Finite Group Databasebeta

← #46845636

Subgroups — #46845636

Order 34688.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576205C211yesmaximal, characteristic
2C2×#22576189C211yesmaximal, characteristic, complemented
3C2×SD16×D271C211yesmaximal, characteristic, complemented
4#31774470C211yesmaximal, characteristic, complemented
5#31774061C211yesmaximal, characteristic, complemented
6#31774060C211yesmaximal, characteristic, complemented
7#31774057C211yesmaximal, characteristic, complemented
8#31774056C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774048C211yesmaximal, characteristic, complemented
11#31774046C211yesmaximal, characteristic, complemented
12#31773900C211yesmaximal, characteristic, complemented
13#31773890C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#450×C271C211yesmaximal, characteristic, complemented
16#22576205—24no—
17#22576205—24no—
18C2×#16864654C2211yescharacteristic
19C2×#16864654C2211yescharacteristic
20C2×#16864653—21no—
21C2×#16864653C2211yescharacteristic
22C2×#16864653C2211yescharacteristic
23C2×Q8×D271C2211yescharacteristic, complemented
24C22×C22⋊D271—21no—
25C22×C22⋊D271—21no—
26C2×D4×D271—21no—
27C2×D4×D271C2211yescharacteristic
28C2×D4×D271C2211yescharacteristic
29#22576189—24no—
30C2×Dic1084C2211yescharacteristic, complemented
31C2×#16864644C2211yescharacteristic, complemented
32C2×#16864643C2211yescharacteristic, complemented
33C2×#16864642C2211yescharacteristic, complemented
34C2×#16864642C2211yescharacteristic, complemented
35C2×#16864641C2211yescharacteristic, complemented
36SD16×D271—24no—
37C2×#16864637C2211yescharacteristic, complemented
38C2×C8×D271C2211yescharacteristic, complemented
39#22576159—21no—
40#22576146—21no—
41#22576145—21no—
42#22576142—21no—
43#22576140C2211yescharacteristic, complemented
44#22576139C2211yescharacteristic, complemented
45#22576136C2211yescharacteristic, complemented
46#22576118C2211yescharacteristic, complemented
47#22576085C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52#22576080C2211yescharacteristic, complemented
53C22⋊(C271⋊C8)C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
56C22×D4×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic, complemented
58C2×SD16×C271C2211yescharacteristic, complemented
59C22⋊Q8×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62C22⋊C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271—21no—
65#16864654—22no—
66#16864654—22no—
67#16864654—22no—
68#16864654—22no—
69#16864653—28no—
70#16864653—22no—
71#16864653—22no—
72#16864653—22no—
73#16864653D414yes—
74C2×Dic542—21no—
75C2×Dic542D411yescharacteristic
76C2×Dic542C2311yescharacteristic
77C2×Dic542C2311yescharacteristic
78Q8×D271—22no—
79Q8×D271—22no—
80C2×D1084D411yescharacteristic
81C2×D1084C2311yescharacteristic
82C2×C22⋊D271—28no—
83C2×C22⋊D271—28no—
84C2×C22⋊D271—21no—
85C2×C22⋊D271—21no—
86C2×C22⋊D271—21no—
87C2×C22⋊D271—21no—
88C2×C22⋊D271—21no—
89C2×C22⋊D271D411yescharacteristic
90C2×C22⋊D271D411yescharacteristic
