Finite Group Databasebeta

← #214294

Subgroups — #214294

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C4×#380C211yesmaximal, characteristic, complemented
4#9148C214yesmaximal, complemented
5#9114C214yesmaximal, complemented
6#8712C212yesmaximal, complemented
7#8328C212yesmaximal
8C22×#123—21no—
9C22×#121C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431C2212yescomplemented
12#2854—21no—
13#2854—21no—
14#2854—21no—
15#2854—21no—
16#2854C412yescomplemented
17#2854C412yescomplemented
18#2854C2212yes—
19#2854C2212yes—
20C2×#390C411yescharacteristic, complemented
21C2×#390C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#389C411yescharacteristic, complemented
24C2×#388—21no—
25C2×#388—21no—
26C2×#386—21no—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yescomplemented
30C2×C4×C22⋊C4—21no—
31#1817C2212yescomplemented
32#1797—21no—
33#1797—21no—
34#1779C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C2×#327C2218yes—
38C4⋊#95—21no—
39#1434C2212yescomplemented
40C22⋊#95C2212yes—
41C4×#95—21no—
42#1294C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#123—22no—
49C2×#123—22no—
50C2×#123—22no—
51C2×#123—22no—
52C2×#123—44no—
53C2×#123—44no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—44no—
57C2×#121C2×C412yes—
58C2×#121C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—22no—
73C22×C42—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#431—44no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8C2314yes—
91C2×C4×C8—22no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2×C412yes—
98#390C2×C412yes—
99#390C2×C412yes—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#380—28no—
121#380—21no—
122#380—21no—
123#380C2×C412yes—
124#380C2312yes—
125C4×C4⋊C4—44no—
126C4×C22⋊C4—44no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134#336—22no—
135#336—22no—
136#327—48no—
137C25—21no—
138C25C2×D411yescharacteristic
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4C22×C412yes—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4D814yes—
171C23×C4SD1614yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—24no—
183#123—48no—
184#123—48no—
185#123—44no—
186#123—44no—
187#123—44no—
188#123—44no—
189#121—28no—
190#121—44no—
191#121—44no—
192#121—44no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—42no—
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—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C4⋊C8—44no—
265C22⋊C8—28no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268C4×C8—44no—
269#95—28no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—28no—
277#95—44no—
278#95—44no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C24C2×D812yes—
296C24C2×SD1612yes—
297C24C22×D411yescharacteristic
298C2×D4—24no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—88no—
312C2×D4—88no—
313C2×D4—88no—
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—28no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
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—42no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4#10214yes—
384C22×C4#10214yes—
385C22×C4#10214yes—
386C22×C4#10214yes—
387C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×D812yes—
399C22×C4C2×D812yes—
400C22×C4C2×SD1612yes—
401C22×C4C2×SD1612yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4H(5,𝔽2)14yes—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—416no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—416no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—88no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23#41012yes—
489C23C2×#10212yes—
490C23C2×#10212yes—
491C23C2×#10412yes—
492C23C22×C22⋊C411yescharacteristic
493C23#53412yes—
494C23C22×D811yescharacteristic
495C23C22×SD1611yescharacteristic
496C23#57512yes—
497D4—84no—
498D4—84no—
499D4—88no—
500D4—84no—
501D4—84no—
502C2×C4—28no—
503C2×C4—28no—
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—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
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—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4#41012yes—
568C2×C4C2×#10212yes—
569C2×C4C2×#10212yes—
570C2×C4C2×#10212yes—
571C2×C4C2×#10212yes—
572C2×C4C2×#10412yes—
573C2×C4C2×#10412yes—
574C2×C4#42112yes—
575C2×C4#51812yes—
576C2×C4#53412yes—
577C8—84no—
578C22—22no—
579C22—22no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—22no—
584C22—21no—
585C22—22no—
586C22—21no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—48no—
596C22—48no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—168no—
609C22#150214yes—
610C22#230811yescharacteristic
611C22C2×#41011yescharacteristic
612C22C2×#41011yescharacteristic
613C22C22×#10211yescharacteristic
614C22#289611yescharacteristic
615C22C22×#10411yescharacteristic
616C22C2×#42111yescharacteristic
617C22#318612yes—
618C22#318712yes—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—88no—
632C4—82no—
633C4—82no—
634C4—82no—
635C2—22no—
636C2—24no—
637C2—22no—
638C2—42no—
639C2—162no—
640C2C2×#150211yescharacteristic
641C2#1753711yescharacteristic
642C2#1757411yescharacteristic
643C2#1758011yescharacteristic
644C2#2174711yescharacteristic
645C2#2175811yescharacteristic
646C2#2177311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ 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. C4×#380C2semidirect
4. #9148C2semidirect
5. #9114C2semidirect
6. #8712C2semidirect
11. C2×#431C22semidirect
16. #2854C4semidirect
17. #2854C4semidirect
20. C2×#390C4semidirect
21. C2×#390C4semidirect
22. C2×#389C4semidirect
23. C2×#389C4semidirect
29. C2×C4×C4⋊C4C22semidirect
31. #1817C22semidirect
39. #1434C22semidirect
42. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#207142
C4×#3802
#91482
#91142
#87122
#83282