Finite Group Databasebeta

← #365219

Subgroups — #365219

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5C4⋊C42C211yesmaximal, characteristic
6#11257C212yesmaximal
7#11122C212yesmaximal
8#10932C212yesmaximal
9#10841C211yesmaximal, characteristic
10#10834C211yesmaximal, characteristic
11#10806C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1948—24no—
27#1803C418yescomplemented
28C4×#103—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1441—22no—
38#1436—22no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42—22no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#106—24no—
72C2×#106—42no—
73C2×#106—42no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2×C414yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103C2×C418yes—
85C2×#103Q814yes—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#391—24no—
90#391—42no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384C2×C414yes—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—42no—
100C4×C4⋊C4—44no—
101C4×C4⋊C4D414yes—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—22no—
114#340—22no—
115#340—42no—
116#340C2×C418yes—
117#336—42no—
118#336—42no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8C22×C412yes—
131C22×Q8C2×Q812yes—
132C22×Q8#4112yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8C22×C414yes—
163C22×C8C2×Q812yes—
164C22×C8#4112yes—
165C4×Q8—28no—
166C4×Q8—28no—
167C4×Q8—44no—
168C4×Q8—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4SD1614yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4#4112yes—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#106—48no—
242C4⋊C8—28no—
243#103—28no—
244#103—28no—
245#103—48no—
246C4×C8—44no—
247#95—24no—
248#95—24no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—82no—
254#95—82no—
255#95—82no—
256C24C22×D411yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8C4×Q814yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
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—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
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—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12312yes—
349C22×C4#12312yes—
350C22×C4C2×SD1614yes—
351C22×C4C2×SD1612yes—
352C22×C4#13512yes—
353C22×C4#13712yes—
354C22×C4#13712yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+411yescharacteristic
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—28no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4C22⋊Q814yes—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—48no—
422C42—48no—
423C42—88no—
424C42#11714yes—
425C42C4×D414yes—
426C42#12414yes—
427C42#12614yes—
428C23C2×C4×D411yescharacteristic
429C23C2×#12111yescharacteristic
430C23C2×#12311yescharacteristic
431C23#54611yescharacteristic
432C23#54611yescharacteristic
433C23D4×Q811yescharacteristic
434C23#55011yescharacteristic
435C23C22×SD1611yescharacteristic
436C23C2×#13511yescharacteristic
437C23C2×#13711yescharacteristic
438C23C2×#13711yescharacteristic
439C23#57611yescharacteristic
440C23#57711yescharacteristic
441C23#57711yescharacteristic
442C23#57811yescharacteristic
443Q8—24no—
444Q8—24no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4#38814yes—
514C2×C4#38814yes—
515C2×C4C4×SD1614yes—
516C2×C4#44114yes—
517C2×C4#46114yes—
518C2×C4#46414yes—
519C2×C4#48314yes—
520C2×C4#48514yes—
521C2×C4C2×#11711yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×C4×Q811yescharacteristic
524C2×C4C4×#4111yescharacteristic
525C2×C4#51811yescharacteristic
526C2×C4#52011yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4#52511yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12611yescharacteristic
531C2×C4#52911yescharacteristic
532C2×C4#53311yescharacteristic
533C2×C4#53511yescharacteristic
534C2×C4#53711yescharacteristic
535C2×C4#54011yescharacteristic
536C2×C4#54211yescharacteristic
537C2×C4#55211yescharacteristic
538C2×C4#55211yescharacteristic
539C2×C4#55211yescharacteristic
540C2×C4#55311yescharacteristic
541C2×C4#55311yescharacteristic
542C2×C4#55311yescharacteristic
543C2×C4#55411yescharacteristic
544C2×C4#55411yescharacteristic
545C2×C4#55411yescharacteristic
546C2×C4#55511yescharacteristic
547C2×C4#55511yescharacteristic
548C2×C4#55511yescharacteristic
549C8—42no—
550C8—42no—
551C8—44no—
552C22C2×#38811yescharacteristic
553C22#236011yescharacteristic
554C22#236311yescharacteristic
555C22#241411yescharacteristic
556C22#241711yescharacteristic
557C22#242011yescharacteristic
558C22#242311yescharacteristic
559C22C2×C4×SD1611yescharacteristic
560C22C2×#44111yescharacteristic
561C22#294611yescharacteristic
562C22#294711yescharacteristic
563C22C2×#46111yescharacteristic
564C22C2×#46411yescharacteristic
565C22#304511yescharacteristic
566C22#304611yescharacteristic
567C22C2×#48311yescharacteristic
568C22C2×#48511yescharacteristic
569C22#309811yescharacteristic
570C22#309911yescharacteristic
571C22#329511yescharacteristic
572C22#329811yescharacteristic
573C22#330411yescharacteristic
574C22#330411yescharacteristic
575C22#332811yescharacteristic
576C22#332911yescharacteristic
577C22#333111yescharacteristic
578C22#333311yescharacteristic
579C22Q8×SD1611yescharacteristic
580C22#338311yescharacteristic
581C22#338511yescharacteristic
582C22#338711yescharacteristic
583C22#339211yescharacteristic
584C22#339611yescharacteristic
585C22#339811yescharacteristic
586C22#340011yescharacteristic
587C4—24no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—162no—
596C4#192718yes—
597C4C4×C22⋊Q811yescharacteristic
598C4#232411yescharacteristic
599C4#234511yescharacteristic
600C4#234611yescharacteristic
601C4#249511yescharacteristic
602C4#249711yescharacteristic
603C4#256211yescharacteristic
604C4#256311yescharacteristic
605C2#1397311yescharacteristic
606C2C2×#192711yescharacteristic
607C2#2106211yescharacteristic
608C2#2296411yescharacteristic
609C2#2297111yescharacteristic
610C2#2309511yescharacteristic
611C2#2310211yescharacteristic
612C2#2463011yescharacteristic
613C2#2463311yescharacteristic
614C2#2481511yescharacteristic
615C2#2481811yescharacteristic
616C2#2496911yescharacteristic
617C2#2497211yescharacteristic
618C2#2506011yescharacteristic
619C2#2506311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1803C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19482
C2×#18032
C2×C4×#1032
#139262
C4⋊C422
#112572
#111222
#109322
#108412
#108342
#108062