Finite Group Databasebeta

← #4414369

Subgroups — #4414369

Order 512.

2486 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2820C212yesmaximal
3C2×C4×#124C211yesmaximal, characteristic
4#16629C212yesmaximal, complemented
5#16596C212yesmaximal
6#16469C214yesmaximal, complemented
7#16466C214yesmaximal, complemented
8#14880C212yesmaximal, complemented
9#14833C214yesmaximal, complemented
10#14824C212yesmaximal
11#14771C214yesmaximal, complemented
12#14709C212yesmaximal, complemented
13#14677C211yesmaximal, characteristic, complemented
14C22×#125C2212yes—
15C22×#124C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2822—28no—
19#2820—216no—
20C2×#400C2214yes—
21C2×#399C2214yes—
22C2×#398C2212yes—
23C2×#398C2214yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#396C2214yes—
30C2×#395C2214yes—
31C2×#394C2214yes—
32C4×#124—28no—
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#388C2214yes—
37C2×#382C2212yes—
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C43C2211yescharacteristic
42#1444C2214yes—
43#1444C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2212yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#125—28no—
74C2×#125—28no—
75C2×#125—28no—
76C2×#124—24no—
77C2×#124—24no—
78C2×#124D418yes—
79C2×C4×Q8—24no—
80C2×C4×Q8—42no—
81C2×C4×Q8—42no—
82C2×C4×D4—24no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#400—216no—
114#399—216no—
115#398—28no—
116#398—28no—
117#398—216no—
118#398—44no—
119#397—216no—
120#397—44no—
121#396—216no—
122#396—216no—
123#395—216no—
124#394—216no—
125#391—24no—
126#390—24no—
127#389—44no—
128#388—216no—
129#382—28no—
130#382D418yes—
131#380—44no—
132C4×C22⋊C4—216no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C43—24no—
156C25C2412yes—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8C2×D414yes—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4C2×D414yes—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—28no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188#125—432no—
189#124—232no—
190C4×Q8—416no—
191C4×D4—416no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C2×D4116yes—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42C2×D414yes—
272C2×C42C2×D414yes—
273C2×C42#4114yes—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—42no—
295#95C2×D4116yes—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—22no—
305C24—22no—
306C24C22×D414yes—
307C24C2511yescharacteristic
308C2×Q8—28no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×D4—28no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—48no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—84no—
389C22×C4—88no—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D414yes—
393C22×C4C22×D412yes—
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396C22×C4C22×D412yes—
397C22×C4C2×#4114yes—
398C22×C4C2×#4112yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+414yes—
402C22×C42−1+414yes—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—416no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—416no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—84no—
418C22⋊C4—232no—
419C22⋊C4—416no—
420C22⋊C4—48no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C42—28no—
426C42—48no—
427C42—48no—
428C42—48no—
429C42—44no—
430C42#121116yes—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—28no—
435C23—28no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23C23×D411yescharacteristic
455C23C23×D411yescharacteristic
456C23C22×#4111yescharacteristic
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×2−1+412yes—
460C23C2×2−1+412yes—
461C23#58511yescharacteristic
462C23#58511yescharacteristic
463C23#58512yes—
464C23#58512yes—
465Q8—416no—
466D4—416no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—816no—
541C2×C4C2×#12118yes—
542C2×C4C2×#12114yes—
543C2×C4#56314yes—
544C22—44no—
545C22—48no—
546C22—44no—
547C22—44no—
548C22—48no—
549C22—84no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—84no—
554C22—82no—
555C22C22×#12111yescharacteristic
556C22#346011yescharacteristic
557C22#346011yescharacteristic
558C22#346112yes—
559C22#346112yes—
560C22#348411yescharacteristic
561C22#348711yescharacteristic
562C22#348912yes—
563C22#348912yes—
564C22#349611yescharacteristic
565C22#349611yescharacteristic
566C22#349912yes—
567C22#350012yes—
568C22C2×#56311yescharacteristic
569C22#352911yescharacteristic
570C22#353112yes—
571C22#354511yescharacteristic
572C22#355011yescharacteristic
573C22#355312yes—
574C22#355912yes—
575C22#356112yes—
576C22#356312yes—
577C22#356712yes—
578C4—24no—
579C4—24no—
580C4—28no—
581C4—44no—
582C4—44no—
583C4—44no—
584C4—42no—
585C4—44no—
586C4—84no—
587C4—84no—
588C4—84no—
589C4—84no—
590C2—84no—
591C2#3516811yescharacteristic
592C2#3530812yes—
593C2#3669811yescharacteristic
594C2#3760012yes—
595C2#3761211yescharacteristic
596C2#4613512yes—
597C2#4613611yescharacteristic
598C2#4614711yescharacteristic
599C2#4614912yes—
600C2#4622811yescharacteristic
601C2#4623611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2822C2semidirect
4. #16629C2semidirect
6. #16469C2semidirect
7. #16466C2semidirect
8. #14880C2semidirect
9. #14833C2semidirect
11. #14771C2semidirect
12. #14709C2semidirect
13. #14677C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#28202
C2×C4×#1242
#166292
#165962
#164692
#164662
#148802
#148332
#148242
#147712
#147092
#146772