Finite Group Databasebeta

← #46839894

Subgroups — #46839894

Order 34688.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576134C211yesmaximal, characteristic, complemented
2C2×#22576132C211yesmaximal, characteristic, complemented
3C2×#22576130C211yesmaximal, characteristic, complemented
4C2×#22576128C211yesmaximal, characteristic, complemented
5C22×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
6#31773819C211yesmaximal, characteristic, complemented
7#31773814C211yesmaximal, characteristic, complemented
8#31773813C211yesmaximal, characteristic, complemented
9#31773806C211yesmaximal, characteristic, complemented
10#31773799C211yesmaximal, characteristic, complemented
11#31773788C211yesmaximal, characteristic, complemented
12#31773776C211yesmaximal, characteristic, complemented
13#31773776C211yesmaximal, characteristic, complemented
14#31773772C211yesmaximal, characteristic, complemented
15C2×C4×D4×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C22⋊D271C2211yescharacteristic
21C22×C22⋊D271C2211yescharacteristic
22C22×C22⋊D271C2211yescharacteristic, complemented
23C23×Dic271C2211yescharacteristic
24#22576134—24no—
25#22576132—24no—
26#22576130—24no—
27#22576128—24no—
28C2×C4⋊Dic271C2211yescharacteristic, complemented
29C2×C4⋊Dic271C2211yescharacteristic, complemented
30C2×Dic271⋊C4C2211yescharacteristic, complemented
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic, complemented
33C2×(C2×D271)⋊C4—24no—
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C271⋊#95C2211yescharacteristic, complemented
48C271⋊#95C2211yescharacteristic, complemented
49C22×D4×C271C2211yescharacteristic
50C23×C4×C271C2211yescharacteristic
51C23×C4×C271C2211yescharacteristic, complemented
52C4×D4×C271—24no—
53C2×C4⋊C4×C271C2211yescharacteristic, complemented
54C2×C22⋊C4×C271C2211yescharacteristic
55C2×C22⋊C4×C271C2211yescharacteristic, complemented
56C2×C42×C271C2211yescharacteristic, complemented
57C23×D271—22no—
58C23×D271—22no—
59C23×D271—22no—
60C23×D271—22no—
61C23×D271—22no—
62C23×D271—41no—
63C23×D271—41no—
64C23×D271C2311yescharacteristic
65C23×D271C2311yescharacteristic
66C23×D271C2311yescharacteristic
67C2×D1084—22no—
68C2×D1084—22no—
69C2×D1084—22no—
70C2×D1084—22no—
71C2×D1084—41no—
72C2×D1084—41no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271—22no—
75C2×C22⋊D271—22no—
76C2×C22⋊D271—22no—
77C2×C22⋊D271—22no—
78C2×C22⋊D271—22no—
79C2×C22⋊D271—22no—
80C2×C22⋊D271—22no—
81C2×C22⋊D271—22no—
82C2×C22⋊D271—41no—
83C2×C22⋊D271—41no—
84C2×C22⋊D271—41no—
85C22×Dic271—22no—
86C22×Dic271—22no—
87C22×Dic271—41no—
88C22×Dic271C2311yescharacteristic
89C22×Dic271C2311yescharacteristic
90C22×Dic271C2311yescharacteristic
91C22×Dic271C2311yescharacteristic
92C22×Dic271C2311yescharacteristic
93C4⋊Dic271—22no—
94C4⋊Dic271—41no—
95Dic271⋊C4—22no—
96C22⋊Dic271—22no—
97C22⋊Dic271—22no—
98(C2×D271)⋊C4—22no—
99(C2×D271)⋊C4—22no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—22no—
102(C2×D271)⋊C4—22no—
103(C2×D271)⋊C4—22no—
104(C2×D271)⋊C4—22no—
105(C2×D271)⋊C4—22no—
106(C2×D271)⋊C4—22no—
107(C2×D271)⋊C4—41no—
108(C2×D271)⋊C4D414yescomplemented
109C24×C271C2311yescharacteristic
110C24×C271C2311yescharacteristic
111C2×D4×C271—22no—
112C2×D4×C271—22no—
113C2×D4×C271D414yescomplemented
114C22×C4×C271—22no—
115C22×C4×C271—22no—
