Finite Group Databasebeta

← #46845602

Subgroups — #46845602

Order 34688.

758 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D271C211yesmaximal, characteristic, complemented
2C2×#22576189C211yesmaximal, characteristic, complemented
3C2×#22576184C211yesmaximal, characteristic, complemented
4#31774407C211yesmaximal, characteristic, complemented
5#31774062C211yesmaximal, characteristic, complemented
6#31774061C211yesmaximal, characteristic, complemented
7#31774060C211yesmaximal, characteristic, complemented
8#31774059C211yesmaximal, characteristic, complemented
9#31774041C211yesmaximal, characteristic, complemented
10#31774040C211yesmaximal, characteristic, complemented
11#31774039C211yesmaximal, characteristic, complemented
12#31773904C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14#31773859C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654—21no—
18C2×#16864654C2211yescharacteristic
19C2×#16864653—21no—
20C2×#16864653C2211yescharacteristic
21C2×#16864653C2211yescharacteristic
22#41×D271—24no—
23#41×D271—24no—
24C2×Q8×D271C2211yescharacteristic
25C2×D4×D271—21no—
26C2×D4×D271C2211yescharacteristic
27C22×C4×D271—21no—
28C22×C4×D271C2211yescharacteristic
29#22576189—24no—
30#22576184—24no—
31C2×Dic1084C2211yescharacteristic, complemented
32C2×#16864644C2211yescharacteristic, complemented
33C2×#16864643C2211yescharacteristic, complemented
34C2×#16864642C2211yescharacteristic, complemented
35C2×#16864642C2211yescharacteristic, complemented
36C2×#16864639C2211yescharacteristic, complemented
37C2×#16864637C2211yescharacteristic
38C2×#16864637C2211yescharacteristic
39#22576142—21no—
40#22576140C2211yescharacteristic, complemented
41#22576139C2211yescharacteristic, complemented
42#22576131C2211yescharacteristic, complemented
43#22576128—21no—
44D4×Dic271—21no—
45(C22⋊D271)⋊C4—21no—
46#22576118C2211yescharacteristic
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, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
56C2×#41×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic, complemented
58C2×D8×C271C2211yescharacteristic, complemented
59#121×C271C2211yescharacteristic, complemented
60#103×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62C22⋊C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271D411yescharacteristic
65#16864655—22no—
66#16864655—22no—
67#16864654—28no—
68#16864654—22no—
69#16864654—22no—
70#16864653—28no—
71#16864653—22no—
72#16864653—22no—
73#16864653—22no—
74#16864653—22no—
75C2×Dic542—21no—
76C2×Dic542C2311yescharacteristic
77C2×Dic542C2311yescharacteristic
78Q8×D271—22no—
79Q8×D271D414yescomplemented
80C2×D1084—21no—
81C2×D1084C2311yescharacteristic
82C2×C22⋊D271—21no—
83C2×C22⋊D271—21no—
84C2×C22⋊D271—21no—
85C2×C22⋊D271—21no—
86C2×C22⋊D271—21no—
87D4×D271—28no—
88D4×D271—22no—
89D4×D271D414yescomplemented
90C22×Dic271—21no—
91C22×Dic271—21no—
92C22×Dic271D411yescharacteristic
93C2×C4×D271—28no—
94C2×C4×D271—21no—
95C2×C4×D271—21no—
96C2×C4×D271—22no—
97C2×C4×D271—22no—
98C2×C4×D271—21no—
99C2×C4×D271D411yescharacteristic
100C2×C4×D271D411yescharacteristic
101C2×C4×D271C2311yescharacteristic
102C2×C4×D271C2311yescharacteristic
103Dic1084—22no—
104#16864644—22no—
105#16864643—22no—
106#16864642—22no—
107#16864642—22no—
108#16864639—22no—
109#16864637—22no—
110#16864637—22no—
111C2×C271⋊C8C2311yescharacteristic
112C2×C271⋊C8C2311yescharacteristic
113C4⋊Dic271—21no—
114C4⋊Dic271C2311yescharacteristic
115Dic271⋊C4—21no—
116C22⋊Dic271—21no—
117C22⋊Dic271—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—22no—
138C22×D271—24no—
