Finite Group Databasebeta

← #46845621

Subgroups — #46845621

Order 34688.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D271C211yesmaximal, characteristic
2C2×#22576192C211yesmaximal, characteristic, complemented
3C2×#22576189C211yesmaximal, characteristic, complemented
4C22⋊Q8×D271C211yesmaximal, characteristic, complemented
5#31774061C211yesmaximal, characteristic, complemented
6#31774059C211yesmaximal, characteristic, complemented
7#31774057C211yesmaximal, characteristic, complemented
8#31774054C211yesmaximal, characteristic, complemented
9#31774051C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31774046C211yesmaximal, characteristic, complemented
12#31773897C211yesmaximal, characteristic, complemented
13#31773895C211yesmaximal, characteristic, complemented
14#31773859C211yesmaximal, characteristic, complemented
15#452×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654—21no—
18C2×#16864654C2211yescharacteristic
19C2×#16864653—21no—
20C2×#16864653C2211yescharacteristic
21#41×D271—24no—
22#41×D271—24no—
23C2×Q8×D271C2211yescharacteristic
24C2×Q8×D271C2211yescharacteristic, complemented
25C2×D4×D271—21no—
26C2×D4×D271C2211yescharacteristic
27C22×C4×D271—21no—
28C22×C4×D271C2211yescharacteristic
29#22576192—24no—
30#22576189—24no—
31C2×Dic1084C2211yescharacteristic, complemented
32C2×#16864644C2211yescharacteristic, complemented
33C2×#16864644C2211yescharacteristic, complemented
34C2×#16864643C2211yescharacteristic, complemented
35C2×#16864642C2211yescharacteristic, complemented
36C2×#16864641C2211yescharacteristic, complemented
37C2×#16864637C2211yescharacteristic, complemented
38C2×#16864637C2211yescharacteristic, complemented
39#22576140C2211yescharacteristic
40#22576139C2211yescharacteristic, complemented
41#22576138—21no—
42#22576137—21no—
43#22576136C2211yescharacteristic
44C4⋊C4×D271—21no—
45C4⋊C4×D271C2211yescharacteristic, complemented
46C22⋊C4×D271—21no—
47#22576085C2211yescharacteristic, complemented
48#22576084C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52#22576080C2211yescharacteristic, complemented
53C22⋊(C271⋊C8)C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic
55(C2×D271)⋊C8C2211yescharacteristic
56C2×#41×C271C2211yescharacteristic
57C2×Q16×C271C2211yescharacteristic, complemented
58C2×SD16×C271C2211yescharacteristic, complemented
59C22⋊Q8×C271C2211yescharacteristic, complemented
60#103×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62C22⋊C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271D411yescharacteristic
65#16864655—22no—
66#16864655D414yes—
67#16864654—28no—
68#16864654—22no—
69#16864654—22no—
70#16864653—28no—
71#16864653—22no—
72#16864653D414yes—
73C2×Dic542—21no—
74C2×Dic542—21no—
75C2×Dic542C2311yescharacteristic
76C2×Dic542C2311yescharacteristic
77Q8×D271—22no—
78Q8×D271—22no—
79Q8×D271—22no—
80Q8×D271—22no—
81C2×D1084—21no—
82C2×D1084C2311yescharacteristic
83C2×C22⋊D271—21no—
84C2×C22⋊D271—21no—
85C2×C22⋊D271—21no—
86D4×D271—28no—
87D4×D271—22no—
88D4×D271—22no—
89C22×Dic271—21no—
