Finite Group Databasebeta

← #391362

Subgroups — #391362

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2093C211yesmaximal, characteristic
2#14712C211yesmaximal, characteristic, complemented
3#14700C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic, complemented
5#11267C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11189C212yesmaximal
9#11066C212yesmaximal
10#11049C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C2×#390C2211yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#389—21no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#388—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#2093C418yescomplemented
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1437—21no—
38#1437—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#107—42no—
72C2×#107—42no—
73C2×#107C2×C418yes—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102C2×C418yes—
80#390—22no—
81#390—42no—
82#390—42no—
83#390C2×C414yes—
84#389—42no—
85#389—42no—
86#389—42no—
87#389—44no—
88#389—44no—
89#388—44no—
90#388—44no—
91#384—22no—
92#384—42no—
93#384C2×C414yes—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C43—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—22no—
109#336—42no—
110#336—42no—
111#336C2×C414yes—
112C25—21no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4C22×C412yes—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8C22×C414yes—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42D814yes—
231C2×C42C22×C412yes—
232C2×C42#4112yes—
233#107—48no—
234#102—48no—
235#102—48no—
236#95—42no—
237#95—44no—
238#95—42no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—44no—
243#95—42no—
244C24—22no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24C22×D411yescharacteristic
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4C2×D812yes—
346C22×C4C2×D812yes—
347C22×C4C2×D812yes—
348C22×C4#13512yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4AGL(1,ℤ/8)12yes—
351C22×C4#13712yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)12yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+411yescharacteristic
360C22×C42−1+411yescharacteristic
361C22×C42−1+412yes—
362C22×C42−1+411yescharacteristic
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—88no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C42—24no—
404C42—24no—
405C42—44no—
406C42—44no—
407C42—44no—
408C42—44no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23C2×C4×D411yescharacteristic
449C23#53411yescharacteristic
450C23#53511yescharacteristic
451C23#53511yescharacteristic
452C23#53611yescharacteristic
453C23#53611yescharacteristic
454C23#53711yescharacteristic
455C23C22×D811yescharacteristic
456C23C2×#13511yescharacteristic
457C23C2×AGL(1,ℤ/8)11yescharacteristic
458C23C2×AGL(1,ℤ/8)11yescharacteristic
459C23C2×#13711yescharacteristic
460C23C2×#13711yescharacteristic
461C23#57511yescharacteristic
462C23#57511yescharacteristic
463D4—44no—
464D4—44no—
465D4—82no—
466D4—82no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4C4×D814yes—
543C2×C4#44214yes—
544C2×C4#51812yes—
545C2×C4#51911yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4#52012yes—
548C2×C4#53912yes—
549C2×C4#54012yes—
550C2×C4#54111yescharacteristic
551C2×C4#54211yescharacteristic
552C2×C4#56012yes—
553C2×C4#56212yes—
554C2×C4#56212yes—
555C2×C4#56212yes—
556C2×C4#56212yes—
557C2×C4#56312yes—
558C2×C4#56312yes—
559C2×C4#56312yes—
560C8—44no—
561C8—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—161no—
570C22—161no—
571C22—161no—
572C22—161no—
573C22—161no—
574C22—161no—
575C22—161no—
576C22—161no—
577C22#232211yescharacteristic
578C22#232311yescharacteristic
579C22#232811yescharacteristic
580C22#281311yescharacteristic
581C22#281611yescharacteristic
582C22#282011yescharacteristic
583C22#282111yescharacteristic
584C22C2×C4×D811yescharacteristic
585C22C2×#44211yescharacteristic
586C22C4×#13711yescharacteristic
587C22C4×#13711yescharacteristic
588C22#318611yescharacteristic
589C22#319011yescharacteristic
590C22#319311yescharacteristic
591C22#319411yescharacteristic
592C22#320511yescharacteristic
593C22#320511yescharacteristic
594C22#320711yescharacteristic
595C22#320711yescharacteristic
596C22#321511yescharacteristic
597C22#321611yescharacteristic
598C22#321811yescharacteristic
599C22#321911yescharacteristic
600C22#322611yescharacteristic
601C22#322611yescharacteristic
602C22#323211yescharacteristic
603C22#323311yescharacteristic
604C22#323411yescharacteristic
605C22#323511yescharacteristic
606C22#323811yescharacteristic
607C22#323911yescharacteristic
608C22#324511yescharacteristic
609C22#324811yescharacteristic
610C22#325111yescharacteristic
611C22#325411yescharacteristic
612C4—24no—
613C4—24no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—82no—
619C4—84no—
620C4—161no—
621C4—161no—
622C4—162no—
623C4#237612yes—
624C4#237812yes—
625C4#283612yes—
626C4#283712yes—
627C2—162no—
628C2#1462811yescharacteristic
629C2#2207511yescharacteristic
630C2#2208111yescharacteristic
631C2#2211011yescharacteristic
632C2#2211611yescharacteristic
633C2#2226911yescharacteristic
634C2#2227311yescharacteristic
635C2#3271811yescharacteristic
636C2#3272311yescharacteristic
637C2#3277011yescharacteristic
638C2#3277511yescharacteristic
639C2#3281811yescharacteristic
640C2#3282511yescharacteristic
641C2#3283411yescharacteristic
642C2#3284111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14712C2semidirect
4. #11373C2semidirect
5. #11267C2semidirect
6. #11219C2semidirect
7. #11219C2semidirect
11. #10795C2semidirect
27. #2093C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20932
#147122
#147002
#113732
#112672
#112192
#111892
#110662
#110492
#107952