116C22×C4×C271—22no—
117C22×C4×C271—22no—
118C22×C4×C271—41no—
119C22×C4×C271—41no—
120C22×C4×C271C2311yescharacteristic
121C22×C4×C271C2311yescharacteristic
122C22×C4×C271C2311yescharacteristic
123C22×C4×C271C2311yescharacteristic
124C22×C4×C271C2311yescharacteristic
125C4⋊C4×C271—22no—
126C22⋊C4×C271—22no—
127C22⋊C4×C271D414yescomplemented
128C42×C271—22no—
129C22×D271—28no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—21no—
134C22×D271—21no—
135C22×D271—21no—
136C22×D271—21no—
137C22×D271—21no—
138C22×D271—21no—
139C22×D271—21no—
140C22×D271—21no—
141C22×D271—21no—
142C22×D271—21no—
143C22×D271—21no—
144C22×D271—21no—
145C22×D271—21no—
146C22×D271—21no—
147C22×D271—21no—
148C22×D271—21no—
149C22×D271—44no—
150C22×D271—44no—
151C22×D271—44no—
152C22×D271—44no—
153C22×D271—44no—
154C22×D271—44no—
155C22×D271C2×D412yes—
156C22×D271#4112yes—
157D1084—44no—
158D1084—44no—
159C22⋊D271—44no—
160C22⋊D271—44no—
161C22⋊D271—44no—
162C22⋊D271—44no—
163C2×Dic271—21no—
164C2×Dic271—21no—
165C2×Dic271—21no—
166C2×Dic271—21no—
167C2×Dic271—21no—
168C2×Dic271—21no—
169C2×Dic271—21no—
170C2×Dic271—21no—
171C2×Dic271—21no—
172C2×Dic271—21no—
173C2×Dic271—21no—
174C2×Dic271—21no—
175C2×Dic271—21no—
176C2×Dic271—21no—
177C2×Dic271—44no—
178C2×Dic271C2×D412yes—
179C23×C271—21no—
180C23×C271—21no—
181C23×C271—21no—
182C23×C271—21no—
183C23×C271—21no—
184C23×C271—21no—
185C23×C271—21no—
186C23×C271—21no—
187C23×C271—21no—
188C23×C271—21no—
189C23×C271C2×D412yes—
190C23×C271C2×D412yes—
191C23×C271C2×D412yes—
192C23×C271#4112yes—
193C23×C271C2411yescharacteristic
194D4×C271—28no—
195C2×C4×C271—21no—
196C2×C4×C271—21no—
197C2×C4×C271—21no—
198C2×C4×C271—21no—
199C2×C4×C271—21no—
200C2×C4×C271—21no—
201C2×C4×C271—21no—
202C2×C4×C271—21no—
203C2×C4×C271—21no—
204C2×C4×C271—21no—
205C2×C4×C271—21no—
206C2×C4×C271—44no—
207C2×C4×C271—44no—
208C2×C4×C271C2×D412yes—
209C2×C4×C271C2×D412yes—
210C2×C4×C271C2×D412yes—
211C2×C4×C271C2×D412yes—
212C22×C271—22no—
213C22×C271—22no—
214C22×C271—22no—
215C22×C271—22no—
216C22×C271—22no—
217C22×C271—22no—
218C22×C271—22no—
219C22×C271—22no—
220C22×C271—22no—
221C22×C271—41no—
222C22×C271—41no—
223C22×C271—41no—
224C22×C271—41no—
225C22×C271#12114yescomplemented
226C22×C271C22×D411yescharacteristic
227C22×C271C22×D411yescharacteristic
228C22×C271C22×D411yescharacteristic
229C22×C271C2×#4111yescharacteristic
230C22×C271C2×#4111yescharacteristic
231C22×C271H(5,𝔽2)11yescharacteristic
232C22×C271H(5,𝔽2)11yescharacteristic
233C2×D271—22no—
234C2×D271—22no—
235C2×D271—22no—
236C2×D271—22no—
237C2×D271—22no—
238C2×D271—22no—
239C2×D271—416no—
240C2×D271—41no—
241C2×D271—41no—
242C2×D271—41no—
243C2×D271—41no—
244C2×D271—41no—
245C2×D271—41no—
246C2×D271—41no—
247C2×D271—41no—
248C2×D271—41no—
249C2×D271—41no—
250C2×D271—41no—
251C2×D271—41no—
252C2×D271—41no—
253C2×D271—41no—
254C2×D271—41no—
255C4×C271—22no—
256C4×C271—22no—
257C4×C271—22no—
258C4×C271—22no—
259C4×C271—41no—
260Dic271—22no—
261Dic271—41no—
262Dic271—41no—
263Dic271—41no—
264Dic271—41no—
265C2×C271—24no—
266C2×C271—42no—
267C2×C271C2×#12111yescharacteristic
268C2×C271#52511yescharacteristic
269C2×C271#53811yescharacteristic
270C2×C271D4211yescharacteristic
271C2×C271D4211yescharacteristic
272C2×C271#54611yescharacteristic
273C2×C271#54611yescharacteristic
