Finite Group Databasebeta

← #213499

Subgroups — #213499

Order 512.

1660 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×C4×#121C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4#9080C214yesmaximal, complemented
5#9079C214yesmaximal, complemented
6#8701C212yesmaximal, complemented
7#8302C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4C412yes—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431C2212yes—
13C22×C22⋊C8C2212yes—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C4×#121—21no—
22C2×#390C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#388C2212yes—
25C2×#386C2212yes—
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C412yes—
29#1817—21no—
30#1817—21no—
31#1817C2212yes—
32#1817C2212yes—
33#1817C2212yes—
34#1797—21no—
35#1797—21no—
36#1779C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39C2×#336—21no—
40C2×#336—21no—
41C2×#323C2218yes—
42#1280C2218yes—
43C23×D4C2×C412yes—
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—24no—
55C2×#121C818yescomplemented
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—44no—
65C2×C4×D4—48no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2312yes—
75C22×C42—44no—
76C22×C42C2×C412yes—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#431—44no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—44no—
95C2×C22⋊C8C2314yes—
96C2×C22⋊C8C2314yes—
97C2×C4×C8—22no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2312yes—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#323—48no—
136C25C22×C411yescharacteristic
137C25C22×C412yes—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4C2×C818yes—
149C22×D4C2×C814yescomplemented
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C2×C814yes—
172C23×C4M1614yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8—44no—
191#121—28no—
192#121—28no—
193#121—28no—
194#121—28no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4C2×C814yescomplemented
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4C2×C818yescomplemented
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—44no—
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C4⋊C8—44no—
264C22⋊C8—28no—
265C22⋊C8—28no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269C4×C8—44no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24C22×C812yes—
293C24C22×C814yes—
294C24C2×M1612yes—
295C24C23×C411yescharacteristic
296C2×D4—28no—
297C2×D4—28no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—48no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—48no—
306C2×D4—44no—
307C2×D4—416no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—416no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—416no—
367C22×C4—416no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—816no—
381C22×C4C22⋊C814yes—
382C22×C4C22⋊C814yes—
383C22×C4C22⋊C814yes—
384C22×C4C22⋊C814yes—
385C22×C4#10014yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C22×C812yes—
397C22×C4C22×C812yes—
398C22×C4C22×C814yes—
399C22×C4C2×M1612yes—
400C22×C4C2×M1612yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4H(5,𝔽2)14yes—
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—88no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—44no—
446C42—44no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23C2×C22⋊C812yes—
489C23C2×C22⋊C812yes—
490C23#41012yes—
491C23C2×#10012yes—
492C23C22×C22⋊C411yescharacteristic
493C23#51812yes—
494C23C23×C811yescharacteristic
495C23C22×M1611yescharacteristic
496C23#56812yes—
497D4—48no—
498D4—48no—
499D4—416no—
500D4—416no—
501C2×C4—28no—
502C2×C4—28no—
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—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—88no—
559C2×C4—88no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4C2×C22⋊C812yes—
563C2×C4C2×C22⋊C812yes—
564C2×C4C2×C22⋊C812yes—
565C2×C4C2×C22⋊C812yes—
566C2×C4#41012yes—
567C2×C4C2×#10012yes—
568C2×C4C2×#10012yes—
569C2×C4#41312yes—
570C2×C4#53412yes—
571C2×C4#53412yes—
572C8—84no—
573C8—84no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—48no—
592C22—48no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—88no—
601C22—88no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#146014yes—
607C22#230811yescharacteristic
608C22C22×C22⋊C811yescharacteristic
609C22#287911yescharacteristic
610C22C2×#41011yescharacteristic
611C22C2×#41011yescharacteristic
612C22C22×#10011yescharacteristic
613C22C2×#41311yescharacteristic
614C22#297212yes—
615C22#297312yes—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—44no—
627C4—41no—
628C4—41no—
629C4—82no—
630C4—82no—
631C2—22no—
632C2—22no—
633C2—24no—
634C2—42no—
635C2—84no—
636C2C2×#146011yescharacteristic
637C2#1734911yescharacteristic
638C2#1740211yescharacteristic
639C2#1740311yescharacteristic
640C2#2173811yescharacteristic
641C2#2174811yescharacteristic
642C2#2175211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C2×#1817C2semidirect
4. #9080C2semidirect
5. #9079C2semidirect
6. #8701C2semidirect
7. #8302C2semidirect
55. C2×#121C8semidirect
149. C22×D4C2×C8semidirect
215. C2×C4⋊C4C2×C8semidirect
231. C2×C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#207142
C2×#18172
#90802
#90792
#87012
#83022