90C22×Dic271D411yescharacteristic
91C2×C4×D271—28no—
92C2×C4×D271—21no—
93C2×C4×D271—21no—
94C2×C4×D271—22no—
95C2×C4×D271—22no—
96C2×C4×D271—21no—
97C2×C4×D271—21no—
98C2×C4×D271—21no—
99C2×C4×D271D411yescharacteristic
100C2×C4×D271D411yescharacteristic
101C2×C4×D271C2311yescharacteristic
102C2×C4×D271C2311yescharacteristic
103Dic1084—22no—
104#16864644—22no—
105#16864644—22no—
106#16864643—22no—
107#16864642—22no—
108#16864641—22no—
109#16864637—22no—
110#16864637—22no—
111C2×C271⋊C8C2311yescharacteristic
112C2×C271⋊C8C2311yescharacteristic
113C4⋊Dic271—21no—
114C4⋊Dic271C2311yescharacteristic
115Dic271⋊C4—21no—
116Dic271⋊C4—21no—
117Dic271⋊C4—21no—
118C22⋊Dic271—21no—
119(C2×D271)⋊C4—21no—
120(C2×D271)⋊C4—21no—
121#41×C271—22no—
122#41×C271—22no—
123C2×Q8×C271C2311yescharacteristic
124C2×Q8×C271C2311yescharacteristic
125C2×D4×C271—21no—
126C2×D4×C271C2311yescharacteristic
127C22×C4×C271—21no—
128C22×C4×C271C2311yescharacteristic
129Q16×C271—22no—
130SD16×C271—22no—
131C2×C8×C271C2311yescharacteristic
132C2×C8×C271C2311yescharacteristic
133C4⋊C4×C271—21no—
134C4⋊C4×C271C2311yescharacteristic
135C22⋊C4×C271—21no—
136C22×D271—24no—
137C22×D271—21no—
138C22×D271—22no—
139C22×D271—22no—
140C22×D271—41no—
141C22×D271—42no—
142C22×D271—42no—
143C22×D271C2×D411yescharacteristic
144C22×D271C2×D411yescharacteristic
145Dic542—24no—
146Dic542—21no—
147Dic542—21no—
148Dic542—21no—
149Dic542—42no—
150Dic542C2×D412yes—
151D1084—24no—
152D1084—21no—
153D1084C2×D412yes—
154C22⋊D271—24no—
155C22⋊D271—24no—
156C22⋊D271—24no—
157C2×Dic271—21no—
158C2×Dic271—24no—
159C2×Dic271—22no—
160C2×Dic271—21no—
161C2×Dic271—21no—
162C2×Dic271C2×D411yescharacteristic
163C2×Dic271C2×D411yescharacteristic
164C4×D271—24no—
165C4×D271—24no—
166C4×D271—22no—
167C4×D271—21no—
168C4×D271—21no—
169C4×D271—22no—
170C4×D271—24no—
171C4×D271—42no—
172C4×D271—42no—
173C4×D271C2×D412yes—
174C4×D271C2×D412yes—
175C271⋊C8—21no—
176C271⋊C8—21no—
177C23×C271—21no—
178C23×C271C2×D411yescharacteristic
179Q8×C271—21no—
180Q8×C271—21no—
181Q8×C271—21no—
182Q8×C271C2×D412yes—
183D4×C271—24no—
184D4×C271—21no—
185D4×C271C2×D412yes—
186C2×C4×C271—24no—
187C2×C4×C271—21no—
188C2×C4×C271—21no—
189C2×C4×C271—21no—
190C2×C4×C271—21no—
191C2×C4×C271—21no—
192C2×C4×C271C2×D411yescharacteristic
193C2×C4×C271C2411yescharacteristic
194C8×C271—21no—
195C8×C271—21no—
196C22×C271—21no—
197C22×C271—22no—
198C22×C271—21no—
199C22×C271—21no—
200C22×C271—41no—
201C22×C271—41no—
202C22×C271C22×D411yescharacteristic
203C2×D271—21no—
204C2×D271—21no—
205C2×D271—21no—
206C2×D271—21no—
207C2×D271—22no—
208C2×D271—44no—
209C2×D271—48no—
210C2×D271—41no—
211C2×D271—41no—
212C2×D271#12012yes—
213C2×D271#13712yescomplemented
214C4×C271—21no—
215C4×C271—22no—
216C4×C271—41no—
217C4×C271—41no—
218C4×C271C22×D411yescharacteristic
219C4×C271C22×D411yescharacteristic
220Dic271—22no—
221Dic271—21no—
222Dic271—41no—
223Dic271—41no—
224Dic271#12012yes—
225C2×C271—41no—
226C2×C271—42no—
227C2×C271C2×#12011yescharacteristic
228C2×C271C2×#13711yescharacteristic
