Finite Group Databasebeta

← #367054

Subgroups — #367054

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1971C211yesmaximal, characteristic, complemented
3C2×#1971C211yesmaximal, characteristic, complemented
4C2×#1943C211yesmaximal, characteristic, complemented
5C2×#1807C211yesmaximal, characteristic
6C2×#1776C211yesmaximal, characteristic, complemented
7C2×#1752C211yesmaximal, characteristic
8#13997C212yesmaximal
9#11067C212yesmaximal, complemented
10#10970C212yesmaximal
11#10932C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#467—24no—
16C2×#467—24no—
17C2×#467C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1971—24no—
34#1971—24no—
35#1943—24no—
36#1807—24no—
37#1776—24no—
38#1752—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44#1445—22no—
45#1439—22no—
46#1437—22no—
47C4⋊#95—22no—
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16C2×C414yes—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8C2×C414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—42no—
72C2×C22⋊Q8—42no—
73C2×C22⋊Q8C2×C418yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—22no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#467—232no—
86C2×#107C2×C414yes—
87C2×#107D414yes—
88C2×#107D414yes—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106D414yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103C2×C418yes—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97D414yes—
113#391—24no—
114#391—24no—
115#391—24no—
116#391—42no—
117#389—24no—
118#389—42no—
119#380—24no—
120#380—42no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—42no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D412yes—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—22no—
137#336—22no—
138C25C2×D411yescharacteristic
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C22×C414yes—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C22×C412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×Q16—28no—
176C2×Q16—28no—
177C2×Q16—28no—
178C2×Q16—44no—
179C2×Q16—44no—
180C2×Q16—44no—
181C2×Q16—44no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—28no—
201C22×C8—28no—
202C22×C8—28no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8—44no—
207C22×C8C22×C412yes—
208C22×C8C22×C412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8#4112yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217C22⋊Q8—216no—
218C22⋊Q8—48no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—82no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—82no—
267C2×C42C2×D412yes—
268C2×C42#4112yes—
269#107C22⋊C4116yescomplemented
270#106—28no—
271#106—44no—
272#103—216no—
273#103—48no—
274#103—48no—
275C22⋊C8—28no—
276#97—28no—
277#97—44no—
278#95—24no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24C4×D412yes—
296C24#13714yes—
297C24C22×D411yescharacteristic
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4C4×D412yes—
402C22×C4#12114yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C42−1+411yescharacteristic
417Q16—416no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—416no—
440C2×C8C2×C22⋊C414yes—
441C2×C8C4×D414yes—
442C2×C8C4×D414yes—
443C2×C8#12114yes—
444C2×C8#12114yes—
445C2×C8#12414yes—
446C2×C8#12714yes—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—416no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—88no—
460C4⋊C4C2×C22⋊C418yes—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—88no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—88no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23#44114yes—
496C23C2×C4×D411yescharacteristic
497C23C2×#12111yescharacteristic
498C23#53211yescharacteristic
499C23D4211yescharacteristic
500C23D4211yescharacteristic
501C23D4211yescharacteristic
502C23#54711yescharacteristic
503C23C2×#13712yes—
504C23#57512yes—
505C23#57712yes—
506C23#57812yes—
507Q8—44no—
508Q8—44no—
509Q8—44no—
510Q8—44no—
511Q8—82no—
512Q8—82no—
513Q8—82no—
514Q8—82no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C4×#4111yescharacteristic
576C2×C4#51811yescharacteristic
577C2×C4#52011yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4C2×#12111yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4C2×#12411yescharacteristic
582C2×C4#52911yescharacteristic
583C2×C4C2×#12711yescharacteristic
584C2×C4#53511yescharacteristic
585C2×C4#53711yescharacteristic
586C2×C4#54011yescharacteristic
587C2×C4#54211yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54611yescharacteristic
590C2×C4#54611yescharacteristic
591C2×C4#54811yescharacteristic
592C2×C4#54811yescharacteristic
593C2×C4#54811yescharacteristic
594C2×C4#55011yescharacteristic
595C2×C4#55011yescharacteristic
596C2×C4#55011yescharacteristic
597C2×C4#55311yescharacteristic
598C2×C4#55311yescharacteristic
599C2×C4#55511yescharacteristic
600C8—42no—
601C8—42no—
602C8—42no—
603C8#39018yes—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22C4×#12111yescharacteristic
616C22D4×C22⋊C411yescharacteristic
617C22#236111yescharacteristic
618C22#241611yescharacteristic
619C22#242811yescharacteristic
620C22#249111yescharacteristic
621C22#254111yescharacteristic
622C22C2×#44111yescharacteristic
623C22#294311yescharacteristic
624C22#295011yescharacteristic
625C22#295211yescharacteristic
626C22#305612yes—
627C22#306312yes—
628C22#317812yes—
629C22#318112yes—
630C22#326612yes—
631C22#326712yes—
632C22#326712yes—
633C22#326912yes—
634C22#327012yes—
635C22#327012yes—
636C22#334412yes—
637C22#334912yes—
638C4—48no—
639C4—42no—
640C4—44no—
641C4—82no—
642C4—84no—
643C4—84no—
644C4—162no—
645C4C2×#39011yescharacteristic
646C4#232411yescharacteristic
647C4#234111yescharacteristic
648C4#236311yescharacteristic
649C4#241411yescharacteristic
650C4#242611yescharacteristic
651C4#249011yescharacteristic
652C4#254011yescharacteristic
653C2—44no—
654C2#1399811yescharacteristic
655C2#2152711yescharacteristic
656C2#2153011yescharacteristic
657C2#2243811yescharacteristic
658C2#2244011yescharacteristic
659C2#2302811yescharacteristic
660C2#2303011yescharacteristic
661C2#2479112yes—
662C2#2542312yes—
663C2#2779712yes—
664C2#2849212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1971C2semidirect
3. C2×#1971C2semidirect
4. C2×#1943C2semidirect
6. C2×#1776C2semidirect
9. #11067C2semidirect
11. #10932C2semidirect
17. C2×#467C4semidirect
269. #107C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#19712
C2×#19432
C2×#18072
C2×#17762
C2×#17522
#139972
#110672
#109702
#109322