Finite Group Databasebeta

← #46845647

Subgroups — #46845647

Order 34688.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576207C211yesmaximal, characteristic, complemented
2C2×#22576184C211yesmaximal, characteristic, complemented
3C2×SD16×D271C211yesmaximal, characteristic, complemented
4#31774463C211yesmaximal, characteristic, complemented
5#31774043C211yesmaximal, characteristic, complemented
6#31774042C211yesmaximal, characteristic, complemented
7#31774039C211yesmaximal, characteristic, complemented
8#31774038C211yesmaximal, characteristic, complemented
9#31774037C211yesmaximal, characteristic, complemented
10#31774035C211yesmaximal, characteristic, complemented
11#31774034C211yesmaximal, characteristic, complemented
12#31773904C211yesmaximal, characteristic, complemented
13#31773890C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16#22576207—24no—
17#22576207—24no—
18C2×#16864654—21no—
19C2×#16864654C2211yescharacteristic
20C2×#16864653—21no—
21C2×#16864653—21no—
22C2×#16864653C2211yescharacteristic
23C2×#16864653C2211yescharacteristic, complemented
24C22×Dic542—21no—
25C22×Dic542C2211yescharacteristic, complemented
26C2×Q8×D271C2211yescharacteristic
27C2×Q8×D271C2211yescharacteristic, complemented
28C2×D4×D271C2211yescharacteristic
29#22576184—24no—
30C2×#16864643C2211yescharacteristic
31C2×#16864643C2211yescharacteristic
32C2×#16864642C2211yescharacteristic
33C2×#16864642C2211yescharacteristic
34C2×#16864641C2211yescharacteristic
35SD16×D271—24no—
36C2×#16864639C2211yescharacteristic
37C2×#16864637C2211yescharacteristic
38C2×C8×D271C2211yescharacteristic
39#22576156—21no—
40#22576146—21no—
41#22576135C2211yescharacteristic
42#22576134C2211yescharacteristic
43#22576131C2211yescharacteristic
44#22576130—21no—
45#22576128—21no—
46#22576118C2211yescharacteristic
47#22576085C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576084C2211yescharacteristic, complemented
50#22576083C2211yescharacteristic, complemented
51#22576083C2211yescharacteristic, complemented
52#22576081C2211yescharacteristic, complemented
53C22⋊(C271⋊C8)C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
56C2×#41×C271C2211yescharacteristic, complemented
57C2×SD16×C271C2211yescharacteristic
58C2×D8×C271C2211yescharacteristic
59#121×C271C2211yescharacteristic
60#103×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62C22⋊C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64#16864654—28no—
65#16864654—22no—
66#16864654—22no—
67#16864653—28no—
68#16864653—28no—
69#16864653—22no—
70#16864653—22no—
71#16864653—22no—
72#16864653D414yescomplemented
73C2×Dic542—28no—
74C2×Dic542—22no—
75C2×Dic542—22no—
76C2×Dic542—21no—
77C2×Dic542—21no—
78C2×Dic542—21no—
79C2×Dic542D411yescharacteristic
80C2×Dic542C2311yescharacteristic
81C2×Dic542C2311yescharacteristic
82Q8×D271—22no—
83Q8×D271—22no—
84Q8×D271—22no—
85Q8×D271D414yescomplemented
86C2×D1084—21no—
87C2×D1084D411yescharacteristic