229C2×C271#57511yescharacteristic
230D271—22no—
231D271—22no—
232D271—42no—
233C271#300711yescharacteristic, complemented, Sylow
234#3007—2711nomaximal, Sylow
235C22×#41—2711no—
236C2×#137—2711no—
237C2×#137—2711no—
238C2×C22⋊Q8—2711no—
239#452—2711no—
240#452—2711no—
241#452—2711no—
242#452D27111yescharacteristic, complemented
243#451—2711no—
244#451—2711no—
245#450—2711no—
246#450—2711no—
247#417—2711no—
248#417—2711no—
249#407—2711no—
250C2×#41—2711no—
251C2×#41—2711no—
252C2×#41—2711no—
253C2×#41—5421no—
254C2×#41—5421no—
255C2×#41—5424no—
256C2×#41—5424no—
257C2×#41C2×D27111yescharacteristic
258C22×Q8—2711no—
259C22×Q8—2711no—
260C22×D4—2711no—
261C22×D4—5421no—
262C23×C4—2711no—
263C23×C4—5421no—
264#137—5424no—
265#137—5424no—
266C2×Q16—2711no—
267C2×Q16—2711no—
268C2×Q16—2711no—
269C2×Q16C2×D27111yescharacteristic, complemented
270C2×SD16—2711no—
271C2×SD16—2711no—
272C2×SD16—2711no—
273C2×SD16C2×D27111yescharacteristic, complemented
274C2×M16—2711no—
275C2×M16—2711no—
276C22⋊Q8—2711no—
277C22⋊Q8—2711no—
278C22⋊Q8—2711no—
279C22⋊Q8—5421no—
280C22⋊Q8—5421no—
281C22⋊Q8C2×D27111yescharacteristic, complemented
282C2×C4⋊C4—2711no—
283C2×C4⋊C4—5421no—
284C2×C22⋊C4—5421no—
285#103—2711no—
286#103—2711no—
287#103—2711no—
288#103C2×D27111yescharacteristic, complemented
289#102—2711no—
290#102—2711no—
291#102—2711no—
292#102C2×D27111yescharacteristic, complemented
293C22⋊C8—2711no—
294C22⋊C8—2711no—
295C22⋊C8—2711no—
296C22⋊C8C2×D27111yescharacteristic, complemented
297C24—2711no—
298C24—5421no—
299#41—22no—
300#41—22no—
301#41—2714no—
302#41—2714no—
303#41—5422no—
304#41—5422no—
305#41—5428no—
306#41—5422no—
307#41—5422no—
308#41—5428no—
309C2×Q8—2711no—
310C2×Q8—2711no—
311C2×Q8—5421no—
312C2×Q8—5421no—
313C2×Q8—5422no—
314C2×Q8—5422no—
315C2×Q8—5422no—
316C2×Q8—5422no—
317C2×Q8C22×D27111yescharacteristic
318C2×Q8C22×D27111yescharacteristic
319C2×D4—21no—
320C2×D4—2711no—
321C2×D4—5421no—
322C2×D4—5421no—
323C2×D4—5422no—
324C2×D4—5428no—
325C2×D4—5421no—
326C2×D4—5421no—
327C2×D4—5422no—
328C2×D4C22×D27111yescharacteristic
329C22×C4—21no—
330C22×C4—2711no—
331C22×C4—2711no—
332C22×C4—2711no—
333C22×C4—2711no—
334C22×C4—2711no—
335C22×C4—5422no—
336C22×C4—5422no—
337C22×C4—5421no—
338C22×C4—5421no—
339C22×C4—5428no—
340C22×C4—5421no—
341C22×C4—5421no—
342C22×C4—5421no—
343C22×C4—5421no—
344C22×C4C22×D27111yescharacteristic
345Q16—22no—
346Q16—5422no—
347Q16—5422no—
348Q16—5422no—
349SD16—22no—
350SD16—5422no—
351SD16—5422no—
352SD16—5422no—
353M16—5422no—
354M16—5422no—
355C2×C8—2711no—
356C2×C8—2711no—
357C2×C8C22×D27111yescharacteristic
358C2×C8C22×D27111yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—2711no—
361C4⋊C4—5421no—
362C4⋊C4—5421no—
363C4⋊C4—5421no—
364C4⋊C4—5421no—
365C4⋊C4C22×D27111yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—5421no—
368C22⋊C4—5421no—
369C22⋊C4—5421no—
370C23—21no—
371C23—2711no—
372C23—2711no—
373C23—5422no—
374C23—5422no—
375C23—5424no—
376C23—5421no—
377C23—10841no—
378C23—10842no—
379C23—10842no—