139C22×D271—41no—
140C22×D271—41no—
141C22×D271—42no—
142C22×D271—42no—
143C22×D271C2×D411yescharacteristic
144C22×D271C2×D411yescharacteristic
145Dic542—21no—
146Dic542—24no—
147Dic542—21no—
148Dic542—21no—
149Dic542C2×D412yes—
150D1084—24no—
151D1084—21no—
152D1084C2×D412yes—
153C22⋊D271—24no—
154C22⋊D271—24no—
155C22⋊D271—24no—
156C22⋊D271—42no—
157C22⋊D271—42no—
158C2×Dic271—24no—
159C2×Dic271—21no—
160C2×Dic271—22no—
161C2×Dic271—21no—
162C2×Dic271—21no—
163C2×Dic271—42no—
164C2×Dic271C2×D411yescharacteristic
165C2×Dic271C2×D411yescharacteristic
166C4×D271—24no—
167C4×D271—24no—
168C4×D271—22no—
169C4×D271—24no—
170C4×D271—21no—
171C4×D271—21no—
172C4×D271—22no—
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—42no—
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—22no—
211C2×D271—21no—
212C2×D271—21no—
213C2×D271—44no—
214C2×D271—48no—
215C2×D271—41no—
216C2×D271—41no—
217C2×D271#12012yes—
218C4×C271—21no—
219C4×C271—22no—
220C4×C271—41no—
221C4×C271C22×D411yescharacteristic
222C4×C271C22×D411yescharacteristic
223Dic271—21no—
224Dic271—22no—
225Dic271—41no—
226Dic271—41no—
227Dic271#12012yes—
228C2×C271—41no—
229C2×C271—42no—
230C2×C271—81no—
231C2×C271C2×#12011yescharacteristic
232C2×C271#57511yescharacteristic
233C2×C271#57511yescharacteristic
234D271—22no—
235D271—41no—
236D271—42no—
237C271#300511yescharacteristic, complemented, Sylow
238#3005—2711nomaximal, Sylow
239C22×#41—2711no—
240C2×#137—2711no—
241C2×AGL(1,ℤ/8)—2711no—
242#525—2711no—
243#452—2711no—
244#452—2711no—
245#452—2711no—
246#452—2711no—
247#449—2711no—
248#449—2711no—
249#449—2711no—
250#449D27111yescharacteristic, complemented
251#419—2711no—
252#418—2711no—
253#407—2711no—
254C2×#41—2711no—
255C2×#41—2711no—
256C2×#41—2711no—
257C2×#41—2711no—
258C2×#41—5421no—
259C2×#41—5421no—
260C2×#41—5424no—
261C2×#41—5424no—
262C2×#41C2×D27111yescharacteristic
263C22×Q8—2711no—
264C22×D4—2711no—
265C22×D4—5421no—
266C23×C4—2711no—
267C23×C4—5421no—
268#137—5424no—
269AGL(1,ℤ/8)—5424no—
270C2×Q16—2711no—
271C2×Q16—2711no—
272C2×SD16—2711no—
273C2×SD16—2711no—
274C2×SD16—2711no—
275C2×SD16C2×D27111yescharacteristic, complemented
276C2×D8—2711no—
277C2×D8C2×D27111yescharacteristic, complemented
278C2×M16—2711no—
279C2×M16—2711no—
280#123—5421no—
281C22⋊Q8—2711no—
282C22⋊Q8—2711no—
283#121—2711no—
284#121C2×D27111yescharacteristic, complemented
285#120—5421no—
286C4×D4—5421no—
287C4×D4—5421no—
288#117—2711no—
289#103—2711no—
290#103—2711no—
291#103—2711no—
292#103C2×D27111yescharacteristic, complemented
293#102—2711no—
294#102—2711no—
295#102—2711no—
296#102C2×D27111yescharacteristic, complemented
297C22⋊C8—2711no—
298C22⋊C8—2711no—
299C22⋊C8—2711no—
300C22⋊C8C2×D27111yescharacteristic, complemented
301C24—2711no—
302C24—5421no—
303#41—22no—
304#41—22no—
305#41—5422no—
306#41—5422no—
307#41—5428no—
308#41—5422no—
309#41—5422no—
310#41—5422no—
311#41—5428no—
312#41—5422no—
313#41—5422no—
314#41—5422no—
315C2×Q8—2714no—
316C2×Q8—2711no—
317C2×Q8—2711no—
318C2×Q8—5421no—
319C2×Q8—5422no—
320C2×Q8C22×D27111yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—2714no—
324C2×D4—2711no—
325C2×D4—5421no—
326C2×D4—5422no—
327C2×D4—5421no—
328C2×D4—5428no—
329C2×D4—5421no—
330C2×D4—5421no—
331C2×D4—5421no—
332C2×D4—5421no—
333C2×D4C22×D27111yescharacteristic
334C2×D4C22×D27111yescharacteristic
335C22×C4—21no—
336C22×C4—2711no—
337C22×C4—2711no—
338C22×C4—2711no—
339C22×C4—2711no—
340C22×C4—2711no—
341C22×C4—5422no—
342C22×C4—5422no—
343C22×C4—5421no—
344C22×C4—5421no—
345C22×C4—5428no—
346C22×C4—5421no—
347C22×C4—5421no—
