Finite Group Databasebeta

← #213601

Subgroups — #213601

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic, complemented
4#9120C214yesmaximal, complemented
5#9119C214yesmaximal, complemented
6#8719C212yesmaximal, complemented
7#8334C212yesmaximal, complemented
8C22×#124C2211yescharacteristic
9C22×C22⋊Q8C411yescharacteristic
10C22×#121C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#407—21no—
13C22×C22⋊C8—21no—
14C2×C4×M16C2212yescomplemented
15C22×#97C2212yescomplemented
16#2850—21no—
17#2850—21no—
18#2850—21no—
19#2850—21no—
20#2850C412yes—
21#2850C412yes—
22#2850C2212yes—
23#2850C2212yes—
24C2×#391C411yescharacteristic
25C2×#390C411yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#386—21no—
29C2×#386—21no—
30C2×#382C2211yescharacteristic
31#1749—21no—
32#1749—21no—
33#1749C2212yes—
34#1749C2212yes—
35#1749C2212yescomplemented
36#1749C2212yescomplemented
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39#1728—21no—
40C2×#336—21no—
41C2×#329C2218yescomplemented
42#1295C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—22no—
50C23×C8—22no—
51C2×#124—21no—
52C2×#124—21no—
53C2×#124—44no—
54C2×#124C2×C412yes—
55C2×#124C2312yes—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—44no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—44no—
64C2×#121C2×C412yes—
65C2×#121C2×C412yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#407—28no—
89C2×C22⋊C8—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C4×M16—44no—
95C2×#97—22no—
96C2×#97—44no—
97C2×#97C2314yes—
98C2×#97C2314yes—
99C2×C4×C8—22no—
100#391—21no—
101#391—21no—
102#391C2×C412yes—
103#391C2×C412yes—
104#390—21no—
105#390—21no—
106#390C2×C412yes—
107#390C2×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#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#382—21no—
125#382—21no—
126#382C2×C412yes—
127#382C2312yes—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#329—48no—
137C25—21no—
138C25C22⋊C411yescharacteristic
139C22×Q8—21no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8C22×C412yes—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4C22×C412yes—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—28no—
185C2×M16—44no—
186C2×M16—44no—
187C22×C8—28no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—44no—
191C22×C8—44no—
192#124—28no—
193#124—44no—
194#124—44no—
195#124—44no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200#121—44no—
201#121—44no—
202#121—44no—
203#121—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
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—22no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—42no—
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×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2411yescharacteristic
259C22⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262C22⋊C8—28no—
263#97—28no—
264#97—28no—
265#97—44no—
266C4×C8—44no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24#10414yes—
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24C2×C22⋊C411yescharacteristic
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—88no—
300C2×D4—24no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—88no—
313C2×D4—816no—
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—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
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—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—416no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4#10014yes—
382C22×C4#10414yes—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395M16—416no—
396C2×C8—28no—
397C2×C8—28no—
398C2×C8—416no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42H(5,𝔽2)14yes—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—28no—
452C23—28no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23C2×#10012yes—
473C23#41312yes—
474C23C2×#10412yes—
475C23C2×#10412yes—
476C23C2×#10412yes—
477C23C2×#10412yes—
478C23#42112yes—
479C23C22×C22⋊C411yescharacteristic
480Q8—84no—
481Q8—84no—
482D4—84no—
483D4—84no—
484D4—88no—
485D4—88no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
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—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4C2×#10012yes—
545C2×C4C2×#10012yes—
546C2×C4C2×#10412yes—
547C2×C4C2×#10412yes—
548C2×C4#42112yes—
549C2×C4#51812yes—
550C2×C4#53412yes—
551C2×C4#53412yes—
552C8—48no—
553C8—48no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—416no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—168no—
582C22#152414yes—
583C22#152714yes—
584C22#230811yescharacteristic
585C22C22×#10011yescharacteristic
586C22C2×#41311yescharacteristic
587C22C22×#10411yescharacteristic
588C22C22×#10411yescharacteristic
589C22C2×#42111yescharacteristic
590C22C2×#42111yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—82no—
604C4—82no—
605C4—82no—
606C2—24no—
607C2—24no—
608C2—42no—
609C2—162no—
610C2C2×#152411yescharacteristic
611C2C2×#152711yescharacteristic
612C2#1764711yescharacteristic
613C2#1766911yescharacteristic
614C2#2175211yescharacteristic
615C2#2177311yescharacteristic
616C2#2177311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. C2×#1749C2semidirect
4. #9120C2semidirect
5. #9119C2semidirect
6. #8719C2semidirect
7. #8334C2semidirect
14. C2×C4×M16C22semidirect
15. C22×#97C22semidirect
35. #1749C22semidirect
36. #1749C22semidirect
41. C2×#329C22semidirect
42. #1295C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
C4×#4072
C2×#17492
#91202
#91192
#87192
#83342