Finite Group Databasebeta

← #219813

Subgroups — #219813

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#550C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3#21264C211yesmaximal, characteristic, complemented
4C2×#1488C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic, complemented
7#9076C212yesmaximal, complemented
8#8943C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8312C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic
13C2×C4×#41C2212yes—
14C2×#432C2212yes—
15C2×#431C2212yes—
16#2366—21no—
17#2366C411yescharacteristic
18#2363—22no—
19#2363C412yes—
20#2354—21no—
21#2354C411yescharacteristic
22Q8×C4⋊C4C411yescharacteristic
23#2341C412yes—
24C4×#127—21no—
25C4×#127C2211yescharacteristic
26C4×#121—22no—
27C4×#121C2212yes—
28C42×Q8C2211yescharacteristic
29C42×D4—21no—
30C42×D4C2211yescharacteristic
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1924—22no—
35#1923—22no—
36#1842C2211yescharacteristic
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1488—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54C22×#41C2×C412yes—
55#550—24no—
56#550—24no—
57C2×#127—21no—
58C2×#127C2×C411yescharacteristic
59C2×#121—22no—
60C2×#121C2×C412yes—
61C4×#41—24no—
62C4×#41—24no—
63C4×#41—42no—
64C4×#41—42no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4C2312yes—
81C2×#117—22no—
82C2×#117C2×C412yes—
83C22×C42—22no—
84C22×C42C2312yes—
85#432—24no—
86#432—42no—
87#431—24no—
88#431—42no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2311yescharacteristic
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×#97—22no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#390—21no—
108#390—21no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390—21no—
113#390—21no—
114#390C2×C412yes—
115#390C2×C412yes—
116#389—22no—
117#389C2×C412yes—
118#388—22no—
119#388—22no—
120#388—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#384—21no—
125#384C2×C411yescharacteristic
126#380—22no—
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
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—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4C2×C412yes—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4D412yes—
145C43—21no—
146C43C2311yescharacteristic
147#336—22no—
148#336—21no—
149#336—22no—
150#336—21no—
151Q8⋊C8—24no—
152Q8⋊C8—22no—
153Q8⋊C8D414yescomplemented
154D4⋊C8—24no—
155D4⋊C8—24no—
156D4⋊C8—42no—
157D4⋊C8—42no—
158C2×#41—24no—
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—42no—
163C2×#41—42no—
164C22×Q8C22×C411yescharacteristic
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22⋊C411yescharacteristic
170C22×D4C22×C412yes—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4C22×C412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—22no—
183#127—28no—
184#127—22no—
185#127M1614yes—
186#121—216no—
187#121—24no—
188#121—24no—
189C4×Q8—22no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—24no—
193C4×Q8—22no—
194C4×Q8—41no—
195C4×Q8—42no—
196C4×Q8M1614yes—
197C4×Q8C2×D412yes—
198C4×Q8#4112yes—
199C4×D4—28no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—81no—
219C4×D4—81no—
220C4×D4—82no—
221C4×D4—82no—
222#117—24no—
223#117—44no—
224#117—44no—
225#117—42no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—22no—
249C2×C22⋊C4C22⋊C412yes—
250C2×C22⋊C4C22⋊C412yes—
251C2×C22⋊C4C22⋊C412yes—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42C22×C412yes—
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2411yescharacteristic
276C4⋊C8—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—22no—
281C4⋊C8—44no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284C4⋊C8—81no—
285C4⋊C8C2×D412yes—
286C22⋊C8—44no—
287C22⋊C8—44no—
288C22⋊C8—44no—
289C22⋊C8—82no—
290#97—44no—
291C4×C8—22no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8—42no—
297C4×C8—42no—
298C4×C8—42no—
299C4×C8C2×D412yes—
300#95—22no—
301#95—22no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95C22⋊C412yes—
306#95C22⋊C412yes—
307#95C22⋊C412yes—
308C24—22no—
309C24C2×C22⋊C412yes—
310#41—48no—
311#41—48no—
312#41—48no—
313#41—48no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8C4×D412yes—
318C2×Q8C2×M1612yes—
319C2×Q8#13112yes—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4C2×M1614yes—
341C22×C4—21no—
342C22×C4—28no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—82no—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—42no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—81no—
418C4⋊C4—82no—
419C4⋊C4C4×D412yes—
420C4⋊C4C2×M1612yes—
421C22⋊C4—28no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—24no—
447C42—22no—
448C42—41no—
449C42—44no—
450C42—44no—
451C42—42no—
452C42—41no—
453C42—41no—
454C42—42no—
455C42—81no—
456C42C2×M1612yes—
457C42C2×M1612yes—
458C42#13512yes—
459C42#13512yes—
460C42C22×D411yescharacteristic
461C42C2×#4111yescharacteristic
462C42H(5,𝔽2)11yescharacteristic
463C42H(5,𝔽2)11yescharacteristic
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23C22×C22⋊C411yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—22no—
482Q8#43614yes—
483D4—24no—
484D4—24no—
485D4—416no—
486D4—42no—
487D4—42no—
488D4—44no—
489D4—48no—
490D4—84no—
491D4—88no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—416no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—88no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4#40712yes—
542C2×C4#40712yes—
543C2×C4#41612yes—
544C2×C4#41612yes—
545C2×C4#42112yes—
546C2×C4C2×C4×D411yescharacteristic
547C2×C4#51811yescharacteristic
548C2×C4D4211yescharacteristic
549C2×C4D4211yescharacteristic
550C2×C4#54611yescharacteristic
551C2×C4#54611yescharacteristic
552C2×C4C22×M1611yescharacteristic
553C2×C4C22×M1611yescharacteristic
554C2×C4C2×#13111yescharacteristic
555C2×C4#56811yescharacteristic
556C2×C4C2×#13511yescharacteristic
557C2×C4C2×#13511yescharacteristic
558C2×C4#57611yescharacteristic
559C2×C4#57711yescharacteristic
560C8—42no—
561C8—42no—
562C8—42no—
563C8—82no—
564C8—82no—
565C8—81no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22D4×C22⋊C411yescharacteristic
581C22C2×#40711yescharacteristic
582C22#288011yescharacteristic
583C22C2×#41611yescharacteristic
584C22#289711yescharacteristic
585C22C2×#42111yescharacteristic
586C22#290111yescharacteristic
587C4—22no—
588C4—22no—
589C4—24no—
590C4—21no—
591C4—21no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—81no—
599C4—82no—
600C4#148814yescomplemented
601C4C2×#43611yescharacteristic
602C4D4×M1611yescharacteristic
603C4#297411yescharacteristic
604C4#297511yescharacteristic
605C4#328411yescharacteristic
606C4#328711yescharacteristic
607C4#330411yescharacteristic
608C4#330511yescharacteristic
609C2—82no—
610C2—84no—
611C2C2×#148811yescharacteristic
612C2#1745311yescharacteristic
613C2#1751011yescharacteristic
614C2#1752611yescharacteristic
615C2#2240111yescharacteristic
616C2#2241811yescharacteristic
617C2#2243511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#550C4
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. #21264C2semidirect
4. C2×#1488C2semidirect
5. #9674C2semidirect
6. #9638C2semidirect
7. #9076C2semidirect
8. #8943C2semidirect
9. #8564C2semidirect
10. #8312C2semidirect
11. C4×Q8⋊C8C2semidirect
153. Q8⋊C8D4semidirect
600. C4#1488semidirect

Maximal subgroups

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

subgroupindex
C4×#5502
#227992
#212642
C2×#14882
#96742
#96382
#90762
#89432
#85642
#83122
C4×Q8⋊C82