Finite Group Databasebeta

← #188411

Subgroups — #188411

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34129C211yesmaximal, characteristic
2#21974C211yesmaximal, characteristic, complemented
3C4×#424C211yesmaximal, characteristic, complemented
4C2×#1489C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8566C212yesmaximal, complemented
9#8564C212yesmaximal, complemented
10#7897C211yesmaximal, characteristic, complemented
11#7896C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#424—21no—
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2354—21no—
19#2354—21no—
20#2354C411yescharacteristic
21#2354C411yescharacteristic
22Q8×C4⋊C4C411yescharacteristic
23Q8×C4⋊C4C411yescharacteristic
24#2346—22no—
25#2346C412yes—
26#2341C412yes—
27#2293—22no—
28#2293C2212yes—
29#2285—21no—
30#2285C2211yescharacteristic
31C42×Q8C2212yes—
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34#1939—21no—
35#1939—21no—
36#1851—22no—
37#1842C2212yes—
38#1831C2212yes—
39#1768—22no—
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2212yes—
42#1729—21no—
43#1489—24no—
44#1476—24no—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×#41C2×C411yescharacteristic
56C22×M16—21no—
57C22×M16—21no—
58C23×C8—21no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41D414yes—
62C4×#41Q814yes—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—22no—
76C2×#117C2×C411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#424—28no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C22⋊C8—21no—
95C4×M16—22no—
96C4×M16—22no—
97C2×#97—21no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2312yes—
105#390—22no—
106#390—22no—
107#390C2×C412yes—
108#390C2×C412yes—
109#389—22no—
110#389—22no—
111#389C2×C412yes—
112#389C2×C412yes—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#384—21no—
120#384—21no—
121#384C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#382—21no—
124#382—22no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—22no—
129#380C2×C412yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4C2×C412yes—
137C4×C4⋊C4C2×C412yes—
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4C2312yes—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4C2×C412yes—
146C4×C22⋊C4D412yes—
147#376—22no—
148#376D412yes—
149C43—22no—
150C43C2312yes—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—22no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157Q8⋊C8—24no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—24no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—41no—
165C2×#41—41no—
166C2×#41C4⋊C414yes—
167C22×Q8C22×C411yescharacteristic
168C22×D4—21no—
169C22×D4C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—24no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16—81no—
181C2×M16—81no—
182C22×C8—24no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—42no—
191C4×Q8—22no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8C4⋊C414yes—
199C4×Q8C2×D412yes—
200C4×Q8C2×Q812yes—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4C4⋊C414yes—
209C4×D4C2×D412yes—
210C4×D4C2×Q812yes—
211#117—22no—
212#117—24no—
213#117—24no—
214#117—24no—
215#117—42no—
216#117—44no—
217#117—44no—
218#117—82no—
219#117—82no—
220#117—81no—
221#117—81no—
222#117C4⋊C414yes—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
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×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C22×C412yes—
262C2×C42C22×C412yes—
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×Q812yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274C2×C42C2411yescharacteristic
275C4⋊C8—28no—
276C4⋊C8—28no—
277C4⋊C8—22no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—42no—
284C4⋊C8—82no—
285C4⋊C8—82no—
286C22⋊C8—42no—
287#97—42no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—22no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—22no—
294C4×C8—42no—
295C4×C8—42no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95C22⋊C412yes—
303#95C22⋊C412yes—
304#95C22⋊C412yes—
305#95C22⋊C412yes—
306C24—21no—
307C24C2×C22⋊C411yescharacteristic
308#41—28no—
309#41—44no—
310#41—44no—
311#41—44no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8C2×C4⋊C412yes—
316C2×Q8#13112yes—
317C2×Q8#13112yes—
318C2×D4—24no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4C2×C4⋊C412yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C4⋊C412yes—
362C22×C4#11712yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4C22⋊Q812yes—
366C22×C4C22⋊Q812yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370M16—48no—
371M16—84no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—48no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—41no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—81no—
408C4⋊C4—81no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4#10414yes—
412C4⋊C4C2×C4⋊C412yes—
413C4⋊C4C2×C4⋊C412yes—
414C22⋊C4—24no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—42no—
439C42—44no—
440C42—44no—
441C42—44no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42C2×C4⋊C412yes—
446C42#11712yes—
447C42#12112yes—
448C42#12112yes—
449C42C22⋊Q812yes—
450C42C22⋊Q812yes—
451C42C22×D411yescharacteristic
452C42C22×Q811yescharacteristic
453C42C2×#4111yescharacteristic
454C42C2×#4111yescharacteristic
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23#38012yes—
469C23C22×C22⋊C411yescharacteristic
470Q8—22no—
471Q8—22no—
472Q8—22no—
473Q8#42414yescomplemented
474D4—22no—
475D4—22no—
476D4—48no—
477D4—41no—
478D4—41no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4#38012yes—
528C2×C4#38012yes—
529C2×C4#38012yes—
530C2×C4C2×#10414yes—
531C2×C4C2×#10412yes—
532C2×C4#42112yes—
533C2×C4C22×C4⋊C411yescharacteristic
534C2×C4C2×#11711yescharacteristic
535C2×C4C2×#12111yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×#13111yescharacteristic
540C2×C4C2×#13111yescharacteristic
541C2×C4#56811yescharacteristic
542C2×C4#56811yescharacteristic
543C2×C4#57611yescharacteristic
544C2×C4#57711yescharacteristic
545C2×C4#57711yescharacteristic
546C2×C4#57811yescharacteristic
547C8—42no—
548C8—42no—
549C8—44no—
550C8—81no—
551C8—82no—
552C8—81no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C2×#38011yescharacteristic
568C22#288011yescharacteristic
569C22#288011yescharacteristic
570C22#289711yescharacteristic
571C22#289811yescharacteristic
572C22C22×#10411yescharacteristic
573C22C2×#42111yescharacteristic
574C4—24no—
575C4—22no—
576C4—24no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—82no—
586C4—82no—
587C4#180614yes—
588C4C2×#42411yescharacteristic
589C4#290511yescharacteristic
590C4#292411yescharacteristic
591C4#292411yescharacteristic
592C4#304511yescharacteristic
593C4#304611yescharacteristic
594C4#308311yescharacteristic
595C4#308411yescharacteristic
596C2—42no—
597C2—82no—
598C2#1751911yescharacteristic
599C2#1752011yescharacteristic
600C2#1752511yescharacteristic
601C2#1752611yescharacteristic
602C2#2062811yescharacteristic
603C2#2065211yescharacteristic
604C2C2×#180611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21974C2semidirect
3. C4×#424C2semidirect
6. #8569C2semidirect
7. #8567C2semidirect
8. #8566C2semidirect
9. #8564C2semidirect
10. #7897C2semidirect
11. #7896C2semidirect
473. Q8#424semidirect

Maximal subgroups

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

subgroupindex
#341292
#219742
C4×#4242
C2×#14892
C2×#14762
#85692
#85672
#85662
#85642
#78972
#78962