Finite Group Databasebeta

← #417112

Subgroups — #417112

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3C2×#2562C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic, complemented
5C2×#1987C211yesmaximal, characteristic, complemented
6C2×#1955C211yesmaximal, characteristic
7#11096C214yesmaximal, complemented
8#11095C214yesmaximal, complemented
9#10703C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C22×#124—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2214yes—
21C22×#97C2211yescharacteristic
22#2562—24no—
23#2562—24no—
24#2547—24no—
25#2547—24no—
26C2×#400—21no—
27C2×#397—22no—
28C2×#394—21no—
29C2×#391—21no—
30C2×#388—22no—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1989—24no—
37#1987—24no—
38#1955—24no—
39C2×#340C2214yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2214yes—
43C23×Q8—21no—
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125D414yes—
60C2×#124—28no—
61C2×#124—48no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#501—232no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C22⋊C8—28no—
95C2×C22⋊C8—28no—
96C2×C22⋊C8—28no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97D414yes—
100#400—28no—
101#397—216no—
102#394—28no—
103#391—28no—
104#388—216no—
105#380—24no—
106#380—24no—
107#380—24no—
108#380—24no—
109#380—28no—
110#380—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—24no—
118C2×#95—22no—
119C2×#95—22no—
120#340—28no—
121#340—28no—
122#340—24no—
123#340—24no—
124C25C2×D412yes—
125C22×Q8—24no—
126C22×Q8—44no—
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—42no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8C2×D414yes—
163C22×C8#4118yes—
164C4⋊Q8—28no—
165C4⋊Q8—28no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168#125—28no—
169#125—28no—
170#125—44no—
171#125—44no—
172#124—432no—
173C4×D4—44no—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C22⋊C4—216no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4—84no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—44no—
229C2×C42—82no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42#4112yes—
236#107—216no—
237#106—216no—
238C4⋊C8—44no—
239C22⋊C8—416no—
240#97—28no—
241#95—28no—
242#95—28no—
243#95—216no—
244#95—24no—
245#95—24no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—24no—
250C24—24no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—24no—
255C24—24no—
256C24AGL(1,ℤ/8)14yes—
257C24C22×D411yescharacteristic
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×D4—28no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—44no—
267C2×D4—44no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4#12012yes—
321C22×C4#12012yes—
322C22×C4#12012yes—
323C22×C4#12012yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4114yes—
333C22×C42−1+414yes—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—28no—
337C2×C8—28no—
338C2×C8—44no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—44no—
343C2×C8#12418yescomplemented
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C22⋊C4—416no—
367C22⋊C4—432no—
368C22⋊C4—48no—
369C22⋊C4—88no—
370C42—48no—
371C42—84no—
372C42#12014yes—
373C42#12114yes—
374C42#12114yes—
375C42#12714yes—
376C23—28no—
377C23—28no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—44no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23C2×#12011yescharacteristic
400C23C2×#12011yescharacteristic
401C23C2×#12111yescharacteristic
402C23#54714yes—
403C23C2×AGL(1,ℤ/8)12yes—
404C23#57512yes—
405C23#57712yes—
406C23#57812yes—
407Q8—816no—
408D4—416no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4#39214yes—
454C2×C4#39214yes—
455C2×C4C2×#12011yescharacteristic
456C2×C4C2×#12111yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×#12412yes—
459C2×C4C2×#12711yescharacteristic
460C2×C4#53212yes—
461C2×C4#53212yes—
462C2×C4#53612yes—
463C2×C4#53612yes—
464C2×C4#53612yes—
465C2×C4#54814yes—
466C2×C4#54814yes—
467C2×C4#55614yes—
468C8—28no—
469C8—82no—
470C22—48no—
471C22—44no—
472C22—48no—
473C22—44no—
474C22—44no—
475C22—84no—
476C22—82no—
477C22—84no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22C2×#39211yescharacteristic
482C22#243212yes—
483C22#243212yes—
484C22#254712yes—
485C22#300311yescharacteristic
486C22#300511yescharacteristic
487C22#300612yes—
488C22#300911yescharacteristic
489C22#301111yescharacteristic
490C22#301312yes—
491C22#305512yes—
492C22#306312yes—
493C22#334314yes—
494C22#334514yes—
495C22#334914yes—
496C22#335014yes—
497C4—24no—
498C4—24no—
499C4—82no—
500C4—84no—
501C4—82no—
502C4—84no—
503C4—84no—
504C4—82no—
505C4#243612yes—
506C4#254512yes—
507C4#254512yes—
508C4#265811yescharacteristic
509C4#285811yescharacteristic
510C2—84no—
511C2#1524311yescharacteristic
512C2#2339111yescharacteristic
513C2#2339411yescharacteristic
514C2#2570912yes—
515C2#2571112yes—
516C2#2571412yes—
517C2#2571612yes—
518C2#2871414yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
4. C2×#1989C2semidirect
5. C2×#1987C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10703C2semidirect
343. C2×C8#124semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#25472
C2×#25622
C2×#19892
C2×#19872
C2×#19552
#110962
#110952
#107032