Finite Group Databasebeta

← #219837

Subgroups — #219837

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34934C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3C4×C8×D4C211yesmaximal, characteristic, complemented
4C2×#1494C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic, complemented
7#9072C212yesmaximal, complemented
8#8943C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8315C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#529C2212yes—
13C2×C8×D4—21no—
14C2×#432C2212yes—
15C22×C4×C8C2212yes—
16#2554—22no—
17#2552C411yescharacteristic
18#2548C411yescharacteristic
19#2541C411yescharacteristic
20#2538C411yescharacteristic
21#2512—21no—
22#2490C412yes—
23#2445—21no—
24#2426C412yes—
25#2392—22no—
26C4×#127—21no—
27C4×#127C2211yescharacteristic
28C4×#121C2212yes—
29#2293—21no—
30C42×Q8C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1924—22no—
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1770—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1494—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yes—
51#1279C2211yescharacteristic
52#1279C2212yes—
53#1279C2211yescharacteristic
54C23×C8—22no—
55C2×#127—21no—
56C2×#127C2×C411yescharacteristic
57#529—24no—
58#529—24no—
59#529—42no—
60#529—42no—
61C2×#126—22no—
62C2×#125C2×C412yes—
63C2×#124C2×C412yes—
64C2×#123—22no—
65C2×C22⋊Q8C2×C412yes—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4C2312yes—
79C2×#117—22no—
80C22×C42C2312yes—
81C8×D4—28no—
82#432—24no—
83#432—42no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2311yescharacteristic
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×#97—22no—
94C2×C4×C8—21no—
95C2×C4×C8—24no—
96C2×C4×C8—22no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—42no—
100C2×C4×C8C2312yes—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#398—22no—
104#397—22no—
105#396—22no—
106#396—22no—
107#394—22no—
108#394—22no—
109#394—22no—
110#392—22no—
111#390—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390—22no—
116#390—22no—
117#390—21no—
118#390—21no—
119#390C2×C412yes—
120#390C2×C412yes—
121#389—21no—
122#389—21no—
123#389C2×C412yes—
124#388—22no—
125#388—22no—
126#388—22no—
127#386—22no—
128#384—21no—
129#384C2×C411yescharacteristic
130#380C2×C412yes—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—22no—
141C4×C22⋊C4C2×C412yes—
142C4×C22⋊C4D412yes—
143C4×C22⋊C4D412yes—
144C43—21no—
145C43C2311yescharacteristic
146#336—22no—
147#336—21no—
148#336—22no—
149#336—21no—
150Q8⋊C8—24no—
151Q8⋊C8—22no—
152Q8⋊C8D414yescomplemented
153D4⋊C8—24no—
154D4⋊C8—24no—
155D4⋊C8—42no—
156D4⋊C8—42no—
157C22×Q8C22×C411yescharacteristic
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22⋊C411yescharacteristic
163C22×D4C22×C412yes—
164C23×C4—22no—
165C23×C4C22×C412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C22×C8—21no—
169C22×C8—28no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—44no—
176C22×C8—44no—
177#127—22no—
178#127—22no—
179#127—44no—
180#126—44no—
181#126—44no—
182#125—24no—
183#125—42no—
184#124—24no—
185#124—42no—
186#123—44no—
187#123—44no—
188C22⋊Q8—24no—
189C22⋊Q8—42no—
190#121—24no—
191#121—24no—
192#121—42no—
193#121—42no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—41no—
200C4×Q8C2×D412yes—
201C4×Q8#4112yes—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—81no—
216C4×D4—81no—
217C4×D4—82no—
218C4×D4—82no—
219#117—44no—
220#117—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4C22⋊C412yes—
249C2×C22⋊C4C22⋊C412yes—
250C2×C22⋊C4C22⋊C412yes—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C412yes—
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2411yescharacteristic
271C4⋊C8—24no—
272C4⋊C8—22no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—22no—
