Finite Group Databasebeta

← #106967

Subgroups — #106967

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2C2×#1826C211yesmaximal, characteristic
3C2×#1825C211yesmaximal, characteristic
4#8019C211yesmaximal, characteristic
5C2×#1293C211yesmaximal, characteristic
6C2×#1282C211yesmaximal, characteristic
7#7915C211yesmaximal, characteristic
8C22×#117C2211yescharacteristic
9#2924—22no—
10#2924—22no—
11#2924—22no—
12#2924C414yescomplemented
13C2×#433—22no—
14C2×#432—21no—
15C2×#431—21no—
16C2×#424—21no—
17C2×#407C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#405—21no—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1826—22no—
25#1826C414yescomplemented
26#1825—21no—
27#1825—21no—
28#1825—21no—
29#1825—21no—
30C2×#340C412yescomplemented
31C2×#340C412yescomplemented
32C2×#339—21no—
33C2×#339—21no—
34#1294—22no—
35#1293—22no—
36#1282—22no—
37#1280—21no—
38#1280—21no—
39C24×C4C2311yescharacteristic
40C22×M16—21no—
41C22×M16—21no—
42C23×C8—21no—
43C23×C8C2×C412yes—
44C2×#117—21no—
45C2×#117—21no—
46C2×#117—21no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117C2×C412yes—
51C2×#117C2×C412yes—
52C2×#117D411yescharacteristic
53C2×#117D411yescharacteristic
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C2×#117D412yes—
57C2×#117Q812yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42—21no—
65C22×C42—21no—
66#433—28no—
67#433—28no—
68#432—24no—
69#432—24no—
70#431—24no—
71#431—24no—
72#424—24no—
73#424—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76#407—21no—
77#407—21no—
78#407—21no—
79#407—21no—
80#407—21no—
81#407C2×C412yes—
82#407C2×C412yes—
83#407D412yes—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2×C412yes—
97C2×C22⋊C8D411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C22⋊C8D412yes—
100C2×C22⋊C8D412yes—
101#405—24no—
102#405—24no—
103C2×#97—22no—
104C2×C4×C8—22no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4C2×C412yes—
113C2×#95—21no—
114C2×#95—21no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#339—24no—
120#339—24no—
121C25#4111yescharacteristic
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—44no—
126C23×C4—41no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×Q811yescharacteristic
138C23×C4#4111yescharacteristic
139C23×C4#4111yescharacteristic
140C23×C4#4111yescharacteristic
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—44no—
145C2×M16—41no—
146C2×M16—44no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—44no—
156C22×C8—41no—
157C22×C8—44no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—21no—
163#117—21no—
164#117—22no—
165#117—21no—
166#117—21no—
167#117—21no—
168#117—21no—
169#117—22no—
170#117—22no—
171#117—21no—
172#117—21no—
173#117—22no—
174#117C22⋊C412yes—
175#117C22⋊C412yes—
176#117C22⋊C412yes—
177#117C22⋊C412yes—
178#117C4⋊C412yes—
179#117C4⋊C412yes—
180#117C4⋊C412yes—
181#117C4⋊C412yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4C4212yes—
192C2×C4⋊C4C4212yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C4⋊C412yes—
198C2×C4⋊C4C4⋊C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—82no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—44no—
224C2×C42—41no—
225C4⋊C8—28no—
226C4⋊C8—28no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241C22⋊C8—41no—
242C22⋊C8C22⋊C412yes—
243C22⋊C8C22⋊C412yes—
244C22⋊C8C22⋊C414yes—
245#97—28no—
246C4×C8—28no—
247#95—22no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—22no—
252C24—22no—
253C24—41no—
254C24—42no—
255C24—41no—
256C24#11711yescharacteristic
257C24C4×D411yescharacteristic
258C24C4×D411yescharacteristic
259C24#12111yescharacteristic
260C24C22⋊Q811yescharacteristic
261C24#12311yescharacteristic
262C24#12311yescharacteristic
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C4211yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4#11711yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C22⋊Q811yescharacteristic
323C22×C4C22⋊Q811yescharacteristic
324C22×C4C22⋊Q811yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12311yescharacteristic
327M16—44no—
328M16—44no—
329M16—44no—
330M16—82no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—44no—
342C2×C8—82no—
343C2×C8—82no—
344C2×C8—82no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4#9512yes—
356C4⋊C4#9512yes—
357C4⋊C4#9512yes—
358C4⋊C4#9512yes—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—24no—
368C22⋊C4—44no—
369C22⋊C4—42no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—44no—
381C42—82no—
382C23—22no—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23C4×C22⋊C411yescharacteristic
412C23#38011yescharacteristic
413C23#38111yescharacteristic
414C23#38511yescharacteristic
415C23#38511yescharacteristic
416C23#38611yescharacteristic
417C23#38611yescharacteristic
418C23#44312yes—
419C23#44412yes—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—162no—
471C2×C4—162no—
472C2×C4C2×#9511yescharacteristic
473C2×C4C4×C22⋊C411yescharacteristic
474C2×C4#37911yescharacteristic
475C2×C4#38011yescharacteristic
476C2×C4#38511yescharacteristic
477C2×C4#38511yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#41014yes—
481C8—42no—
482C8—42no—
483C8—44no—
484C8—82no—
485C8—82no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—82no—
511C22—82no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22C22⋊#9511yescharacteristic
517C22#176411yescharacteristic
518C22#176511yescharacteristic
519C22#186612yes—
520C22#187612yes—
521C22#189912yes—
522C22#190211yescharacteristic
523C22#190311yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—84no—
535C4—162no—
536C4#174012yes—
537C4#189912yes—
538C2—22no—
539C2—21no—
540C2—21no—
541C2—41no—
542C2—41no—
543C2—41no—
544C2—41no—
545C2#924012yes—
546C2#924212yes—
547C2#1073611yescharacteristic
548C2#1078411yescharacteristic
549C2#1079111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2924C4semidirect
25. #1826C4semidirect
30. C2×#340C4semidirect
31. C2×#340C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
C2×#18262
C2×#18252
#80192
C2×#12932
C2×#12822
#79152