274D271—42no—
275D271—42no—
276D271—81no—
277C271#242411yescharacteristic, complemented, Sylow
278#2424—2711nomaximal, Sylow
279C2×#121—2711no—
280C2×#121—2711no—
281C2×#120—2711no—
282C2×#120—2711no—
283C2×C4×D4D27111yescharacteristic, complemented
284C22×C22⋊C4—2711no—
285#396—2711no—
286#394—2711no—
287#394—2711no—
288#392—2711no—
289#390—2711no—
290#388—2711no—
291#386—2711no—
292#386—2711no—
293#385—2711no—
294C25—2711no—
295C22×D4—2711no—
296C22×D4—2711no—
297C22×D4—2711no—
298C22×D4—2711no—
299C22×D4—2711no—
300C22×D4—2711no—
301C22×D4C2×D27111yescharacteristic
302C23×C4—2711no—
303C23×C4C2×D27111yescharacteristic
304C23×C4C2×D27111yescharacteristic, complemented
305#121—5424no—
306#121—5424no—
307#120—5424no—
308#120—5424no—
309C4×D4—24no—
310C2×C4⋊C4—2711no—
311C2×C4⋊C4—2711no—
312C2×C4⋊C4—2711no—
313C2×C4⋊C4C2×D27111yescharacteristic, complemented
314C2×C22⋊C4—2711no—
315C2×C22⋊C4—2711no—
316C2×C22⋊C4—2711no—
317C2×C22⋊C4—2711no—
318C2×C22⋊C4—2711no—
319C2×C22⋊C4—2711no—
320C2×C22⋊C4—2711no—
321C2×C22⋊C4—2711no—
322C2×C22⋊C4—2711no—
323C2×C22⋊C4—2711no—
324C2×C22⋊C4—2711no—
325C2×C22⋊C4—2711no—
326C2×C22⋊C4—2711no—
327C2×C22⋊C4—5424no—
328C2×C22⋊C4C2×D27111yescharacteristic
329C2×C22⋊C4C2×D27111yescharacteristic, complemented
330C2×C42C2×D27111yescharacteristic, complemented
331#95—2711no—
332#95—2711no—
333#95—2711no—
334#95—2711no—
335C24—2711no—
336C24—2711no—
337C24—2711no—
338C24—5422no—
339C24—5422no—
340C24—5422no—
341C24—5422no—
342C24—5422no—
343C24—10841no—
344C24—10841no—
345C24C22×D27111yescharacteristic
346C24C22×D27111yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—5422no—
350C2×D4—5422no—
351C2×D4—5422no—
352C2×D4—5422no—
353C2×D4—5422no—
354C2×D4—5422no—
355C2×D4—5422no—
356C2×D4—5422no—
357C2×D4—5422no—
358C2×D4—5422no—
359C2×D4—5422no—
360C2×D4—5422no—
361C2×D4—5422no—
362C2×D4—10841no—
363C2×D4—10841no—
364C2×D4—10841no—
365C2×D4—10841no—
366C2×D4—10841no—
367C2×D4D108414yescomplemented
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—2711no—
375C22×C4—2711no—
376C22×C4—2711no—
377C22×C4—2711no—
378C22×C4—2711no—
379C22×C4—5422no—
380C22×C4—5422no—
381C22×C4—10841no—
382C22×C4C22×D27111yescharacteristic
383C22×C4C22×D27111yescharacteristic
384C22×C4C22×D27111yescharacteristic
385C22×C4C22×D27111yescharacteristic
386C22×C4C22×D27111yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—5422no—
389C4⋊C4—5422no—
390C4⋊C4—10841no—
391C22⋊C4—22no—
392C22⋊C4—2714no—
393C22⋊C4—5422no—
394C22⋊C4—5422no—
395C22⋊C4—5422no—
396C22⋊C4—5422no—
397C22⋊C4—5422no—
398C22⋊C4—5422no—
399C22⋊C4—5422no—
400C22⋊C4—5422no—
401C22⋊C4—5422no—
402C22⋊C4—5422no—
403C22⋊C4—5422no—
404C22⋊C4—10841no—
405C22⋊C4C22⋊D27114yescomplemented
406C42—22no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—2712no—
418C23—2712no—
419C23—5428no—
420C23—5421no—
421C23—5421no—
422C23—5421no—
423C23—5421no—
424C23—5421no—
425C23—5421no—
426C23—5421no—
427C23—5421no—
428C23—5421no—
429C23—5421no—
430C23—5421no—
431C23—5421no—
432C23—5421no—
433C23—5421no—
434C23—5421no—
435C23—5421no—
436C23—5421no—
437C23—5421no—
438C23—5421no—
439C23—10844no—
440C23—10844no—
441C23—10844no—
442C23—10844no—
443C23—10844no—
444C23—10844no—
445C23C2×C22⋊D27112yes—
446C23C2×D108412yes—
447C23C2×D108412yes—
448C23#1686465412yes—