88C2×D1084C2311yescharacteristic
89C2×C22⋊D271—21no—
90C2×C22⋊D271—21no—
91C2×C22⋊D271—21no—
92C2×C22⋊D271—21no—
93C2×C22⋊D271—21no—
94C2×C22⋊D271D411yescharacteristic
95C2×C22⋊D271D411yescharacteristic
96D4×D271—22no—
97D4×D271—22no—
98C22×Dic271—21no—
99C22×Dic271—21no—
100C22×Dic271—21no—
101C2×C4×D271—21no—
102C2×C4×D271—21no—
103C2×C4×D271C2311yescharacteristic
104C2×C4×D271C2311yescharacteristic
105#16864643—22no—
106#16864643—22no—
107#16864642—22no—
108#16864642—22no—
109#16864641—22no—
110#16864639—22no—
111#16864637—22no—
112C2×C271⋊C8C2311yescharacteristic
113C2×C271⋊C8C2311yescharacteristic
114C8×D271—22no—
115C4⋊Dic271C2311yescharacteristic
116C22⋊Dic271—21no—
117(C2×D271)⋊C4—21no—
118(C2×D271)⋊C4—21no—
119(C2×D271)⋊C4—21no—
120C4×Dic271—21no—
121#41×C271—22no—
122#41×C271—22no—
123C2×Q8×C271C2311yescharacteristic
124C2×D4×C271—21no—
125C2×D4×C271—21no—
126C2×D4×C271C2311yescharacteristic
127C2×D4×C271C2311yescharacteristic
128C22×C4×C271—21no—
129C22×C4×C271C2311yescharacteristic
130SD16×C271—22no—
131D8×C271—22no—
132C2×C8×C271C2311yescharacteristic
133C2×C8×C271C2311yescharacteristic
134C4⋊C4×C271C2311yescharacteristic
135C22⋊C4×C271—21no—
136C22×D271—21no—
137C22×D271—42no—
138C22×D271—81no—
139C22×D271—81no—
140C22×D271C2×D411yescharacteristic
141C22×D271C2×D411yescharacteristic
142Dic542—24no—
143Dic542—24no—
144Dic542—21no—
145Dic542—21no—
146Dic542—24no—
147Dic542—22no—
148Dic542C2×D412yes—
149Dic542C2×D412yes—
150D1084—22no—
151D1084—21no—
152D1084—21no—
153D1084—81no—
154C22⋊D271—24no—
155C22⋊D271—24no—
156C22⋊D271—22no—
157C22⋊D271—22no—
158C22⋊D271—42no—
159C22⋊D271—81no—
160C22⋊D271—81no—
161C2×Dic271—24no—
162C2×Dic271—24no—
163C2×Dic271—21no—
164C2×Dic271—21no—
165C2×Dic271—24no—
166C2×Dic271—21no—
167C2×Dic271C2×D411yescharacteristic
168C2×Dic271C2×D411yescharacteristic
169C4×D271—24no—
170C4×D271—24no—
171C4×D271—21no—
172C4×D271—21no—
173C4×D271C2×D412yes—
174C4×D271C2×D412yes—
175C271⋊C8—21no—
176C271⋊C8—21no—
177C23×C271—21no—
178C23×C271—21no—
179C23×C271C2×D411yescharacteristic
180Q8×C271—21no—
181Q8×C271C2×D412yes—
182D4×C271—24no—
183D4×C271—21no—
184D4×C271—21no—
185D4×C271—21no—
186D4×C271—81no—
187D4×C271C2×D412yes—
188C2×C4×C271—24no—
189C2×C4×C271—21no—
190C2×C4×C271—21no—
191C2×C4×C271—21no—
192C2×C4×C271—21no—
193C2×C4×C271C2×D411yescharacteristic
194C2×C4×C271C2411yescharacteristic
195C8×C271—21no—
196C8×C271—21no—
197C22×C271—21no—
198C22×C271—22no—
199C22×C271—21no—
200C22×C271—21no—
201C22×C271—41no—
202C22×C271—41no—
203C22×C271—41no—
204C22×C271—41no—
205C22×C271—41no—
206C22×C271C22×D411yescharacteristic
207C2×D271—21no—
208C2×D271—21no—
209C2×D271—21no—
210C2×D271—21no—
211C2×D271—21no—
212C2×D271—41no—
213C2×D271—41no—
214C2×D271—41no—
215C2×D271—84no—
216C2×D271#12012yes—
217C4×C271—21no—
218C4×C271—22no—
219C4×C271—41no—
220C4×C271C22×D411yescharacteristic
221C4×C271C22×D411yescharacteristic
