Finite Group Databasebeta

← #192760

Subgroups — #192760

Order 512.

1084 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#117C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8895C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic
11#7891C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#103—21no—
18#2355—21no—
19#2355—21no—
20#2355C2211yescharacteristic
21#2355C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24#2345—22no—
25#2345—21no—
26Q8×C22⋊C4—21no—
27#2295—22no—
28#2295—21no—
29#2285—21no—
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C42×Q8—21no—
33C4×#117—21no—
34C4×#117C2211yescharacteristic
35#1948—21no—
36#1948C2212yes—
37#1947—21no—
38#1947—22no—
39#1947C2212yes—
40#1852—22no—
41#1845C2212yes—
42#1834C2212yes—
43#1803—21no—
44#1803—21no—
45#1803C2212yes—
46#1803C2212yes—
47#1736—21no—
48C2×#340—21no—
49#1487C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C23×Q8D411yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8Q814yes—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431C2×C414yes—
83#428—44no—
84C2×#107—22no—
85C2×#107—24no—
86C2×#106—22no—
87#423—22no—
88#423C2×C414yes—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—44no—
99C2×C22⋊C8—21no—
100C2×C4×C8C2312yes—
101#391—22no—
102#391—22no—
103#391—21no—
104#391—21no—
105#389—21no—
106#389—21no—
107#389—22no—
108#389—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384C2312yes—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#382—21no—
125#382—21no—
126#380—22no—
127#380—22no—
128#380—21no—
129#380D412yes—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D412yes—
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150#376—22no—
151#376—21no—
152C43—22no—
153C43—21no—
154#340—22no—
155#340—22no—
156#340—24no—
157#340—22no—
158#340—22no—
159Q8⋊C8C2×C418yescomplemented
160Q8⋊C8C2×C418yescomplemented
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8C4⋊C414yes—
168C22×Q8C2×D411yescharacteristic
169C22×Q8C2×D411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—24no—
177C2×M16—42no—
178C2×M16—42no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—42no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8C4⋊C414yes—
200C4×Q8C4⋊C414yes—
201C4×Q8C22×C412yes—
202C4×Q8C22×C412yes—
203C4×Q8C2×Q812yes—
204C4×Q8C2×Q812yes—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—42no—
209#117—42no—
210#117Q1614yes—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C4⋊C414yes—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C42—24no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—24no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42C22×C412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×Q812yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274C2×C42C2411yescharacteristic
275#107—44no—
276#107—84no—
277#106—44no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—42no—
284C4⋊C8C22×C414yes—
285#103—44no—
286#103—44no—
287C22⋊C8—24no—
288C4×C8—22no—
289C4×C8C22×C414yes—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95—22no—
301#95—22no—
302#95—21no—
303#95—21no—
304C24#12011yescharacteristic
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—24no—
310C2×Q8—28no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8C2×C4⋊C412yes—
321C2×Q8C2×C4⋊C412yes—
322C2×Q8AGL(1,ℤ/8)12yes—
323C2×Q8#13712yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4C2×C4⋊C412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4#12312yes—
359C22×C4C2×Q1612yes—
360C22×C4#13512yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366M16—48no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—84no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4C2×C4⋊C412yes—
415C4⋊C4C2×C4⋊C412yes—
416C4⋊C4C2×Q1612yes—
417C4⋊C4C2×Q1612yes—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—44no—
429C42—44no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—42no—
435C42—44no—
436C42—81no—
437C42—81no—
438C42—82no—
439C42—82no—
440C42C2×C4⋊C412yes—
441C42C4×D412yes—
442C42C4×D412yes—
443C42C22⋊Q812yes—
444C42C22⋊Q812yes—
445C42#12312yes—
446C42C23×C411yescharacteristic
447C42C22×Q811yescharacteristic
448C42C2×#4111yescharacteristic
449C42C2×#4111yescharacteristic
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23#38512yes—
455C23#44912yes—
456C23#45112yes—
457C23C2×#12011yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—22no—
463Q8—48no—
464Q8—41no—
465Q8—41no—
466Q8#42814yescomplemented
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—164no—
515C2×C4#38512yes—
516C2×C4#38512yes—
517C2×C4#38512yes—
518C2×C4C4×Q1614yes—
519C2×C4#44014yes—
520C2×C4#44912yes—
521C2×C4#45112yes—
522C2×C4#47714yes—
523C2×C4#47914yes—
524C2×C4C22×C4⋊C411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C2×C4C22×Q1611yescharacteristic
531C2×C4C2×#13511yescharacteristic
532C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
533C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
534C2×C4C2×#13711yescharacteristic
535C2×C4C2×#13711yescharacteristic
536C2×C4#57511yescharacteristic
537C2×C4#57511yescharacteristic
538C8—44no—
539C8—44no—
540C8—82no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22#186314yes—
548C22C2×#38511yescharacteristic
549C22C2×#45111yescharacteristic
550C22C2×#44911yescharacteristic
551C22#300611yescharacteristic
552C22#300711yescharacteristic
553C22#302111yescharacteristic
554C22#302211yescharacteristic
555C4—22no—
556C4—24no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—82no—
567C4—81no—
568C4—82no—
569C4—162no—
570C4#186314yes—
571C4C2×#42811yescharacteristic
572C4#291111yescharacteristic
573C4C2×C4×Q1611yescharacteristic
574C4C2×#44011yescharacteristic
575C4C2×#47711yescharacteristic
576C4C2×#47911yescharacteristic
577C4#309411yescharacteristic
578C4#309511yescharacteristic
579C2—24no—
580C2#1878911yescharacteristic
581C2#1879011yescharacteristic
582C2#1880111yescharacteristic
583C2#1880211yescharacteristic
584C2#2084611yescharacteristic
585C2#2086511yescharacteristic
586C2C2×#186311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8896C2semidirect
7. #8895C2semidirect
8. #8884C2semidirect
9. #8883C2semidirect
49. #1487C4semidirect
159. Q8⋊C8C2×C4semidirect
160. Q8⋊C8C2×C4semidirect
466. Q8#428semidirect

Maximal subgroups

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

subgroupindex
Q8×#1172
#219822
#211712
C2×#18032
C2×#14872
#88962
#88952
#88842
#88832
#80452
#78912