Finite Group Databasebeta

← #4414344

Subgroups — #4414344

Order 512.

2462 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2844C211yesmaximal, characteristic, complemented
2C22×#382C211yesmaximal, characteristic
3#16689C212yesmaximal, complemented
4#16686C212yesmaximal, complemented
5#16572C214yesmaximal, complemented
6#16524C214yesmaximal, complemented
7#14822C212yesmaximal
8#14767C214yesmaximal
9#14758C214yesmaximal
10#14720C212yesmaximal
11#14682C212yesmaximal
12#14677C211yesmaximal, characteristic, complemented
13#14677C212yesmaximal, complemented
14C22×C22⋊Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2844—28no—
19C2×#400C2212yes—
20C2×#400C2214yes—
21C2×#400C2212yes—
22C2×#395C2212yes—
23C2×#395C2214yes—
24C2×#395C2212yes—
25C2×#385C2214yes—
26C2×#382—28no—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2212yes—
30C2×#382C2212yes—
31C2×#382C2211yescharacteristic
32C2×#382C2214yes—
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C22×#95C2214yes—
38#1446C2214yes—
39#1445C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62#1434C2214yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8—48no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C42—24no—
101C22×C42—24no—
102C22×C42Q818yescomplemented
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106#400—28no—
107#400—216no—
108#400—44no—
109#395—28no—
110#395—216no—
111#395—44no—
112#385—48no—
113#382—24no—
114#382—24no—
115#382—28no—
116#382—24no—
117#382—28no—
118#382—216no—
119#382D418yes—
120#380—44no—
121#380—44no—
122#380—44no—
123C2×#95—216no—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2411yescharacteristic
152C22×Q8—24no—
153C22×Q8—44no—
154C22×Q8—42no—
155C22×Q8—44no—
156C22×Q8—42no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C2×Q814yes—
180C23×C4C2×Q818yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C22⋊Q8—432no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—232no—
253C2×C42C2×D414yes—
254C2×C42C2×Q814yes—
255C2×C42C2×Q818yes—
256C2×C42C2×Q814yes—
257C2×C42#4114yes—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—24no—
264#95—28no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—44no—
284#95C2×D4116yes—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24C22×Q812yes—
292C24H(5,𝔽2)14yes—
293C24H(5,𝔽2)12yes—
294C242−1+412yes—
295C242−1+414yes—
296C24C2511yescharacteristic
297C2×Q8—48no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×Q8—84no—
301C2×Q8—88no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—216no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—216no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
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—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—416no—
383C22×C4—416no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D414yes—
393C22×C4C22×D412yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q812yes—
396C22×C4C22×Q814yes—
397C22×C4C22×Q812yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4114yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+412yes—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—416no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—88no—
416C4⋊C4—84no—
417C4⋊C4—88no—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C42—216no—
421C42—28no—
422C42C22⋊Q8116yescomplemented
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23#54414yes—
442C23#54418yes—
443C23H(3,𝔽4)14yes—
444C23#56318yes—
445C23#56314yes—
446C23C23×D411yescharacteristic
447C23C23×Q811yescharacteristic
448C23C22×#4111yescharacteristic
449C23C2×H(5,𝔽2)11yescharacteristic
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×H(5,𝔽2)12yes—
452C23C2×2−1+411yescharacteristic
453C23C2×2−1+411yescharacteristic
454C23C2×2−1+412yes—
455C23#58512yes—
456C23#58512yes—
457Q8—816no—
458C2×C4—232no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—432no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
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—816no—
534C2×C4—816no—
535C2×C4C2×C22⋊Q814yes—
536C2×C4C2×C22⋊Q818yes—
537C2×C4#54414yes—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—28no—
543C22—28no—
544C22—24no—
545C22—28no—
546C22—28no—
547C22—24no—
548C22#284418yescomplemented
549C22C22×C22⋊Q811yescharacteristic
550C22C2×#54411yescharacteristic
551C22C2×#54412yes—
552C22C2×#54411yescharacteristic
553C22#345812yes—
554C22#348411yescharacteristic
555C22#348611yescharacteristic
556C22#348812yes—
557C22#348912yes—
558C22#349211yescharacteristic
559C22#349211yescharacteristic
560C22#349412yes—
561C22#349512yes—
562C22C2×H(3,𝔽4)11yescharacteristic
563C22C2×#56311yescharacteristic
564C22C2×#56312yes—
565C22#354511yescharacteristic
566C22#354511yescharacteristic
567C22#355312yes—
568C22#355812yes—
569C22#355912yes—
570C22#356112yes—
571C22#356312yes—
572C4—24no—
573C4—24no—
574C4—28no—
575C4—44no—
576C4—44no—
577C4—44no—
578C4—82no—
579C4—82no—
580C4—84no—
581C4—84no—
582C4—84no—
583C4—84no—
584C2—28no—
585C2#3522411yescharacteristic
586C2#3525312yes—
587C2C2×#284411yescharacteristic
588C2#3756112yes—
589C2#3760911yescharacteristic
590C2#4597711yescharacteristic
591C2#4597912yes—
592C2#4598912yes—
593C2#4599111yescharacteristic
594C2#4613611yescharacteristic
595C2#4614511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2844C2semidirect
3. #16689C2semidirect
4. #16686C2semidirect
5. #16572C2semidirect
6. #16524C2semidirect
12. #14677C2semidirect
13. #14677C2semidirect
102. C22×C42Q8semidirect
422. C42C22⋊Q8semidirect
548. C22#2844semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
C22×#3822
#166892
#166862
#165722
#165242
#148222
#147672
#147582
#147202
#146822
#146772