Finite Group Databasebeta

← #923688

Subgroups — #923688

Order 512.

3018 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C212yesmaximal, complemented
2C2×#2445C212yesmaximal
3#17103C214yesmaximal, complemented
4#16269C214yesmaximal, complemented
5#16257C214yesmaximal, complemented
6#15269C214yesmaximal, complemented
7#15218C211yesmaximal, characteristic, complemented
8#15149C212yesmaximal, complemented
9#15147C212yesmaximal, complemented
10#15127C211yesmaximal, characteristic
11#14260C212yesmaximal, complemented
12#14018C212yesmaximal, complemented
13#13856C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#123C2214yes—
17C22×C4×D4C2212yes—
18#2454—216no—
19#2445—216no—
20C2×#399C2214yes—
21C2×#398C2214yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#386C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#380C2214yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44#1444C2212yes—
45#1444C2212yes—
46#1444C2212yes—
47#1444C2212yes—
48#1444C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2314yes—
71C24×C4C2314yes—
72C2×#127—28no—
73C2×#127—42no—
74C2×#125—28no—
75C2×#125—42no—
76C2×#123—216no—
77C2×#123—48no—
78C2×#123—48no—
79C2×C4×D4—28no—
80C2×C4×D4—28no—
81C2×C4×D4—44no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111#399—216no—
112#398—48no—
113#397—48no—
114#396—216no—
115#394—216no—
116#394—216no—
117#394—48no—
118#390—28no—
119#390—28no—
120#390—28no—
121#390—28no—
122#390D4116yes—
123#389—28no—
124#389—28no—
125#389—28no—
126#386—216no—
127#385—216no—
128#385—48no—
129#380—216no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4—42no—
134#376—24no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C25C2414yes—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4C2×D418yes—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—28no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190#127—416no—
191#125—416no—
192#123—464no—
193C4×D4—432no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D418yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
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⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C2×D4116yes—
265C2×C22⋊C4C2×D4116yes—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×D418yes—
278C2×C42#4114yes—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—22no—
287#95—24no—
288#95—22no—
289#95—28no—
290#95—28no—
291#95—44no—
292#95—41no—
293#95—42no—
294#95—41no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24C22×D418yes—
315C24C2511yescharacteristic
316C2×D4—216no—
317C2×D4—216no—
318C2×D4—216no—
319C2×D4—416no—
320C2×D4—416no—
321C2×D4—416no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—88no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—88no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
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—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—416no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D418yes—
406C22×C4C22×D414yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C4H(5,𝔽2)14yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+412yes—
414C22×C42−1+414yes—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—416no—
420C4⋊C4—416no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C42—28no—
440C42—216no—
441C42—48no—
442C42—44no—
443C42—44no—
444C23—28no—
445C23—28no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23C23×D412yes—
480C23C22×#4111yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)12yes—
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+411yescharacteristic
485C23C2×2−1+412yes—
486C23#58512yes—
487C23#58512yes—
488D4—432no—
489D4—832no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—832no—
564C2×C4#53618yes—
565C2×C4#53914yes—
566C2×C4D4214yes—
567C2×C4#54718yes—
568C2×C4#56314yes—
569C22—48no—
570C22—48no—
571C22—48no—
572C22—48no—
573C22—48no—
574C22—48no—
575C22—48no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22C2×#53612yes—
585C22C2×#53911yescharacteristic
586C22#346012yes—
587C22C2×D4211yescharacteristic
588C22C2×#54712yes—
589C22#348412yes—
590C22#348512yes—
591C22#348812yes—
592C22#348812yes—
593C22#350811yescharacteristic
594C22#351211yescharacteristic
595C22C2×#56311yescharacteristic
596C22#354212yes—
597C22#354212yes—
598C22#354812yes—
599C22#354811yescharacteristic
600C22#355412yes—
601C22#355412yes—
602C22#355412yes—
603C22#355712yes—
604C22#355711yescharacteristic
605C4—48no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—44no—
610C4—44no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—84no—
615C4—84no—
616C4#268018yes—
617C2—88no—
618C2#3731811yescharacteristic
619C2#3752011yescharacteristic
620C2#3779212yes—
621C2#3779812yes—
622C2#4247912yes—
623C2#4267812yes—
624C2#4493212yes—
625C2C2×#268011yescharacteristic
626C2#5311212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2454C2semidirect
3. #17103C2semidirect
4. #16269C2semidirect
5. #16257C2semidirect
6. #15269C2semidirect
7. #15218C2semidirect
8. #15149C2semidirect
9. #15147C2semidirect
11. #14260C2semidirect
12. #14018C2semidirect
13. #13856C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#24452
#171032
#162692
#162572
#152692
#152182
#151492
#151472
#151272
#142602
#140182
#138562