348C22×C4—5421no—
349C22×C4C22×D27111yescharacteristic
350Q16—5422no—
351Q16—5422no—
352SD16—22no—
353SD16—5422no—
354SD16—5422no—
355SD16—5422no—
356D8—22no—
357D8—5422no—
358M16—5422no—
359M16—5422no—
360C2×C8—2711no—
361C2×C8—2711no—
362C2×C8C22×D27111yescharacteristic
363C2×C8C22×D27111yescharacteristic
364C4⋊C4—2711no—
365C4⋊C4—5421no—
366C4⋊C4—5421no—
367C4⋊C4C22×D27111yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—5421no—
370C22⋊C4—5421no—
371C22⋊C4—5421no—
372C22⋊C4—5421no—
373C42—5421no—
374C23—21no—
375C23—21no—
376C23—2711no—
377C23—2711no—
378C23—5424no—
379C23—5422no—
380C23—5421no—
381C23—10841no—
382C23—10841no—
383C23—10842no—
384C23—10842no—
385C23D4×D27111yescharacteristic
386Q8—21no—
387Q8—2712no—
388Q8—5424no—
389Q8—5421no—
390Q8—5421no—
391Q8—5421no—
392Q8D4×D27112yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—2712no—
399D4—5424no—
400D4—5424no—
401D4—5424no—
402D4—5424no—
403D4—5421no—
404D4—10842no—
405D4—10842no—
406D4D4×D27112yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—2711no—
413C2×C4—2712no—
414C2×C4—2712no—
415C2×C4—2711no—
416C2×C4—5422no—
417C2×C4—5422no—
418C2×C4—5424no—
419C2×C4—5424no—
420C2×C4—5424no—
421C2×C4—5422no—
422C2×C4—5424no—
423C2×C4—5421no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4—10842no—
429C2×C4D4×D27111yescharacteristic
430C2×C4C23×D27111yescharacteristic
431C8—21no—
432C8—21no—
433C8—5421no—
434C8—5421no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—2712no—
445C22—5421no—
446C22—5421no—
447C22—5422no—
448C22—5421no—
449C22—5421no—
450C22—5421no—
451C22—10844no—
452C22—10848no—
453C22—10841no—
454C22—10841no—
455C22C2×D4×D27111yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—2712no—
460C4—5421no—
461C4—5422no—
462C4—10841no—
463C4—10841no—
464C4C2×D4×D27111yescharacteristic
465C4C2×D4×D27111yescharacteristic
466C2—41no—
467C2—42no—
468C2—81no—
469C2—5422no—
470C2—10841no—
471C2—10842no—
472C2#120×D27111yescharacteristic
473C2#3177469311yescharacteristic
474C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3005
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×#41×D271C2semidirect
2. C2×#22576189C2semidirect
3. C2×#22576184C2semidirect
4. #31774407C2semidirect
5. #31774062C2semidirect
6. #31774061C2semidirect
7. #31774060C2semidirect
8. #31774059C2semidirect
9. #31774041C2semidirect
10. #31774040C2semidirect
11. #31774039C2semidirect
12. #31773904C2semidirect
13. #31773900C2semidirect
14. #31773859C2semidirect
15. #449×C271C2semidirect
31. C2×Dic1084C22semidirect
32. C2×#16864644C22semidirect
33. C2×#16864643C22semidirect
34. C2×#16864642C22semidirect
35. C2×#16864642C22semidirect
36. C2×#16864639C22semidirect
40. #22576140C22semidirect
41. #22576139C22semidirect
42. #22576131C22semidirect
47. #22576085C22semidirect
48. #22576084C22semidirect
49. #22576083C22semidirect
50. #22576082C22semidirect
51. #22576081C22semidirect
52. #22576080C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
57. C2×SD16×C271C22semidirect
58. C2×D8×C271C22semidirect
59. #121×C271C22semidirect
60. #103×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
79. Q8×D271D4semidirect
89. D4×D271D4semidirect
237. C271#3005semidirect
250. #449D271semidirect
275. C2×SD16C2×D271semidirect
277. C2×D8C2×D271semidirect
284. #121C2×D271semidirect
292. #103C2×D271semidirect
296. #102C2×D271semidirect
300. 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×#225761892
C2×#225761842
#317744072
#317740622
#317740612
#317740602
#317740592
#317740412
#317740402
#317740392
#317739042
#317739002
#317738592
#449×C2712
#3005271