91D4×D271—28no—
92D4×D271—22no—
93D4×D271—22no—
94D4×D271—22no—
95D4×D271D414yes—
96C22×Dic271—21no—
97C22×Dic271—21no—
98C2×C4×D271—21no—
99C2×C4×D271—21no—
100C2×C4×D271C2311yescharacteristic
101C2×C4×D271C2311yescharacteristic
102Dic1084—22no—
103#16864644—22no—
104#16864643—22no—
105#16864642—22no—
106#16864642—22no—
107#16864641—22no—
108#16864637—22no—
109C2×C271⋊C8C2311yescharacteristic
110C2×C271⋊C8C2311yescharacteristic
111C8×D271—22no—
112C4⋊Dic271—21no—
113C4⋊Dic271C2311yescharacteristic
114Dic271⋊C4—21no—
115Dic271⋊C4—21no—
116C22⋊Dic271—21no—
117(C2×D271)⋊C4—21no—
118(C2×D271)⋊C4—21no—
119(C2×D271)⋊C4—21no—
120C4×Dic271—21no—
121C24×C271—21no—
122C2×Q8×C271C2311yescharacteristic
123C2×D4×C271—22no—
124C2×D4×C271—22no—
125C2×D4×C271—21no—
126C2×D4×C271C2311yescharacteristic
127C2×D4×C271C2311yescharacteristic
128C22×C4×C271C2311yescharacteristic
129SD16×C271—22no—
130SD16×C271—22no—
131C2×C8×C271C2311yescharacteristic
132C2×C8×C271C2311yescharacteristic
133C4⋊C4×C271—21no—
134C4⋊C4×C271C2311yescharacteristic
135C22⋊C4×C271—21no—
136C22×D271—21no—
137C22×D271—24no—
138C22×D271—24no—
139C22×D271—81no—
140C22×D271—81no—
141C22×D271—81no—
142C22×D271—81no—
143C22×D271C2×D411yescharacteristic
144C22×D271C2×D411yescharacteristic
145Dic542—21no—
146Dic542—22no—
147Dic542—21no—
148Dic542—21no—
149Dic542—42no—
150Dic542C2×D412yes—
151D1084—22no—
152D1084—21no—
153D1084C2×D412yes—
154C22⋊D271—24no—
155C22⋊D271—24no—
156C22⋊D271—24no—
157C22⋊D271—24no—
158C22⋊D271—22no—
159C22⋊D271—24no—
160C22⋊D271—22no—
161C2×Dic271—24no—
162C2×Dic271—21no—
163C2×Dic271—24no—
164C2×Dic271—21no—
165C2×Dic271—21no—
166C2×Dic271C2×D411yescharacteristic
167C2×Dic271C2×D411yescharacteristic
168C4×D271—24no—
169C4×D271—21no—
170C4×D271—21no—
171C4×D271—42no—
172C4×D271C2×D412yes—
173C4×D271C2×D412yes—
174C271⋊C8—21no—
175C271⋊C8—21no—
176C23×C271—24no—
177C23×C271—21no—
178C23×C271—21no—
179C23×C271—81no—
180C23×C271—81no—
181C23×C271C2×D411yescharacteristic
182Q8×C271—21no—
183Q8×C271—21no—
184D4×C271—24no—
185D4×C271—21no—
186D4×C271—21no—
187D4×C271C2×D412yes—
188D4×C271C2×D412yes—
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—22no—
200C22×C271—21no—
201C22×C271—21no—
202C22×C271—41no—
203C22×C271—41no—
204C22×C271—41no—
205C22×C271—41no—
206C22×C271—84no—
207C22×C271C22×D411yescharacteristic
208C2×D271—21no—
209C2×D271—21no—
210C2×D271—21no—
211C2×D271—22no—
212C2×D271—21no—
213C2×D271—21no—
214C2×D271—41no—
215C2×D271—41no—
216C2×D271—84no—
217C2×D271—84no—
218C2×D271#12012yes—
219C4×C271—21no—
220C4×C271—41no—
221C4×C271—41no—
222C4×C271C22×D411yescharacteristic
223C4×C271C22×D411yescharacteristic
224Dic271—22no—
225Dic271—21no—
226Dic271—41no—
227Dic271—41no—
228Dic271#12012yes—
229C2×C271—42no—
230C2×C271—42no—
231C2×C271—41no—
232C2×C271C2×#12011yescharacteristic
233C2×C271#57711yescharacteristic
234C2×C271#57711yescharacteristic
235D271—22no—
236D271—41no—
237D271—42no—
238C271#300911yescharacteristic, complemented, Sylow
