Finite Group Databasebeta

← #115928

Subgroups — #115928

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21385C211yesmaximal, characteristic
2C2×#1994C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4C4×#389C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#7901C211yesmaximal, characteristic
7C2×#1276C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#389—21no—
13C2×C4×C4⋊C4—21no—
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C43C2211yescharacteristic
17#1994—22no—
18#1994—22no—
19#1987C412yescomplemented
20#1981C412yescomplemented
21#1872—21no—
22#1872—21no—
23C4×#103—22no—
24C2×#339—21no—
25C2×#339—21no—
26#1503—22no—
27#1503—22no—
28C2×#328—22no—
29#1436—22no—
30C4⋊#95—21no—
31C4×#95—21no—
32#1303—22no—
33#1279—22no—
34#1276—22no—
35C22×M16—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533D412yes—
43#533Q812yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42C2311yescharacteristic
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431D412yes—
71#423—21no—
72#423—21no—
73#423—21no—
74#423D412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2×C412yes—
79C2×#103—22no—
80C2×#103—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#389—28no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—42no—
93C4×C4⋊C4—42no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4D412yes—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D412yes—
101C43—21no—
102C43—21no—
103C43—21no—
104C43—21no—
105#340—22no—
106#340—22no—
107#339—24no—
108#339—24no—
109#328—28no—
110C22×Q8—21no—
111C22×Q8—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C22×C8—22no—
127C22×C8—41no—
128C22×C8—41no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8C4⋊C412yescomplemented
136C4⋊Q8C4⋊C412yescomplemented
137#125—22no—
138#125—22no—
139#125—41no—
140#125—41no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157#117—22no—
158#117—22no—
159#117—41no—
160#117—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C22⋊C4—41no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42SD1612yes—
213C2×C42Q1612yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q811yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—41no—
232C4⋊C8—41no—
233C4⋊C8—41no—
234#103—44no—
235#103—44no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C4×C8—22no—
241C4×C8—41no—
242C4×C8—41no—
243C4×C8—41no—
244#95—28no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—44no—
250#95—44no—
251C24#12011yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—81no—
259C2×Q8—81no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4C22⋊Q811yescharacteristic
308C22×C4C22⋊Q811yescharacteristic
309C22×C4C22⋊Q811yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12411yescharacteristic
312C22×C4#12411yescharacteristic
313C22×C4#12411yescharacteristic
314C22×C4#12711yescharacteristic
315C22×C4C2×SD1611yescharacteristic
316C22×C4C2×Q1611yescharacteristic
317C22×C4#13511yescharacteristic
318C22×C4#13511yescharacteristic
319C22×C4AGL(1,ℤ/8)11yescharacteristic
320C22×C4#13711yescharacteristic
321C22×C4#13711yescharacteristic
322C22×C4#13711yescharacteristic
323M16—42no—
324M16—42no—
325M16—81no—
326M16—81no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—81no—
333C2×C8—81no—
334C2×C8—81no—
335C2×C8—81no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—416no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4#10414yes—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—44no—
364C42—44no—
365C42—44no—
366C42—44no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—42no—
375C42—41no—
376C42—41no—
377C42—81no—
378C42—81no—
379C42—81no—
380C42—81no—
381C42C2×C4⋊C411yescharacteristic
382C42#11711yescharacteristic
383C42C4×D411yescharacteristic
384C42C4×D411yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42C22⋊Q811yescharacteristic
389C42C22⋊Q811yescharacteristic
390C42C22⋊Q811yescharacteristic
391C42#12311yescharacteristic
392C42#12311yescharacteristic
393C42#12311yescharacteristic
394C42#12311yescharacteristic
395C42#12611yescharacteristic
396C42#12611yescharacteristic
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23#38611yescharacteristic
402C23#39211yescharacteristic
403C23#39311yescharacteristic
404C23#45011yescharacteristic
405C23#45111yescharacteristic
406C23#45211yescharacteristic
407C23#45211yescharacteristic
408Q8—44no—
409Q8—82no—
410Q8—82no—
411Q8—82no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4#38011yescharacteristic
464C2×C4#38511yescharacteristic
465C2×C4#38511yescharacteristic
466C2×C4#38811yescharacteristic
467C2×C4#38811yescharacteristic
468C2×C4#39011yescharacteristic
469C2×C4#39411yescharacteristic
470C2×C4#39411yescharacteristic
471C2×C4#39611yescharacteristic
472C2×C4#39611yescharacteristic
473C2×C4#39711yescharacteristic
474C2×C4#39911yescharacteristic
475C2×C4C2×#10412yes—
476C2×C4#42112yes—
477C2×C4C4×SD1611yescharacteristic
478C2×C4C4×Q1611yescharacteristic
479C2×C4#44011yescharacteristic
480C2×C4#44111yescharacteristic
481C2×C4#44811yescharacteristic
482C2×C4#44911yescharacteristic
483C2×C4#45111yescharacteristic
484C2×C4#45211yescharacteristic
485C2×C4#46111yescharacteristic
486C2×C4#46211yescharacteristic
487C2×C4#46311yescharacteristic
488C2×C4#46411yescharacteristic
489C2×C4#46511yescharacteristic
490C2×C4#46711yescharacteristic
491C2×C4#46811yescharacteristic
492C2×C4#47011yescharacteristic
493C2×C4#47511yescharacteristic
494C2×C4#47711yescharacteristic
495C2×C4#47911yescharacteristic
496C2×C4#47911yescharacteristic
497C2×C4#48711yescharacteristic
498C2×C4#48811yescharacteristic
499C2×C4#48911yescharacteristic
500C2×C4#49111yescharacteristic
501C8—82no—
502C8—82no—
503C8—82no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22#143911yescharacteristic
517C22#187711yescharacteristic
518C22#187811yescharacteristic
519C22#189611yescharacteristic
520C22#189711yescharacteristic
521C22#190611yescharacteristic
522C22#190611yescharacteristic
523C22#200011yescharacteristic
524C22#200111yescharacteristic
525C22#202411yescharacteristic
526C22#202611yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—44no—
534C4—44no—
535C4—44no—
536C4—84no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4#180612yes—
541C4#186311yescharacteristic
542C4#186511yescharacteristic
543C4#193311yescharacteristic
544C4#193511yescharacteristic
545C4#198012yes—
546C4#204211yescharacteristic
547C4#204311yescharacteristic
548C4#206611yescharacteristic
549C4#206811yescharacteristic
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2#923111yescharacteristic
555C2#923211yescharacteristic
556C2#923511yescharacteristic
557C2#923611yescharacteristic
558C2#1106011yescharacteristic
559C2#1106711yescharacteristic
560C2#1107511yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1987C4semidirect
20. #1981C4semidirect
135. C4⋊Q8C4⋊C4semidirect
136. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#213852
C2×#19942
C2×#15032
C4×#3892
C2×#13032
#79012
C2×#12762