Finite Group Databasebeta

← #370562

Subgroups — #370562

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#1990C212yesmaximal
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11207C212yesmaximal
7#10880C212yesmaximal
8#10849C211yesmaximal, characteristic
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2551—24no—
18#2551C418yescomplemented
19C2×#395—22no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1990—28no—
31C8×C4⋊C4—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1444—22no—
44C4⋊#95C2212yes—
45C23×Q8—22no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—28no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—44no—
56C2×C4⋊Q8—44no—
57C2×C4⋊Q8C2×C418yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8Q814yescomplemented
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8—24no—
84C2×C4×C8—24no—
85C2×C4×C8—42no—
86C2×C4×C8Q814yescomplemented
87#395—216no—
88#389—28no—
89#389—24no—
90#389—44no—
91#389—44no—
92#389C2×C418yes—
93#384—24no—
94#384—22no—
95#384—22no—
96#384—22no—
97#384—42no—
98#384C2×C414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4D414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340D414yes—
120#340Q814yescomplemented
121#336—24no—
122C22×Q8—28no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8—82no—
128C22×Q8—82no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8C2×D412yes—
157C22×C8C2×Q812yes—
158C22×C8C2×Q814yes—
159C22×C8C2×Q812yes—
160C22×C8C2×Q812yes—
161C4⋊Q8—216no—
162C4⋊Q8—48no—
163C4⋊Q8—48no—
164C4⋊Q8—48no—
165C4⋊Q8—432no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4C22⋊C414yes—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4D814yes—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4SD1614yes—
222C2×C4⋊C4Q1614yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q812yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C4⋊C4#4112yes—
231C2×C4⋊C4#4112yes—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C22×C414yes—
243C2×C42C22×C412yes—
244C2×C42C2×D414yes—
245C2×C42C2×D412yes—
246C2×C42C2×Q812yes—
247C2×C42C2×Q812yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C4⋊C8—28no—
251C4×C8—28no—
252C4×C8—48no—
253#95—24no—
254#95—28no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95C22⋊C418yes—
259C24C22×D411yescharacteristic
260C2×Q8—416no—
261C2×Q8—416no—
262C2×Q8—88no—
263C2×Q8—88no—
264C2×Q8—88no—
265C2×Q8—88no—
266C2×Q8—88no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C414yes—
323C22×C4C2×C22⋊C414yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4#12412yes—
326C22×C4#12412yes—
327C22×C4C4⋊Q812yes—
328C22×C4C4⋊Q812yes—
329C22×C4C2×D814yes—
330C22×C4C2×D812yes—
331C22×C4C2×SD1614yes—
332C22×C4C2×SD1614yes—
333C22×C4C2×SD1612yes—
334C22×C4C2×SD1612yes—
335C22×C4C2×Q1614yes—
336C22×C4C2×Q1612yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C42−1+411yescharacteristic
350C22×C42−1+411yescharacteristic
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8C4⋊Q814yescomplemented
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—416no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C4⋊C4—88no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4#102116yes—
403C4⋊C4#103116yes—
404C4⋊C4C4×D414yes—
405C4⋊C4C4×Q814yes—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—48no—
411C42—84no—
412C42C4×Q814yes—
413C42C4×Q814yes—
414C42C22⋊Q814yes—
415C42C22⋊Q814yes—
416C23C22×C22⋊C411yescharacteristic
417C23C2×#12411yescharacteristic
418C23C2×C4⋊Q811yescharacteristic
419C23D4211yescharacteristic
420C23#54611yescharacteristic
421C23#54811yescharacteristic
422C23D4×Q811yescharacteristic
423C23C22×D811yescharacteristic
424C23C22×SD1611yescharacteristic
425C23C22×SD1611yescharacteristic