239#3009—2711nomaximal, Sylow
240C2×H(5,𝔽2)—2711no—
241C2×#137—2711no—
242C22×SD16—2711no—
243#536—2711no—
244#452—2711no—
245#452—2711no—
246#452—2711no—
247#452—2711no—
248#450—2711no—
249#450—2711no—
250#450—2711no—
251#450D27111yescharacteristic, complemented
252#418—2711no—
253#416—2711no—
254#408—2711no—
255H(5,𝔽2)—5424no—
256H(5,𝔽2)—5424no—
257C2×#41—2711no—
258C2×#41—2711no—
259C2×#41—2711no—
260C2×#41—2711no—
261C2×#41—5421no—
262C22×Q8—2711no—
263C22×D4—2711no—
264C22×D4—2711no—
265C22×D4—5421no—
266C22×D4—5421no—
267C22×D4—5421no—
268C22×D4C2×D27111yescharacteristic
269#137—5424no—
270C2×Q16—2711no—
271C2×Q16—2711no—
272C2×SD16—2711no—
273C2×SD16—2711no—
274C2×SD16—2711no—
275C2×SD16—2711no—
276C2×SD16—5424no—
277C2×SD16C2×D27111yescharacteristic, complemented
278C2×SD16C2×D27111yescharacteristic, complemented
279C2×M16—2711no—
280C22×C8—2711no—
281C4⋊Q8—5421no—
282#124—5421no—
283#123—5421no—
284#123—5421no—
285C22⋊Q8—2711no—
286C22⋊Q8—2711no—
287C22⋊Q8—2711no—
288C22⋊Q8C2×D27111yescharacteristic, complemented
289#117—2711no—
290#103—2711no—
291#103—2711no—
292#102—2711no—
293#102—2711no—
294#102—2711no—
295#102—2711no—
296#102C2×D27111yescharacteristic, complemented
297#102C2×D27111yescharacteristic, complemented
298C22⋊C8—2711no—
299C22⋊C8—2711no—
300C22⋊C8—2711no—
301C22⋊C8C2×D27111yescharacteristic, complemented
302C24—21no—
303C24—5421no—
304C24—5421no—
305#41—2714no—
306#41—5422no—
307#41—5422no—
308#41—5422no—
309#41—5422no—
310#41—5422no—
311#41—5422no—
312#41—5428no—
313#41—5422no—
314C2×Q8—2711no—
315C2×Q8—2711no—
316C2×Q8—2711no—
317C2×Q8—5421no—
318C2×Q8—5422no—
319C2×Q8—5422no—
320C2×Q8C22×D27111yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—2714no—
325C2×D4—2711no—
326C2×D4—2711no—
327C2×D4—2711no—
328C2×D4—2711no—
329C2×D4—5422no—
330C2×D4—5421no—
331C2×D4—5421no—
332C2×D4—5422no—
333C2×D4—5428no—
334C2×D4—5421no—
335C2×D4—5421no—
336C2×D4—5428no—
337C2×D4—5428no—
338C2×D4—5421no—
339C2×D4—5422no—
340C2×D4C22×D27111yescharacteristic
341C2×D4C22×D27111yescharacteristic
342C22×C4—2711no—
343C22×C4—2711no—
344C22×C4—5421no—
345C22×C4—5421no—
346C22×C4—5421no—
347C22×C4—5421no—
348C22×C4C22×D27111yescharacteristic
349Q16—5422no—
350Q16—5422no—
351SD16—22no—
352SD16—22no—
353SD16—5422no—
354SD16—5422no—
355SD16—5422no—
356SD16—5422no—
357M16—5422no—
358C2×C8—2711no—
359C2×C8—2711no—
360C2×C8—5422no—
361C2×C8C22×D27111yescharacteristic
362C2×C8C22×D27111yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—2711no—
365C4⋊C4—5421no—
366C4⋊C4—5421no—
367C4⋊C4—5421no—
368C4⋊C4C22×D27111yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—5421no—
371C22⋊C4—5421no—
372C22⋊C4—5421no—
373C22⋊C4—5421no—
374C42—5421no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—81no—
379C23—81no—
380C23—2711no—
381C23—2711no—
382C23—5424no—
383C23—5424no—
384C23—5421no—
385C23—21681no—
386C23—21681no—
387C23—21681no—
388C23—21681no—
389C23D4×D27111yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—2712no—