449C23C23×D27111yescharacteristic
450D4—28no—
451D4—10844no—
452D4—10844no—
453D4—10844no—
454D4—10844no—
455D4—10844no—
456D4—10844no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—2712no—
471C2×C4—5421no—
472C2×C4—5421no—
473C2×C4—5421no—
474C2×C4—5421no—
475C2×C4—5421no—
476C2×C4—5421no—
477C2×C4—5421no—
478C2×C4—5421no—
479C2×C4—5421no—
480C2×C4—5421no—
481C2×C4—5421no—
482C2×C4—5421no—
483C2×C4—5421no—
484C2×C4—5421no—
485C2×C4—10844no—
486C2×C4D4×D27112yes—
487C2×C4C2×C22⋊D27112yes—
488C2×C4C2×C22⋊D27112yes—
489C2×C4C2×D108412yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—5422no—
504C22—5422no—
505C22—5422no—
506C22—5422no—
507C22—5422no—
508C22—5422no—
509C22—108416no—
510C22—10841no—
511C22—10841no—
512C22—10841no—
513C22—10841no—
514C22—10841no—
515C22—10841no—
516C22—10841no—
517C22—10841no—
518C22—10841no—
519C22—10841no—
520C22—10841no—
521C22—10841no—
522C22—10841no—
523C22—10841no—
524C22—10841no—
525C22#2257613414yescomplemented
526C22C2×D4×D27111yescharacteristic
527C22C22×C22⋊D27111yescharacteristic
528C22C22×D108411yescharacteristic
529C22#41×D27111yescharacteristic
530C22C2×#1686465411yescharacteristic
531C22#2257620511yescharacteristic
532C22#2257620611yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—5422no—
539C4—10841no—
540C4—10841no—
541C4—10841no—
542C4—10841no—
543C2—24no—
544C2—42no—
545C2—10842no—
546C2—10842no—
547C2—21681no—
548C2C2×#2257613411yescharacteristic
549C2#3177440611yescharacteristic
550C2#3177448011yescharacteristic
551C2D4×C22⋊D27111yescharacteristic
552C2D4×D108411yescharacteristic
553C2#3177453511yescharacteristic
554C2#3177454211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×C4×D4×C271C2
hypercenterC2×C4×D4D271
hypocenterC271#2424
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#22576134C2semidirect
2. C2×#22576132C2semidirect
3. C2×#22576130C2semidirect
4. C2×#22576128C2semidirect
5. C22×(C2×D271)⋊C4C2semidirect
6. #31773819C2semidirect
7. #31773814C2semidirect
8. #31773813C2semidirect
9. #31773806C2semidirect
10. #31773799C2semidirect
11. #31773788C2semidirect
12. #31773776C2semidirect
13. #31773776C2semidirect
14. #31773772C2semidirect
15. C2×C4×D4×C271C2semidirect
22. C22×C22⋊D271C22semidirect
28. C2×C4⋊Dic271C22semidirect
29. C2×C4⋊Dic271C22semidirect
30. C2×Dic271⋊C4C22semidirect
32. C2×C22⋊Dic271C22semidirect
42. C2×(C2×D271)⋊C4C22semidirect
43. C2×(C2×D271)⋊C4C22semidirect
44. C2×(C2×D271)⋊C4C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
47. C271⋊#95C22semidirect
48. C271⋊#95C22semidirect
51. C23×C4×C271C22semidirect
53. C2×C4⋊C4×C271C22semidirect
55. C2×C22⋊C4×C271C22semidirect
56. C2×C42×C271C22semidirect
108. (C2×D271)⋊C4D4semidirect
113. C2×D4×C271D4semidirect
127. C22⋊C4×C271D4semidirect
225. C22×C271#121semidirect
277. C271#2424semidirect
283. C2×C4×D4D271semidirect
304. C23×C4C2×D271semidirect
313. C2×C4⋊C4C2×D271semidirect
329. C2×C22⋊C4C2×D271semidirect
330. C2×C42C2×D271semidirect
367. C2×D4D1084semidirect
405. C22⋊C4C22⋊D271semidirect
525. C22#22576134semidirect

Maximal subgroups

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

subgroupindex
C2×#225761342
C2×#225761322
C2×#225761302
C2×#225761282
C22×(C2×D271)⋊C42
#317738192
#317738142
#317738132
#317738062
#317737992
#317737882
#317737762
#317737722
C2×C4×D4×C2712
#2424271