222Dic271—22no—
223Dic271—22no—
224Dic271—21no—
225Dic271—41no—
226Dic271#12012yes—
227C2×C271—41no—
228C2×C271—42no—
229C2×C271—81no—
230C2×C271C2×#12011yescharacteristic
231C2×C271#57711yescharacteristic
232C2×C271#57711yescharacteristic
233D271—22no—
234D271—41no—
235D271—81no—
236C271#301011yescharacteristic, complemented, Sylow
237#3010—2711nomaximal, Sylow
238C2×2−1+4—2711no—
239C2×AGL(1,ℤ/8)—2711no—
240C22×SD16—2711no—
241#535—2711no—
242#449—2711no—
243#449—2711no—
244#449—2711no—
245#449D27111yescharacteristic, complemented
246#448—2711no—
247#448—2711no—
248#448—2711no—
249#448—2711no—
250#419—2711no—
251#416—2711no—
252#408—2711no—
2532−1+4—5424no—
2542−1+4—5424no—
255C2×#41—2711no—
256C2×#41—2711no—
257C2×#41—2711no—
258C2×#41—5421no—
259C2×#41—5421no—
260C2×#41—5421no—
261C2×#41C2×D27111yescharacteristic, complemented
262C22×Q8—2711no—
263C22×Q8—2711no—
264C22×Q8—2711no—
265C22×Q8—5421no—
266C22×D4—2711no—
267AGL(1,ℤ/8)—5424no—
268C2×SD16—2711no—
269C2×SD16—2711no—
270C2×SD16—2711no—
271C2×SD16—2711no—
272C2×SD16—2711no—
273C2×SD16—5424no—
274C2×SD16C2×D27111yescharacteristic
275C2×D8—2711no—
276C2×D8C2×D27111yescharacteristic
277C2×M16—2711no—
278C22×C8—2711no—
279#127—5421no—
280#124—5421no—
281#121—2711no—
282#121—2711no—
283#121—2711no—
284#121C2×D27111yescharacteristic
285#120—5421no—
286#120—5421no—
287#117—2711no—
288#103—2711no—
289#103—2711no—
290#103—2711no—
291#103—2711no—
292#103—2711no—
293#103C2×D27111yescharacteristic, complemented
294#102—2711no—
295#102C2×D27111yescharacteristic, complemented
296C22⋊C8—2711no—
297C22⋊C8—2711no—
298C22⋊C8—2711no—
299C22⋊C8C2×D27111yescharacteristic, complemented
300C24—5421no—
301#41—22no—
302#41—22no—
303#41—2714no—
304#41—5422no—
305#41—5422no—
306#41—5428no—
307#41—5422no—
308#41—5422no—
309#41—5428no—
310#41—5428no—
311#41—5422no—
312C2×Q8—2711no—
313C2×Q8—2714no—
314C2×Q8—2711no—
315C2×Q8—2711no—
316C2×Q8—5421no—
317C2×Q8—5421no—
318C2×Q8—5422no—
319C2×Q8—5422no—
320C2×Q8—5428no—
321C2×Q8—5422no—
322C2×Q8—5422no—
323C2×Q8—5422no—
324C2×Q8—5421no—
325C2×Q8C22×D27111yescharacteristic
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—2711no—
329C2×D4—2711no—
330C2×D4—2711no—
331C2×D4—2711no—
332C2×D4—5421no—
333C2×D4—5422no—
334C2×D4—5422no—
335C2×D4—5421no—
336C2×D4—5421no—
337C2×D4—5421no—
338C2×D4—5421no—
339C2×D4—5421no—
340C2×D4C22×D27111yescharacteristic
341C2×D4C22×D27111yescharacteristic
342C22×C4—21no—
343C22×C4—2711no—
344C22×C4—2711no—
345C22×C4—5421no—
346C22×C4—5421no—
347C22×C4—5421no—
348C22×C4—5421no—
349C22×C4—5421no—
350C22×C4C22×D27111yescharacteristic
351SD16—22no—
352SD16—5422no—
353SD16—5422no—
354SD16—5422no—
355SD16—5422no—
356SD16—5422no—
357D8—22no—
358D8—5422no—
359M16—5422no—
360C2×C8—2711no—
361C2×C8—2711no—
362C2×C8—5422no—
363C2×C8C22×D27111yescharacteristic
364C2×C8C22×D27111yescharacteristic
365C4⋊C4—2711no—
