
212Dic271—41no—
213Dic271AGL(1,ℤ/8)12yescomplemented
214C2×C271—42no—
215C2×C271—41no—
216C2×C271—81no—
217C2×C271—81no—
218C2×C271D4211yescharacteristic
219C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C271#57611yescharacteristic
221D271—41no—
222D271—81no—
223D271—81no—
224C271#327411yescharacteristic, complemented, Sylow
225#3274—2711nomaximal, Sylow
226C2×AGL(1,ℤ/8)—2711no—
227C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
228#550—2711no—
229D42—2711no—
230#493—2711no—
231#469—2711no—
232#469—2711no—
233#460—2711no—
234#459—2711no—
235#449—2711no—
236#449—2711no—
237#447—2711no—
238#447—2711no—
239C4×SD16—2711no—
240#436—2711no—
241C2×#41—2711no—
242C2×#41C2×D27111yescharacteristic, complemented
243C22×D4—2711no—
244C22×D4—5421no—
245C22×D4C2×D27111yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—5424no—
248C2×SD16—2711no—
249C2×SD16C2×D27111yescharacteristic
250C2×SD16C2×D27111yescharacteristic, complemented
251C2×D8—2711no—
252C2×D8—2711no—
253C2×D8—2711no—
254C2×D8—2711no—
255C2×D8C2×D27111yescharacteristic
256C2×D8C2×D27111yescharacteristic, complemented
257C2×M16—2711no—
258C2×M16C2×D27111yescharacteristic, complemented
259#127—2711no—
260#127—2711no—
261#127—5421no—
262#121—2711no—
263#121—2711no—
264#121—2711no—
265#121—2711no—
266#121—5421no—
267#121—5421no—
268#121—5421no—
269#120—5421no—
270#120—5421no—
271C4×Q8—2711no—
272C4×D4—2711no—
273C4×D4—2711no—
274C4×D4—5421no—
275#106—2711no—
276C4⋊C8—2711no—
277#103—2711no—
278#102—2711no—
279#102—2711no—
280#102—2711no—
281#102—2711no—
282#102—2711no—
283C22⋊C8—2711no—
284C22⋊C8—2711no—
285C4×C8—2711no—
286C24—21no—
287C24—5421no—
288#41—22no—
289#41—41no—
290#41—5422no—
291#41—10841no—
292C2×Q8C22×D27111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—2711no—
298C2×D4—2711no—
299C2×D4—2711no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—5421no—
303C2×D4—5421no—
304C2×D4—5421no—
305C2×D4—5421no—
306C2×D4—5422no—
307C2×D4—5421no—
308C2×D4—5421no—
309C2×D4—5421no—
310C2×D4—5421no—
311C2×D4—10841no—
312C2×D4—10844no—
313C2×D4C22×D27111yescharacteristic
314C2×D4C22×D27111yescharacteristic
315C2×D4C22×D27111yescharacteristic
316C22×C4—21no—
317C22×C4—2711no—
318C22×C4—5421no—
319C22×C4—5421no—
320C22×C4C22×D27111yescharacteristic
321SD16—22no—
322SD16—5422no—
323SD16C22⋊D27114yescomplemented
324D8—22no—
325D8—22no—
326D8—5422no—
327D8—5422no—
328D8—10841no—
329D8—10841no—
330M16—22no—
331M16—5422no—
332C2×C8—2711no—
333C2×C8—2711no—
334C2×C8C22×D27111yescharacteristic
335C2×C8C22×D27111yescharacteristic
336C4⋊C4—2711no—
337C4⋊C4—2711no—
338C4⋊C4—2711no—
339C4⋊C4—5421no—
340C22⋊C4—2711no—
341C22⋊C4—2711no—
342C22⋊C4—5421no—
343C22⋊C4—5421no—
344C22⋊C4—5421no—
345C22⋊C4—5421no—
346C22⋊C4—5421no—
347C42—2711no—
348C42—5421no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—2711no—
356C23—5421no—
357C23—5421no—
358C23—10842no—
359C23—10842no—
360C23—21681no—
361C23D4×D27111yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D27112yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—5422no—
372D4—5421no—
373D4—5421no—
374D4—5421no—
375D4—5421no—
376D4—10842no—
377D4—10842no—
378D4—10842no—
379D4—10842no—
380D4—10842no—
381D4—10842no—
382D4—21681no—
383D4—21681no—
384D4—21681no—
385D4—21681no—
386D4C2×C22⋊D27112yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—2711no—
392C2×C4—2711no—
393C2×C4—2711no—
394C2×C4—5421no—
395C2×C4—5421no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—10842no—
399C2×C4—10842no—
400C2×C4D4×D27111yescharacteristic
401C2×C4C23×D27111yescharacteristic
402C8—21no—
403C8—5421no—
404C8—5421no—
405C8C2×C22⋊D27112yescomplemented
406C22—21no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—5421no—
420C22—5421no—
421C22—5421no—
422C22—10841no—
423C22—10841no—
424C22—10841no—
425C22—10841no—
426C22—10841no—
427C22—10841no—
428C22—21684no—
429C22C2×D4×D27111yescharacteristic
430C4—21no—
431C4—22no—
432C4—2712no—
433C4—5421no—
434C4—5421no—
435C4—10841no—
436C4—10841no—
437C4C22×C22⋊D27111yescharacteristic
438C4#2257620511yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—10841no—
444C2—21681no—
445C2—21681no—
446C2D4×C22⋊D27111yescharacteristic
447C2AGL(1,ℤ/8)×D27111yescharacteristic
448C2#3177470111yescharacteristic
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#3274
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×#22576187C2semidirect
2. #31774566C2semidirect
3. D4×C22⋊D271C2semidirect
4. #31774272C2semidirect
5. #31774150C2semidirect
6. #31774147C2semidirect
7. #31774095C2semidirect
8. #31774090C2semidirect
9. #31774042C2semidirect
10. #31774041C2semidirect
11. #31774031C2semidirect
12. #31774029C2semidirect
13. SD16×Dic271C2semidirect
14. #31773973C2semidirect
15. C2×AGL(1,ℤ/8)×C271C2semidirect
16. C2×#16864655C22semidirect
20. C2×#16864641C22semidirect
21. C2×D2168C22semidirect
23. C2×#16864639C22semidirect
24. C2×#16864639C22semidirect
25. C2×#16864637C22semidirect
28. #22576156C22semidirect
31. #22576134C22semidirect
35. #22576131C22semidirect
38. Q8×Dic271C22semidirect
42. #22576092C22semidirect
43. Dic271⋊C8C22semidirect
44. #22576084C22semidirect
45. #22576082C22semidirect
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
118. SD16×C271D4semidirect
164. Q8×C271C2×D4semidirect
180. C8×C271C2×D4semidirect
213. Dic271AGL(1,ℤ/8)semidirect
224. C271#3274semidirect
227. C2×AGL(1,ℤ/8)D271semidirect
242. C2×#41C2×D271semidirect
250. C2×SD16C2×D271semidirect
256. C2×D8C2×D271semidirect
258. C2×M16C2×D271semidirect
323. SD16C22⋊D271semidirect
363. Q8C2×C22⋊D271semidirect
405. C8C2×C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
#317745662
D4×C22⋊D2712
#317742722
#317741502
#317741472
#317740952
#317740902
#317740422
#317740412
#317740312
#317740292
SD16×Dic2712
#317739732
C2×AGL(1,ℤ/8)×C2712
#3274271