Finite Group Databasebeta

← #4307975

Subgroups — #4307975

Order 512.

6789 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C2×#2623C212yesmaximal, complemented
4#14843C214yesmaximal, complemented
5#14786C214yesmaximal, complemented
6#14777C214yesmaximal, complemented
7#14776C214yesmaximal, complemented
8#14694C212yesmaximal
9#14693C212yesmaximal, complemented
10#14681C211yesmaximal, characteristic, complemented
11#14342C212yesmaximal, complemented
12#14263C212yesmaximal, complemented
13#14256C212yesmaximal, complemented
14C25×C4C2211yescharacteristic
15C25×C4C2212yes—
16C22×#123C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#120C2211yescharacteristic
19C23×C4⋊C4—28no—
20C23×C4⋊C4—216no—
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2214yes—
23C23×C4⋊C4C2214yes—
24C23×C4⋊C4C2212yes—
25#2849—28no—
26#2623—216no—
27C2×#400C2214yescomplemented
28C2×#399C2214yescomplemented
29C2×#398C2214yescomplemented
30C2×#397C2214yescomplemented
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#386C2218yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2214yes—
38C2×#385C2218yes—
39C2×#381C2211yescharacteristic
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#379C2211yescharacteristic
44C2×#379C2212yes—
45#1445C2218yescomplemented
46#1445C2218yescomplemented
47#1444C2218yescomplemented
48#1442C2218yescomplemented
49#1441C2218yescomplemented
50#1439C2218yescomplemented
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C22⋊#95C2218yes—
56C22⋊#95C2218yes—
57C26C2311yescharacteristic
58C23×D4C2311yescharacteristic
59C23×D4C2312yes—
60C24×C4—28no—
61C24×C4—28no—
62C24×C4—28no—
63C24×C4—28no—
64C24×C4—216no—
65C24×C4—416no—
66C24×C4D418yes—
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2318yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2314yes—
73C2×#123—24no—
74C2×#123—24no—
75C2×#123—28no—
76C2×#123—28no—
77C2×#120—24no—
78C2×#120—28no—
79C22×C4⋊C4—2128no—
80C22×C4⋊C4—216no—
81C22×C4⋊C4—216no—
82C22×C4⋊C4—216no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—232no—
87C22×C4⋊C4—232no—
88C22×C4⋊C4D4116yescomplemented
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2318yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2318yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2318yes—
108#400—216no—
109#399—216no—
110#398—216no—
111#397—216no—
112#386—28no—
113#386—28no—
114#386—216no—
115#386—416no—
116#385—28no—
117#385—28no—
118#385—216no—
119#385—416no—
120#381—24no—
121#381—24no—
122#381—28no—
123#381—28no—
124#379—24no—
125#379—28no—
126C2×#95C2314yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2318yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C23116yes—
136C2×#95C23116yes—
137C25—24no—
138C25—24no—
139C25—28no—
140C25—28no—
141C25C2×D414yes—
142C25#4118yes—
143C25C2411yescharacteristic
144C25C2411yescharacteristic
145C25C2412yes—
146C22×D4—22no—
147C22×D4—24no—
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—28no—
151C22×D4—24no—
152C22×D4—28no—
153C22×D4—28no—
154C23×C4—2128no—
155C23×C4—232no—
156C23×C4—24no—
157C23×C4—28no—
158C23×C4—28no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—232no—
163C23×C4—232no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—28no—
167C23×C4—28no—
168C23×C4—264no—
169C23×C4—216no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—232no—
173C23×C4—216no—
174C23×C4—216no—
175C23×C4—416no—
176C23×C4—416no—
177C23×C4—4128no—
178C23×C4—464no—
179C23×C4C2×D418yes—
180C23×C4C2×D418yes—
181C23×C4C2×D414yes—
182C23×C4C2×D418yes—
183C23×C4C2×D414yes—
184C23×C4C2×D4116yes—
185C23×C4#4118yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2418yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2418yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196#123—416no—
197#123—416no—
198#120—416no—
199C2×C4⋊C4—264no—
200C2×C4⋊C4—2128no—
201C2×C4⋊C4—2128no—
202C2×C4⋊C4—264no—
203C2×C4⋊C4—232no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—416no—
217C2×C4⋊C4C2×D4116yes—
218C2×C4⋊C4C2×D4116yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—216no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—216no—
244C2×C22⋊C4—416no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—28no—
250#95—28no—
251#95—216no—
252#95—216no—
253#95—216no—
254#95—416no—
255#95—416no—
256C24—232no—
257C24—232no—
258C24—264no—
259C24—28no—
260C24—28no—
