Finite Group Databasebeta

← #46845638

Subgroups — #46845638

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×#22576206C211yesmaximal, characteristic
2C2×#22576192C211yesmaximal, characteristic, complemented
3C2×SD16×D271C211yesmaximal, characteristic, complemented
4#31774470C211yesmaximal, characteristic, complemented
5#31774062C211yesmaximal, characteristic, complemented
6#31774059C211yesmaximal, characteristic, complemented
7#31774058C211yesmaximal, characteristic, complemented
8#31774050C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774047C211yesmaximal, characteristic, complemented
11#31774046C211yesmaximal, characteristic, complemented
12#31773900C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#452×C271C211yesmaximal, characteristic, complemented
16#22576206—24no—
17#22576206—24no—
18C2×#16864655C2211yescharacteristic
19C2×#16864655C2211yescharacteristic
20C2×#16864654—21no—
21C2×#16864654C2211yescharacteristic
22C2×Q8×D271C2211yescharacteristic, complemented
23C22×D1084—21no—
24C22×D1084C2211yescharacteristic
25C2×D4×D271—21no—
26C2×D4×D271—21no—
27C2×D4×D271C2211yescharacteristic
28C2×D4×D271C2211yescharacteristic
29#22576192—24no—
30C2×#16864644C2211yescharacteristic, complemented
31C2×#16864643C2211yescharacteristic, complemented
32C2×#16864643C2211yescharacteristic, complemented
33C2×#16864642C2211yescharacteristic, complemented
34C2×#16864641C2211yescharacteristic, 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#22576084C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576080C2211yescharacteristic, complemented
52#22576080C2211yescharacteristic, complemented
53C22⋊(C271⋊C8)C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
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×D271—21no—
65C23×D271—21no—
66#16864655—22no—
67#16864655—22no—
68#16864655—22no—
69#16864655D414yes—
70#16864654—28no—
71#16864654—22no—
72#16864654—22no—
73C2×Dic542—21no—
74C2×Dic542D411yescharacteristic
75C2×Dic542C2311yescharacteristic
76Q8×D271—22no—
77Q8×D271—22no—
78C2×D1084—28no—
79C2×D1084—21no—
80C2×D1084—21no—
81C2×D1084—22no—
82C2×D1084—22no—
83C2×D1084—21no—
84C2×D1084D411yescharacteristic
85C2×D1084C2311yescharacteristic
86C2×D1084C2311yescharacteristic
87C2×C22⋊D271—21no—
88C2×C22⋊D271—21no—
89C2×C22⋊D271D411yescharacteristic
90C2×C22⋊D271D411yescharacteristic
91D4×D271—28no—
92D4×D271—28no—
93D4×D271—22no—
94D4×D271—22no—
95D4×D271—22no—
96D4×D271D414yes—
97C2×C4×D271—21no—
98C2×C4×D271—21no—
99C2×C4×D271—21no—
100C2×C4×D271C2311yescharacteristic
101C2×C4×D271C2311yescharacteristic
102#16864644—22no—
103#16864643—22no—
104#16864643—22no—
105#16864642—22no—
106#16864641—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—
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—21no—
137C22×D271—21no—
138C22×D271—24no—
139C22×D271—24no—
140C22×D271—24no—
141C22×D271—81no—
142C22×D271—81no—
143C22×D271—81no—
144C22×D271—81no—
145C22×D271—81no—
146C22×D271—81no—
147C22×D271C2×D411yescharacteristic
148C22×D271C2×D411yescharacteristic
149Dic542—22no—
150Dic542—21no—
151Dic542—21no—
152Dic542—42no—
153D1084—24no—
154D1084—24no—
155D1084—24no—
156D1084—22no—