393Q8—5422no—
394Q8—5421no—
395Q8—5421no—
396Q8—5421no—
397Q8—10842no—
398D4—24no—
399D4—21no—
400D4—21no—
401D4—2712no—
402D4—5422no—
403D4—5422no—
404D4—5424no—
405D4—5422no—
406D4—5424no—
407D4—5424no—
408D4—5424no—
409D4—5424no—
410D4—5421no—
411D4D4×D27112yes—
412D4D4×D27112yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—2711no—
418C2×C4—2712no—
419C2×C4—2712no—
420C2×C4—2711no—
421C2×C4—5424no—
422C2×C4—5424no—
423C2×C4—5424no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4—5421no—
429C2×C4—10842no—
430C2×C4D4×D27111yescharacteristic
431C2×C4C23×D27111yescharacteristic
432C8—21no—
433C8—21no—
434C8—5421no—
435C8—5421no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—84no—
446C22—2712no—
447C22—5421no—
448C22—5421no—
449C22—5422no—
450C22—5421no—
451C22—5421no—
452C22—5421no—
453C22—10841no—
454C22—10841no—
455C22—21684no—
456C22—21684no—
457C22C2×D4×D27111yescharacteristic
458C4—21no—
459C4—41no—
460C4—41no—
461C4—2712no—
462C4—5421no—
463C4—5422no—
464C4—10841no—
465C4—10841no—
466C4C2×D4×D27111yescharacteristic
467C4C2×D4×D27111yescharacteristic
468C2—41no—
469C2—42no—
470C2—42no—
471C2—5422no—
472C2—10841no—
473C2—10842no—
474C2#120×D27111yescharacteristic
475C2#3177470311yescharacteristic
476C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#450×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576205C2
hypercenter#450D271
hypocenterC271#3009
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#22576189C2semidirect
3. C2×SD16×D271C2semidirect
4. #31774470C2semidirect
5. #31774061C2semidirect
6. #31774060C2semidirect
7. #31774057C2semidirect
8. #31774056C2semidirect
9. #31774049C2semidirect
10. #31774048C2semidirect
11. #31774046C2semidirect
12. #31773900C2semidirect
13. #31773890C2semidirect
14. #31773862C2semidirect
15. #450×C271C2semidirect
23. C2×Q8×D271C22semidirect
30. C2×Dic1084C22semidirect
31. C2×#16864644C22semidirect
32. C2×#16864643C22semidirect
33. C2×#16864642C22semidirect
34. C2×#16864642C22semidirect
35. C2×#16864641C22semidirect
37. C2×#16864637C22semidirect
38. C2×C8×D271C22semidirect
43. #22576140C22semidirect
44. #22576139C22semidirect
45. #22576136C22semidirect
46. #22576118C22semidirect
47. #22576085C22semidirect
48. #22576083C22semidirect
49. #22576082C22semidirect
50. #22576081C22semidirect
51. #22576081C22semidirect
52. #22576080C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
57. C2×SD16×C271C22semidirect
58. C2×SD16×C271C22semidirect
59. C22⋊Q8×C271C22semidirect
60. #102×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
238. C271#3009semidirect
251. #450D271semidirect
277. C2×SD16C2×D271semidirect
278. C2×SD16C2×D271semidirect
288. C22⋊Q8C2×D271semidirect
296. #102C2×D271semidirect
297. #102C2×D271semidirect
301. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762052
C2×#225761892
C2×SD16×D2712
#317744702
#317740612
#317740602
#317740572
#317740562
#317740492
#317740482
#317740462
#317739002
#317738902
#317738622
#450×C2712
#3009271