366C4⋊C4C22×D27111yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—5421no—
369C22⋊C4—5421no—
370C22⋊C4—5421no—
371C22⋊C4—5421no—
372C42—5421no—
373C23—21no—
374C23—21no—
375C23—2711no—
376C23—2711no—
377C23—5421no—
378C23—10842no—
379C23—21681no—
380C23—21681no—
381C23D4×D27111yescharacteristic
382Q8—21no—
383Q8—2712no—
384Q8—2712no—
385Q8—5422no—
386Q8—5424no—
387Q8—5424no—
388Q8—5424no—
389Q8—5421no—
390Q8—5421no—
391Q8D4×D27112yes—
392D4—24no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—81no—
397D4—5422no—
398D4—5422no—
399D4—5422no—
400D4—5424no—
401D4—5424no—
402D4—5421no—
403D4—5421no—
404D4—10842no—
405D4—21681no—
406D4—21681no—
407D4—21681no—
408D4D4×D27112yes—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—2711no—
415C2×C4—2712no—
416C2×C4—2712no—
417C2×C4—2711no—
418C2×C4—5424no—
419C2×C4—5424no—
420C2×C4—5424no—
421C2×C4—5424no—
422C2×C4—5424no—
423C2×C4—5421no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4D4×D27111yescharacteristic
429C2×C4C23×D27111yescharacteristic
430C8—21no—
431C8—21no—
432C8—5421no—
433C8—5421no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—2712no—
444C22—5421no—
445C22—5421no—
446C22—5421no—
447C22—5421no—
448C22—5421no—
449C22—10841no—
450C22—10841no—
451C22—10841no—
452C22—21684no—
453C22C2×D4×D27111yescharacteristic
454C4—21no—
455C4—22no—
456C4—41no—
457C4—2712no—
458C4—5421no—
459C4—5422no—
460C4—5422no—
461C4—10841no—
462C4C2×D4×D27111yescharacteristic
463C4C2×D4×D27111yescharacteristic
464C2—41no—
465C2—42no—
466C2—81no—
467C2—5422no—
468C2—10841no—
469C2—21681no—
470C2#120×D27111yescharacteristic
471C2#3177470411yescharacteristic
472C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3010
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22576207C2semidirect
2. C2×#22576184C2semidirect
3. C2×SD16×D271C2semidirect
4. #31774463C2semidirect
5. #31774043C2semidirect
6. #31774042C2semidirect
7. #31774039C2semidirect
8. #31774038C2semidirect
9. #31774037C2semidirect
10. #31774035C2semidirect
11. #31774034C2semidirect
12. #31773904C2semidirect
13. #31773890C2semidirect
14. #31773862C2semidirect
15. #449×C271C2semidirect
23. C2×#16864653C22semidirect
25. C22×Dic542C22semidirect
27. C2×Q8×D271C22semidirect
47. #22576085C22semidirect
48. #22576085C22semidirect
49. #22576084C22semidirect
50. #22576083C22semidirect
51. #22576083C22semidirect
52. #22576081C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
56. C2×#41×C271C22semidirect
60. #103×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
72. #16864653D4semidirect
85. Q8×D271D4semidirect
236. C271#3010semidirect
245. #449D271semidirect
261. C2×#41C2×D271semidirect
293. #103C2×D271semidirect
295. #102C2×D271semidirect
299. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762072
C2×#225761842
C2×SD16×D2712
#317744632
#317740432
#317740422
#317740392
#317740382
#317740372
#317740352
#317740342
#317739042
#317738902
#317738622
#449×C2712
#3010271