157D1084—21no—
158D1084—21no—
159D1084C2×D412yes—
160D1084C2×D412yes—
161C22⋊D271—24no—
162C22⋊D271—22no—
163C22⋊D271—22no—
164C22⋊D271—24no—
165C2×Dic271—21no—
166C2×Dic271—21no—
167C2×Dic271C2×D411yescharacteristic
168C2×Dic271C2×D411yescharacteristic
169C4×D271—24no—
170C4×D271—21no—
171C4×D271—21no—
172C4×D271—24no—
173C4×D271—42no—
174C4×D271C2×D412yes—
175C4×D271C2×D412yes—
176C271⋊C8—21no—
177C271⋊C8—21no—
178C23×C271—21no—
179C23×C271C2×D411yescharacteristic
180Q8×C271—21no—
181Q8×C271—21no—
182Q8×C271—21no—
183Q8×C271C2×D412yes—
184D4×C271—24no—
185D4×C271—21no—
186D4×C271C2×D412yes—
187C2×C4×C271—24no—
188C2×C4×C271—21no—
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×C271C22×D411yescharacteristic
204C2×D271—21no—
205C2×D271—21no—
206C2×D271—21no—
207C2×D271—22no—
208C2×D271—22no—
209C2×D271—21no—
210C2×D271—21no—
211C2×D271—41no—
212C2×D271—41no—
213C2×D271—41no—
214C2×D271—41no—
215C2×D271—84no—
216C2×D271—84no—
217C2×D271—84no—
218C2×D271#12012yes—
219C4×C271—21no—
220C4×C271—22no—
221C4×C271—41no—
222C4×C271—41no—
223C4×C271C22×D411yescharacteristic
224C4×C271C22×D411yescharacteristic
225Dic271—21no—
226Dic271—41no—
227Dic271—41no—
228Dic271#12012yes—
229C2×C271—41no—
230C2×C271—42no—
231C2×C271C2×#12011yescharacteristic
232C2×C271#57711yescharacteristic
233C2×C271#57711yescharacteristic
234D271—22no—
235D271—41no—
236D271—42no—
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#452D27111yescharacteristic, complemented
248#450—2711no—
249#450—2711no—
250#450—2711no—
251#450—2711no—
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—5421no—
261C2×#41C2×D27111yescharacteristic
262C22×Q8—2711no—
263C22×D4—2711no—
264C22×D4—2711no—
265C22×D4—2711no—
266C22×D4—5421no—
267C22×D4—5421no—
268C22×D4—5421no—
269#137—5424no—
270C2×Q16—2711no—
271C2×Q16C2×D27111yescharacteristic, complemented
272C2×SD16—2711no—
273C2×SD16—2711no—
274C2×SD16—2711no—
275C2×SD16—2711no—
276C2×SD16—2711no—
277C2×SD16—5424no—
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#103C2×D27111yescharacteristic, complemented
292#102—2711no—
293#102—2711no—
294#102—2711no—
295#102—2711no—
296#102—2711no—
297#102C2×D27111yescharacteristic, complemented
298C22⋊C8—2711no—
299C22⋊C8—2711no—
300C22⋊C8—2711no—
301C22⋊C8C2×D27111yescharacteristic, complemented
302C24—5421no—
303C24—5421no—
304C24—5421no—
305#41—22no—
306#41—22no—
307#41—2714no—
308#41—5422no—
309#41—5422no—
310#41—5428no—
311#41—5422no—
312#41—5422no—
313#41—5422no—
314C2×Q8—2711no—
315C2×Q8—2711no—
316C2×Q8—5421no—
317C2×Q8—5422no—
318C2×Q8—5422no—
319C2×Q8C22×D27111yescharacteristic
320C2×Q8C22×D27111yescharacteristic
321C2×D4—21no—
322C2×D4—2714no—
323C2×D4—2711no—
324C2×D4—2711no—
325C2×D4—2711no—
326C2×D4—2711no—
327C2×D4—2711no—
328C2×D4—5422no—
329C2×D4—5422no—
330C2×D4—5421no—
331C2×D4—5421no—
332C2×D4—5428no—
333C2×D4—5422no—
334C2×D4—5421no—
335C2×D4—5421no—
336C2×D4—5422no—
337C2×D4—5428no—
338C2×D4—5428no—
339C2×D4—5422no—
340C2×D4—5421no—
341C2×D4C22×D27111yescharacteristic
342C22×C4—21no—
343C22×C4—2711no—
344C22×C4—2711no—
345C22×C4—5421no—
346C22×C4—5421no—
347C22×C4—5421no—
348C22×C4C22×D27111yescharacteristic
349Q16—22no—
