Finite Group Databasebeta

← #219800

Subgroups — #219800

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#547C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic, complemented
3C4×C8×D4C211yesmaximal, characteristic, complemented
4C2×#1476C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9071C212yesmaximal, complemented
8#8938C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8312C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×#547C411yescharacteristic
13C2×C4×#41C2212yes—
14C2×C8×D4—21no—
15C2×#431C2212yes—
16C22×C4×C8C2212yes—
17#2365C411yescharacteristic
18#2362C412yes—
19#2361—21no—
20#2359—22no—
21#2354C411yescharacteristic
22D4×C4⋊C4C411yescharacteristic
23#2344—21no—
24#2339C412yes—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C22⋊Q8C2212yes—
27C4×#123—22no—
28C4×#121—21no—
29C42×D4C2211yescharacteristic
30C42×D4C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32#1954—21no—
33#1923—22no—
34#1842C2211yescharacteristic
35#1797C2212yes—
36C8×C4⋊C4—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1476—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2211yescharacteristic
51#1279C2211yescharacteristic
52#1279C2212yes—
53#1279C2211yescharacteristic
54C22×#41C2×C412yes—
55C23×C8—22no—
56#547—24no—
57#547C818yescomplemented
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#123—22no—
60C2×C22⋊Q8C2×C412yes—
61C2×#121—21no—
62C4×#41—24no—
63C4×#41—24no—
64C4×#41—42no—
65C4×#41—42no—
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8C2312yes—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—22no—
78C2×#117C2×C412yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42—22no—
82C22×C42—21no—
83C22×C42C2312yes—
84C8×D4—28no—
85#431—24no—
86#431—42no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2311yescharacteristic
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C4×C8—21no—
97C2×C4×C8—24no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—42no—
103C2×C4×C8C2312yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391C2×C412yes—
107#391C2×C412yes—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389C2×C412yes—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#386—22no—
120#386—22no—
121#385—22no—
122#385—22no—
123#384C2×C411yescharacteristic
124#381—22no—
125#380—21no—
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—22no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4C2×C412yes—
144C4×C22⋊C4D412yes—
145C4×C22⋊C4D412yes—
146C2×#95—22no—
147C43C2311yescharacteristic
148#336—22no—
149#336—21no—
150#336—22no—
151#336—21no—
152Q8⋊C8—24no—
153Q8⋊C8—24no—
154Q8⋊C8—42no—
155Q8⋊C8—42no—
156D4⋊C8—24no—
157D4⋊C8—22no—
158D4⋊C8D414yescomplemented
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41C2×C818yescomplemented
165C22×Q8C22×C412yes—
166C22×D4—22no—
167C22×D4C22⋊C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4C22×C412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C22×C8—21no—
178C22×C8—28no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—44no—
185C22×C8—44no—
186C4⋊Q8—22no—
187C4⋊Q8C2×C814yescomplemented
188#123—216no—
189C22⋊Q8—24no—
190C22⋊Q8C2×C818yescomplemented
191#121—28no—
192C4×Q8—24no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—42no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—41no—
207C4×D4—42no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4C2×C814yes—
213C4×D4C2×C814yescomplemented
214C4×D4M1614yes—
215C4×D4C2×D412yes—
216C4×D4#4112yes—
217#117—24no—
218#117—44no—
219#117—44no—
220#117—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C412yes—
247C2×C22⋊C4C22⋊C412yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42C22×C412yes—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2411yescharacteristic
272C4⋊C8—24no—
273C4⋊C8—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—22no—
278C4⋊C8—42no—
279C4⋊C8—81no—
280C4⋊C8C2×D412yes—
281C22⋊C8—28no—
282C22⋊C8—44no—
283C22⋊C8—82no—
284C4×C8—24no—
285C4×C8—22no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—22no—
290C4×C8—44no—
291C4×C8—44no—
292C4×C8—42no—
293C4×C8—42no—