380C23D4×D27111yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—2712no—
385Q8—5424no—
386Q8—5421no—
387Q8—5421no—
388Q8—5421no—
389Q8—10842no—
390Q8D4×D27112yes—
391D4—24no—
392D4—21no—
393D4—2712no—
394D4—5424no—
395D4—5424no—
396D4—5424no—
397D4—5424no—
398D4—5421no—
399D4D4×D27112yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—2711no—
407C2×C4—2712no—
408C2×C4—2712no—
409C2×C4—2711no—
410C2×C4—5422no—
411C2×C4—5424no—
412C2×C4—5422no—
413C2×C4—5424no—
414C2×C4—5424no—
415C2×C4—5422no—
416C2×C4—5424no—
417C2×C4—5421no—
418C2×C4—5421no—
419C2×C4—5421no—
420C2×C4—5421no—
421C2×C4—5421no—
422C2×C4—10842no—
423C2×C4—10842no—
424C2×C4D4×D27111yescharacteristic
425C2×C4C23×D27111yescharacteristic
426C8—21no—
427C8—21no—
428C8—5421no—
429C8—5421no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—2712no—
437C22—2712no—
438C22—5421no—
439C22—5421no—
440C22—5421no—
441C22—5421no—
442C22—5422no—
443C22—10844no—
444C22—10848no—
445C22—10841no—
446C22—10841no—
447C22C2×D4×D27111yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—2712no—
453C4—5421no—
454C4—5422no—
455C4—10841no—
456C4—10841no—
457C4C2×D4×D27111yescharacteristic
458C4C2×D4×D27111yescharacteristic
459C2—41no—
460C2—42no—
461C2—5422no—
462C2—5422no—
463C2—10842no—
464C2#120×D27111yescharacteristic
465C2#137×D27111yescharacteristic
466C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#452×C271C2
socleC22×C271C22×D4
prime radicalC2×#41×D271C2
hypercenter#452D271
hypocenterC271#3007
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#22576192C2semidirect
3. C2×#22576189C2semidirect
4. C22⋊Q8×D271C2semidirect
5. #31774061C2semidirect
6. #31774059C2semidirect
7. #31774057C2semidirect
8. #31774054C2semidirect
9. #31774051C2semidirect
10. #31774049C2semidirect
11. #31774046C2semidirect
12. #31773897C2semidirect
13. #31773895C2semidirect
14. #31773859C2semidirect
15. #452×C271C2semidirect
24. C2×Q8×D271C22semidirect
31. C2×Dic1084C22semidirect
32. C2×#16864644C22semidirect
33. C2×#16864644C22semidirect
34. C2×#16864643C22semidirect
35. C2×#16864642C22semidirect
36. C2×#16864641C22semidirect
37. C2×#16864637C22semidirect
38. C2×#16864637C22semidirect
40. #22576139C22semidirect
45. C4⋊C4×D271C22semidirect
47. #22576085C22semidirect
48. #22576084C22semidirect
49. #22576083C22semidirect
50. #22576082C22semidirect
51. #22576081C22semidirect
52. #22576080C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
57. C2×Q16×C271C22semidirect
58. C2×SD16×C271C22semidirect
59. C22⋊Q8×C271C22semidirect
60. #103×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
213. C2×D271#137semidirect
233. C271#3007semidirect
242. #452D271semidirect
269. C2×Q16C2×D271semidirect
273. C2×SD16C2×D271semidirect
281. C22⋊Q8C2×D271semidirect
288. #103C2×D271semidirect
292. #102C2×D271semidirect
296. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2712
C2×#225761922
C2×#225761892
C22⋊Q8×D2712
#317740612
#317740592
#317740572
#317740542
#317740512
#317740492
#317740462
#317738972
#317738952
#317738592
#452×C2712
#3007271