426C23C22×Q1611yescharacteristic
427C23#57611yescharacteristic
428C23#57711yescharacteristic
429C23#57711yescharacteristic
430C23#57811yescharacteristic
431Q8—832no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#39114yes—
493C2×C4#39114yes—
494C2×C4C2×#10218yes—
495C2×C4C2×#10214yes—
496C2×C4C2×#10318yes—
497C2×C4C2×#10314yes—
498C2×C4#48614yes—
499C2×C4#48714yes—
500C2×C4#49814yes—
501C2×C4#50014yes—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×Q811yescharacteristic
504C2×C4C2×C4×Q811yescharacteristic
505C2×C4C2×C4×Q811yescharacteristic
506C2×C4#51811yescharacteristic
507C2×C4#51911yescharacteristic
508C2×C4C2×C22⋊Q811yescharacteristic
509C2×C4C2×C22⋊Q811yescharacteristic
510C2×C4C2×C4⋊Q811yescharacteristic
511C2×C4#53211yescharacteristic
512C2×C4#53311yescharacteristic
513C2×C4#53311yescharacteristic
514C2×C4#54211yescharacteristic
515C2×C4#54211yescharacteristic
516C2×C4#54411yescharacteristic
517C2×C4#54411yescharacteristic
518C2×C4D4211yescharacteristic
519C2×C4#54611yescharacteristic
520C2×C4#54711yescharacteristic
521C2×C4#54811yescharacteristic
522C2×C4D4×Q811yescharacteristic
523C2×C4D4×Q811yescharacteristic
524C2×C4#55411yescharacteristic
525C2×C4#55411yescharacteristic
526C2×C4#55511yescharacteristic
527C2×C4#55611yescharacteristic
528C2×C4#55711yescharacteristic
529C2×C4Q8211yescharacteristic
530C8—24no—
531C8—42no—
532C8—44no—
533C22C2×#39111yescharacteristic
534C22D4×C22⋊C411yescharacteristic
535C22Q8×C22⋊C411yescharacteristic
536C22#249111yescharacteristic
537C22#249211yescharacteristic
538C22#255011yescharacteristic
539C22#255111yescharacteristic
540C22C22×#10211yescharacteristic
541C22C22×#10311yescharacteristic
542C22#289711yescharacteristic
543C22#289811yescharacteristic
544C22C2×#48611yescharacteristic
545C22C2×#48711yescharacteristic
546C22#313611yescharacteristic
547C22#313711yescharacteristic
548C22C2×#49811yescharacteristic
549C22C2×#50011yescharacteristic
550C22#316311yescharacteristic
551C22#316411yescharacteristic
552C22D4×D811yescharacteristic
553C22D4×SD1611yescharacteristic
554C22D4×SD1611yescharacteristic
555C22D4×Q1611yescharacteristic
556C22#329411yescharacteristic
557C22#329511yescharacteristic
558C22#329811yescharacteristic
559C22#329911yescharacteristic
560C22#335811yescharacteristic
561C22#335911yescharacteristic
562C22#336211yescharacteristic
563C22#336311yescharacteristic
564C22Q8×D811yescharacteristic
565C22Q8×SD1611yescharacteristic
566C22Q8×SD1611yescharacteristic
567C22Q8×Q1611yescharacteristic
568C4—24no—
569C4—28no—
570C4—28no—
571C4—44no—
572C4—44no—
573C4—82no—
574C4—84no—
575C4—82no—
576C4—162no—
577C4#199018yescomplemented
578C4C4×C4⋊Q811yescharacteristic
579C4#233111yescharacteristic
580C4#236511yescharacteristic
581C4#236911yescharacteristic
582C4#243911yescharacteristic
583C4#244011yescharacteristic
584C4#245111yescharacteristic
585C4#245211yescharacteristic
586C2#1405311yescharacteristic
587C2C2×#199011yescharacteristic
588C2#2132611yescharacteristic
589C2D4×#10211yescharacteristic
590C2D4×#10311yescharacteristic
591C2Q8×#10211yescharacteristic
592C2Q8×#10311yescharacteristic
593C2#2776711yescharacteristic
594C2#2776911yescharacteristic
595C2#2781311yescharacteristic
596C2#2781511yescharacteristic
597C2#2878911yescharacteristic
598C2#2879111yescharacteristic
599C2#2880911yescharacteristic
600C2#2881311yescharacteristic
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
18. #2551C4semidirect
80. C2×C4⋊C8Q8semidirect
86. C2×C4×C8Q8semidirect
120. #340Q8semidirect
371. C2×C8C4⋊Q8semidirect
577. C4#1990semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#19902
C2×C8×C4⋊C42
#139102
#113842
#112072
#108802
#108492
#108412
C4×#3402