277C4⋊C8—42no—
278C4⋊C8—81no—
279C4⋊C8C2×D412yes—
280C22⋊C8—28no—
281C22⋊C8—44no—
282C22⋊C8—82no—
283#97—44no—
284C4×C8—24no—
285C4×C8—22no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—22no—
290C4×C8—44no—
291C4×C8—42no—
292C4×C8—42no—
293C4×C8C2×D412yes—
294#95—22no—
295#95—22no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95C22⋊C412yes—
300#95C22⋊C412yes—
301#95C22⋊C412yes—
302C24—22no—
303C24C2×C22⋊C412yes—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8C4×D412yes—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—81no—
327C2×D4—81no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4#10414yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
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—21no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—22no—
374C2×C8—416no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—42no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—81no—
405C4⋊C4—82no—
406C4⋊C4C4×D412yes—
407C4⋊C4#13112yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—22no—
436C42—41no—
437C42—42no—
438C42—41no—
439C42—41no—
440C42—42no—
441C42—42no—
442C42—81no—
443C42#13112yes—
444C42#13112yes—
445C42#13512yes—
446C42#13512yes—
447C42C22×D411yescharacteristic
448C42C2×#4111yescharacteristic
449C42H(5,𝔽2)11yescharacteristic
450C42H(5,𝔽2)11yescharacteristic
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×#10414yes—
465C23C22×C22⋊C411yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—28no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—84no—
476D4—88no—
477D4—88no—
478D4—84no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#40812yes—
527C2×C4#40812yes—
528C2×C4#41612yes—
529C2×C4#41612yes—
530C2×C4C2×#10412yes—
531C2×C4C2×C4×D411yescharacteristic
532C2×C4#51811yescharacteristic
533C2×C4D4211yescharacteristic
534C2×C4D4211yescharacteristic
535C2×C4#54611yescharacteristic
536C2×C4#54611yescharacteristic
537C2×C4C2×#13111yescharacteristic
538C2×C4C2×#13111yescharacteristic
539C2×C4C2×#13111yescharacteristic
540C2×C4#56811yescharacteristic
541C2×C4C2×#13511yescharacteristic
542C2×C4C2×#13511yescharacteristic
543C2×C4#57611yescharacteristic
544C2×C4#57711yescharacteristic
545C8—24no—
546C8—42no—
547C8—44no—
548C8—42no—
549C8—82no—
550C8—81no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22D4×C22⋊C411yescharacteristic
566C22C2×#40811yescharacteristic
567C22#288011yescharacteristic
568C22C2×#41611yescharacteristic
569C22#289711yescharacteristic
570C22C22×#10411yescharacteristic
571C22#290111yescharacteristic
572C4—22no—
573C4—22no—
574C4—21no—
575C4—21no—
576C4—41no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—82no—
585C4#149414yescomplemented
586C4#293111yescharacteristic
587C4#293511yescharacteristic
588C4#297911yescharacteristic
589C4#298111yescharacteristic
590C4#328411yescharacteristic
591C4#328711yescharacteristic
592C4#330411yescharacteristic
593C4#330511yescharacteristic
594C2—44no—
595C2—162no—
596C2C2×#149411yescharacteristic
597C2#1748311yescharacteristic
598C2#1751011yescharacteristic
599C2#1751711yescharacteristic
600C2#2240211yescharacteristic
601C2#2241811yescharacteristic
602C2D4×#10411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2541C4
LCSG ▷ C22×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. #22799C2semidirect
3. C4×C8×D4C2semidirect
4. C2×#1494C2semidirect
5. #9674C2semidirect
6. #9638C2semidirect
7. #9072C2semidirect
8. #8943C2semidirect
9. #8564C2semidirect
10. #8315C2semidirect
11. C4×Q8⋊C8C2semidirect
152. Q8⋊C8D4semidirect
585. C4#1494semidirect

Maximal subgroups

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

subgroupindex
#349342
#227992
C4×C8×D42
C2×#14942
#96742
#96382
#90722
#89432
#85642
#83152
C4×Q8⋊C82