350Q16—5422no—
351SD16—22no—
352SD16—5422no—
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—21no—
376C23—2711no—
377C23—2711no—
378C23—5424no—
379C23—5424no—
380C23—5424no—
381C23—5421no—
382C23—5421no—
383C23—21681no—
384C23—21681no—
385C23—21681no—
386C23—21681no—
387C23—21681no—
388C23—21681no—
389C23D4×D27111yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—5422no—
394Q8—5421no—
395Q8—5421no—
396Q8—10842no—
397Q8D4×D27112yes—
398D4—24no—
399D4—21no—
400D4—2712no—
401D4—2712no—
402D4—5422no—
403D4—5424no—
404D4—5424no—
405D4—5424no—
406D4—5422no—
407D4—5424no—
408D4—5422no—
409D4—5424no—
410D4—5421no—
411D4—5421no—
412D4D4×D27112yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—2711no—
420C2×C4—2712no—
421C2×C4—2712no—
422C2×C4—2711no—
423C2×C4—5424no—
424C2×C4—5424no—
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—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—2712no—
443C22—5421no—
444C22—5421no—
445C22—5422no—
446C22—5422no—
447C22—5421no—
448C22—5421no—
449C22—5421no—
450C22—10841no—
451C22—10841no—
452C22—10841no—
453C22—10841no—
454C22—21684no—
455C22—21684no—
456C22—21684no—
457C22C2×D4×D27111yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—2712no—
463C4—5421no—
464C4—10841no—
465C4—10841no—
466C4C2×D4×D27111yescharacteristic
467C4C2×D4×D27111yescharacteristic
468C2—41no—
469C2—42no—
470C2—5422no—
471C2—10841no—
472C2—10842no—
473C2—10842no—
474C2#120×D27111yescharacteristic
475C2#3177470411yescharacteristic
476C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#452×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576206C2
hypercenter#452D271
hypocenterC271#3009
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×SD16×D271C2semidirect
4. #31774470C2semidirect
5. #31774062C2semidirect
6. #31774059C2semidirect
7. #31774058C2semidirect
8. #31774050C2semidirect
9. #31774049C2semidirect
10. #31774047C2semidirect
11. #31774046C2semidirect
12. #31773900C2semidirect
13. #31773892C2semidirect
14. #31773862C2semidirect
15. #452×C271C2semidirect
22. C2×Q8×D271C22semidirect
30. C2×#16864644C22semidirect
31. C2×#16864643C22semidirect
32. C2×#16864643C22semidirect
33. C2×#16864642C22semidirect
34. C2×#16864641C22semidirect
35. C2×#16864641C22semidirect
37. C2×#16864637C22semidirect
38. C2×C8×D271C22semidirect
43. #22576140C22semidirect
44. #22576139C22semidirect
45. #22576136C22semidirect
46. #22576118C22semidirect
47. #22576084C22semidirect
48. #22576082C22semidirect
49. #22576082C22semidirect
50. #22576081C22semidirect
51. #22576080C22semidirect
52. #22576080C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
57. C2×Q16×C271C22semidirect
58. C2×SD16×C271C22semidirect
59. C22⋊Q8×C271C22semidirect
60. #103×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
238. C271#3009semidirect
247. #452D271semidirect
271. C2×Q16C2×D271semidirect
278. C2×SD16C2×D271semidirect
288. C22⋊Q8C2×D271semidirect
291. #103C2×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×#225762062
C2×#225761922
C2×SD16×D2712
#317744702
#317740622
#317740592
#317740582
#317740502
#317740492
#317740472
#317740462
#317739002
#317738922
#317738622
#452×C2712
#3009271