Finite Group Databasebeta

← #115265

Subgroups — #115265

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2#21765C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1821C211yesmaximal, characteristic
5C2×#1472C211yesmaximal, characteristic
6C2×#1298C211yesmaximal, characteristic
7#7915C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#483—22no—
11C2×#483—22no—
12C2×#483—22no—
13C2×#483—22no—
14C2×#483—22no—
15C2×#483C414yescomplemented
16C22×#106—21no—
17C22×#103—21no—
18C2×#407C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1894—22no—
24#1872—22no—
25#1824—22no—
26#1824—22no—
27#1823—21no—
28#1821—22no—
29#1821—22no—
30#1787—21no—
31C2×#343—22no—
32C2×#340—22no—
33#1472—22no—
34#1472—22no—
35C2×#324—22no—
36#1298—22no—
37#1280—22no—
38C23×Q8D411yescharacteristic
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C22×M16—21no—
42C22×M16D412yes—
43C23×C8—21no—
44C23×C8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8D412yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76#483—28no—
77#483—28no—
78#483—28no—
79#483—28no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87#407—21no—
88#407—21no—
89#407—21no—
90#407—21no—
91#407—21no—
92#407—21no—
93#407—21no—
94#407D412yes—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8C2×C412yes—
108C2×C22⋊C8D412yes—
109C2×C22⋊C8D412yes—
110C2×C22⋊C8D412yes—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115C2×#95—21no—
116#343—28no—
117#340—24no—
118#340—28no—
119#324—28no—
120C25#4111yescharacteristic
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—44no—
127C22×Q8C22⋊C412yes—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—44no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—84no—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—41no—
158C2×M16—44no—
159C2×M16—44no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—82no—
238#106—216no—
239#103—216no—
240#103—84no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253C22⋊C8—41no—
254C22⋊C8—41no—
255C22⋊C8—41no—
256C22⋊C8C22⋊C414yes—
257#95—41no—
258#95—41no—
259C24—22no—
260C24—24no—
261C24—22no—
262C24—42no—
263C24C4×D411yescharacteristic
264C24#12111yescharacteristic
265C24#12111yescharacteristic
266C24#12111yescharacteristic
267C24#12311yescharacteristic
268C24#12311yescharacteristic
269C24#12411yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—44no—
278C2×Q8—82no—
279C2×Q8—82no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12311yescharacteristic
345C22×C4#12311yescharacteristic
346C22×C4#12411yescharacteristic
347C22×C4#12711yescharacteristic
348C22×C4#12711yescharacteristic
349M16—44no—
350M16—44no—
351M16—82no—
352M16—82no—
353C2×C8—24no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—44no—
362C2×C8—82no—
363C2×C8—82no—
364C2×C8—82no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C23—21no—
390C23—22no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—21no—
401C23—22no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—48no—
411C23—48no—
412C23#38611yescharacteristic
413C23#38611yescharacteristic
414C23#39011yescharacteristic
415C23#39211yescharacteristic
416C23#39411yescharacteristic
417C23#39411yescharacteristic
418C23#39411yescharacteristic
419C23#47112yes—
420C23#47114yes—
421C23#47312yes—
422Q8—44no—
423Q8—44no—
424Q8—82no—
425Q8—82no—
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—22no—
440C2×C4—22no—
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—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—84no—
463C2×C4—81no—
464C2×C4—84no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—164no—
474C2×C4#37911yescharacteristic
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#39011yescharacteristic
478C2×C4#39211yescharacteristic
479C2×C4#39211yescharacteristic
480C2×C4#39211yescharacteristic
481C2×C4#39411yescharacteristic
482C2×C4#45814yes—
483C8—44no—
484C8—82no—
485C8—82no—
486C8—82no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—24no—
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—44no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—42no—
510C22—42no—
511C22#143811yescharacteristic
512C22#190812yes—
513C22#197712yes—
514C22#201911yescharacteristic
515C22#201912yes—
516C22#202011yescharacteristic
517C22#202212yes—
518C22#204612yes—
519C22#205212yes—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—82no—
527C4—84no—
528C4—82no—
529C4—162no—
530C4#189012yes—
531C4#202212yes—
532C2—22no—
533C2—22no—
534C2—22no—
535C2—22no—
536C2—41no—
537C2—41no—
538C2#999312yes—
539C2#999512yes—
540C2#1099211yescharacteristic
541C2#1102111yescharacteristic
542C2#1102811yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×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
15. C2×#483C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
#217652
C2×#18242
C2×#18212
C2×#14722
C2×#12982
#79152