294C4×C8C2×D412yes—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—21no—
304#95C22⋊C412yes—
305#95C22⋊C412yes—
306#95C22⋊C412yes—
307C24—21no—
308C24C2×C22⋊C412yes—
309#41—216no—
310#41—48no—
311#41—48no—
312#41—48no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8C22×C814yes—
319C2×D4—28no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4C22⋊C814yes—
328C2×D4C4×D412yes—
329C2×D4C22×C812yes—
330C2×D4#13112yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
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—24no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—81no—
367C22×C4—82no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C22×C814yes—
376C22×C4C2×M1614yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—416no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—42no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4C22⋊C814yes—
416C4⋊C4#10414yes—
417C4⋊C4C4×D412yes—
418C4⋊C4C22×C812yes—
419C22⋊C4—28no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—42no—
428C22⋊C4—82no—
429C22⋊C4C22⋊C818yes—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—22no—
441C42—41no—
442C42—44no—
443C42—44no—
444C42—42no—
445C42—42no—
446C42—41no—
447C42—41no—
448C42—42no—
449C42—42no—
450C42C22×C812yes—
451C42C2×M1612yes—
452C42#13512yes—
453C42#13512yes—
454C42C22×D411yescharacteristic
455C42C2×#4111yescharacteristic
456C42H(5,𝔽2)11yescharacteristic
457C42H(5,𝔽2)11yescharacteristic
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23C2×C22⋊C814yes—
472C23C22×C22⋊C411yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—42no—
476Q8—42no—
477D4—22no—
478D4—22no—
479D4—22no—
480D4—28no—
481D4—416no—
482D4C8×D414yescomplemented
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4—88no—
532C2×C4C2×C22⋊C812yes—
533C2×C4C2×C22⋊C814yes—
534C2×C4C2×C22⋊C812yes—
535C2×C4#41612yes—
536C2×C4#41612yes—
537C2×C4C2×#10414yes—
538C2×C4C2×#10412yes—
539C2×C4C2×C4×D411yescharacteristic
540C2×C4#51811yescharacteristic
541C2×C4D4211yescharacteristic
542C2×C4D4211yescharacteristic
543C2×C4#54611yescharacteristic
544C2×C4#54611yescharacteristic
545C2×C4C23×C811yescharacteristic
546C2×C4C22×M1611yescharacteristic
547C2×C4C2×#13111yescharacteristic
548C2×C4#56811yescharacteristic
549C2×C4C2×#13511yescharacteristic
550C2×C4C2×#13511yescharacteristic
551C2×C4#57711yescharacteristic
552C2×C4#57811yescharacteristic
553C8—24no—
554C8—42no—
555C8—44no—
556C8—42no—
557C8—82no—
558C8—81no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22D4×C22⋊C411yescharacteristic
574C22C22×C22⋊C811yescharacteristic
575C22#288011yescharacteristic
576C22C2×#41611yescharacteristic
577C22#289811yescharacteristic
578C22C22×#10411yescharacteristic
579C22#290111yescharacteristic
580C4—22no—
581C4—22no—
582C4—24no—
583C4—21no—
584C4—21no—
585C4—24no—
586C4—42no—
587C4—41no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—82no—
593C4#147614yescomplemented
594C4C2×C8×D411yescharacteristic
595C4D4×M1611yescharacteristic
596C4#297211yescharacteristic
597C4#297411yescharacteristic
598C4#328311yescharacteristic
599C4#328411yescharacteristic
600C4#330211yescharacteristic
601C4#330311yescharacteristic
602C2—44no—
603C2—44no—
604C2C2×#147611yescharacteristic
605C2#1743911yescharacteristic
606C2#1750811yescharacteristic
607C2#1751811yescharacteristic
608C2D4×C22⋊C811yescharacteristic
609C2#2241711yescharacteristic
610C2D4×#10411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#547C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22795C2semidirect
3. C4×C8×D4C2semidirect
4. C2×#1476C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9071C2semidirect
8. #8938C2semidirect
9. #8563C2semidirect
10. #8312C2semidirect
11. C4×D4⋊C8C2semidirect
57. #547C8semidirect
158. D4⋊C8D4semidirect
164. C2×#41C2×C8semidirect
187. C4⋊Q8C2×C8semidirect
190. C22⋊Q8C2×C8semidirect
213. C4×D4C2×C8semidirect
482. D4C8×D4semidirect
593. C4#1476semidirect

Maximal subgroups

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

subgroupindex
C4×#5472
#227952
C4×C8×D42
C2×#14762
#96732
#96372
#90712
#89382
#85632
#83122
C4×D4⋊C82