261C24—232no—
262C24—216no—
263C24—216no—
264C24—264no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24#123116yes—
278C24C22×D412yes—
279C24C22×D414yes—
280C24C22×D414yes—
281C24C2×#4114yes—
282C24C2×#4114yes—
283C24C2×#4118yes—
284C24H(5,𝔽2)18yes—
285C242−1+418yes—
286C24C2511yescharacteristic
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—48no—
290C2×D4—44no—
291C2×D4—416no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4—432no—
295C22×C4—2256no—
296C22×C4—264no—
297C22×C4—264no—
298C22×C4—28no—
299C22×C4—216no—
300C22×C4—216no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—232no—
305C22×C4—232no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—216no—
313C22×C4—216no—
314C22×C4—216no—
315C22×C4—216no—
316C22×C4—264no—
317C22×C4—232no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—216no—
321C22×C4—216no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—216no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—216no—
344C22×C4—24no—
345C22×C4—432no—
346C22×C4—464no—
347C22×C4—464no—
348C22×C4—4256no—
349C22×C4—432no—
350C22×C4—432no—
351C22×C4—432no—
352C22×C4—432no—
353C22×C4#120116yes—
354C22×C4#123116yes—
355C22×C4C22×D4116yes—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D414yes—
359C22×C4C22×D414yes—
360C22×C4C22×D414yes—
361C22×C4C2×#4118yes—
362C4⋊C4—264no—
363C4⋊C4—232no—
364C4⋊C4—232no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—48no—
368C4⋊C4—816no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—48no—
372C22⋊C4—44no—
373C22⋊C4—416no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—432no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—48no—
380C22⋊C4—816no—
381C23—232no—
382C23—264no—
383C23—232no—
384C23—2256no—
385C23—232no—
386C23—264no—
387C23—24no—
388C23—24no—
389C23—216no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—216no—
395C23—216no—
396C23—28no—
397C23—216no—
398C23—28no—
399C23—216no—
400C23—216no—
401C23—216no—
402C23—216no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—48no—
410C23—44no—
411C23—44no—
412C23—48no—
413C23C2×#12014yes—
414C23C2×#12314yes—
415C23C2×#12314yes—
416C23C2×#12318yes—
417C23#525116yes—
418C23#53518yes—
419C23#53618yes—
420C23#539132yes—
421C23C23×D411yescharacteristic
422C23C23×D412yes—
423C23C22×#4112yes—
424C23C22×#4112yes—
425C23C2×H(5,𝔽2)14yes—
426C23C2×2−1+414yes—
427D4—88no—
428D4—816no—
429C2×C4—264no—
430C2×C4—264no—
431C2×C4—232no—
432C2×C4—232no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—216no—
442C2×C4—216no—
443C2×C4—216no—
444C2×C4—216no—
445C2×C4—464no—
446C2×C4—4128no—
447C2×C4—416no—
448C2×C4—416no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—48no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—416no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—416no—
472C2×C4C2×#12014yes—
473C2×C4C2×#12014yes—
474C2×C4C2×#12318yes—
475C22—2128no—
476C22—28no—
477C22—28no—
478C22—28no—
479C22—24no—
480C22—28no—
481C22—28no—
482C22—28no—
483C22—28no—
484C22—24no—
485C22—28no—
486C22—232no—
487C22—264no—
488C22—81no—
489C22—82no—
490C22—81no—
491C22—82no—
492C22—81no—
493C22—88no—
494C22#2623116yescomplemented
495C22#284918yes—
496C22C22×#12011yescharacteristic
497C22C22×#12311yescharacteristic
498C22C22×#12311yescharacteristic
499C22C2×#52514yes—
500C22C2×#53512yes—
501C22C2×#53612yes—
502C22C2×#53918yes—
503C22#348414yes—
504C22#348614yes—
505C22#349218yes—
506C4—24no—
507C4—24no—
508C4—28no—
509C4—416no—
510C4—81no—
511C4—82no—
512C4—81no—
513C4—82no—
514C4—81no—
515C4—88no—
516C2—28no—
517C2—216no—
518C2—161no—
519C2C2×#262312yes—
520C2#3498918yes—
521C2#3522211yescharacteristic
522C2#3522712yes—
523C2#3524211yescharacteristic
524C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×C4⋊C4C2semidirect
3. C2×#2623C2semidirect
4. #14843C2semidirect
5. #14786C2semidirect
6. #14777C2semidirect
7. #14776C2semidirect
9. #14693C2semidirect
10. #14681C2semidirect
11. #14342C2semidirect
12. #14263C2semidirect
13. #14256C2semidirect
27. C2×#400C22semidirect
28. C2×#399C22semidirect
29. C2×#398C22semidirect
30. C2×#397C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1439C22semidirect
88. C22×C4⋊C4D4semidirect
494. C22#2623semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#28492
C2×#26232
#148432
#147862
#147772
#147762
#146942
#146932
#146812
#143422
#142632
#142562