Finite Group Databasebeta

← #99634

Subgroups — #99634

Order 512.

889 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2023C211yesmaximal, characteristic
2#14445C211yesmaximal, characteristic
3#11384C211yesmaximal, characteristic, complemented
4#11010C211yesmaximal, characteristic, complemented
5C2×#1270C211yesmaximal, characteristic
6#7025C211yesmaximal, characteristic, complemented
7#7025C211yesmaximal, characteristic, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×#103—21no—
10C22×#103—21no—
11C22×#102—21no—
12C2×#390C2211yescharacteristic
13C2×#388—21no—
14C2×#388—21no—
15#2023—22no—
16#2023C414yescomplemented
17C2×#340—21no—
18C2×#340—21no—
19C2×#340—21no—
20C2×#336C2211yescharacteristic
21C2×#336C2211yescharacteristic
22C2×#336C2211yescharacteristic
23#1439—21no—
24#1439—21no—
25#1438—21no—
26#1436—21no—
27C4⋊#95C2211yescharacteristic
28C4⋊#95C2211yescharacteristic
29#1270—22no—
30C23×Q8—21no—
31C23×D4D411yescharacteristic
32C23×C8—21no—
33C23×C8—21no—
34C23×C8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—42no—
39C2×C4⋊Q8—42no—
40C2×C4⋊Q8C2×C412yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56C2×#103—42no—
57C2×#103—42no—
58C2×#103—42no—
59C2×#103—42no—
60C2×#103—42no—
61C2×#103—42no—
62C2×#102—24no—
63C2×#102—42no—
64C2×#102—42no—
65#390—21no—
66#390—21no—
67#390C2×C412yes—
68#390D412yes—
69#388—42no—
70#388—42no—
71#388—42no—
72#388—42no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80#340—24no—
81#340—42no—
82#340—42no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336C2×C412yes—
89C25—21no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—41no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—41no—
110C22×D4—41no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4C22⋊C412yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4D812yes—
179C2×C4⋊C4SD1612yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—81no—
197C2×C22⋊C4—81no—
198C2×C22⋊C4—81no—
199C2×C22⋊C4—81no—
200C2×C22⋊C4—81no—
201C2×C22⋊C4—81no—
202C2×C22⋊C4—81no—
203C2×C22⋊C4—81no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—81no—
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213#103—48no—
214#103—48no—
215#102—48no—
216#95—42no—
217#95—42no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—81no—
227C24—22no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24#12011yescharacteristic
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—81no—
260C2×Q8—81no—
261C2×Q8—81no—
262C2×Q8—81no—
263C2×Q8—81no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×Q8—81no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4—81no—
286C2×D4—81no—
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—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12311yescharacteristic
347C22×C4C2×D811yescharacteristic
348C22×C4C2×SD1611yescharacteristic
349C22×C4#13511yescharacteristic
350C22×C4#13511yescharacteristic
351C22×C4#13511yescharacteristic
352C22×C4#13511yescharacteristic
353C22×C4AGL(1,ℤ/8)11yescharacteristic
354C22×C4AGL(1,ℤ/8)11yescharacteristic
355C22×C4AGL(1,ℤ/8)11yescharacteristic
356C22×C4#13711yescharacteristic
357C22×C4#13711yescharacteristic
358C22×C4#13711yescharacteristic
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—81no—
369C2×C8—81no—
370C2×C8—81no—
371C2×C8—81no—
372C2×C8—81no—
373C2×C8—81no—
374C2×C8—81no—
375C2×C8—81no—
376C2×C8—81no—
377C2×C8—81no—
378C2×C8—81no—
379C2×C8—81no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
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—81no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—84no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
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—
448C23—81no—
449C23—81no—
450C23#38611yescharacteristic
451C23#44711yescharacteristic
452C23#44911yescharacteristic
453C23#45011yescharacteristic
454C23#45211yescharacteristic
455C23#45211yescharacteristic
456C23#45211yescharacteristic
457C23#45311yescharacteristic
458C23#45411yescharacteristic
459C23#45511yescharacteristic
460C23#45511yescharacteristic
461C23#45511yescharacteristic
462C23#45611yescharacteristic
463C23B(4,𝔽2)11yescharacteristic
464C23B(4,𝔽2)11yescharacteristic
465Q8—44no—
466Q8—44no—
467Q8—82no—
468Q8—82no—
469Q8—82no—
470Q8—82no—
471Q8—82no—
472Q8—82no—
473D4—44no—
474D4—82no—
475D4—82no—
476D4—82no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4#41611yescharacteristic
556C2×C4#41911yescharacteristic
557C2×C4C4×D811yescharacteristic
558C2×C4C4×SD1611yescharacteristic
559C2×C4#44011yescharacteristic
560C2×C4#44211yescharacteristic
561C2×C4#45911yescharacteristic
562C2×C4#46111yescharacteristic
563C2×C4#46311yescharacteristic
564C2×C4#46311yescharacteristic
565C2×C4#48511yescharacteristic
566C2×C4#48511yescharacteristic
567C8—82no—
568C8—82no—
569C8—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—161no—
578C22—161no—
579C22—161no—
580C22—161no—
581C22—161no—
582C22—161no—
583C22—161no—
584C22—161no—
585C22#159511yescharacteristic
586C22#159711yescharacteristic
587C22#159811yescharacteristic
588C22#160011yescharacteristic
589C22#160111yescharacteristic
590C22#160311yescharacteristic
591C22#160411yescharacteristic
592C22#160611yescharacteristic
593C22#160611yescharacteristic
594C22#160611yescharacteristic
595C22#160611yescharacteristic
596C22#160611yescharacteristic
597C22#163111yescharacteristic
598C22#163411yescharacteristic
599C22#163511yescharacteristic
600C22#163811yescharacteristic
601C22#163911yescharacteristic
602C22#164011yescharacteristic
603C22#164111yescharacteristic
604C22#164111yescharacteristic
605C22#164211yescharacteristic
606C22#164611yescharacteristic
607C22#164711yescharacteristic
608C22#165211yescharacteristic
609C22#165911yescharacteristic
610C22#166111yescharacteristic
611C22#166211yescharacteristic
612C22#166411yescharacteristic
613C22#187311yescharacteristic
614C22#187711yescharacteristic
615C22#187911yescharacteristic
616C22#188211yescharacteristic
617C22#188311yescharacteristic
618C22#188511yescharacteristic
619C22#188911yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—82no—
633C4—84no—
634C4—82no—
635C4—82no—
636C4—161no—
637C4—161no—
638C2—162no—
639C2#775211yescharacteristic
640C2#775811yescharacteristic
641C2#776211yescharacteristic
642C2#776511yescharacteristic
643C2#777311yescharacteristic
644C2#777611yescharacteristic
645C2#778111yescharacteristic
646C2#778511yescharacteristic
647C2#897611yescharacteristic
648C2#898411yescharacteristic
649C2#899911yescharacteristic
650C2#901611yescharacteristic
651C2#903911yescharacteristic
652C2#904111yescharacteristic
653C2#905711yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11384C2semidirect
4. #11010C2semidirect
6. #7025C2semidirect
7. #7025C2semidirect
16. #2023C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20232
#144452
#113842
#110102
C2×#12702
#70252