Finite Group Databasebeta

← #413659

Subgroups — #413659

Order 512.

1136 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2084C211yesmaximal, characteristic
2C2×#2085C212yesmaximal
3C2×#2083C211yesmaximal, characteristic
4C2×#1949C212yesmaximal
5#14687C211yesmaximal, characteristic
6C4×#389C211yesmaximal, characteristic
7#11358C212yesmaximal
8#11357C211yesmaximal, characteristic
9#11357C211yesmaximal, characteristic
10#10850C212yesmaximal
11#10832C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2212yes—
20C2×#382C2212yes—
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#2085—28no—
26#2084—24no—
27#2083—24no—
28#1949—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1436—21no—
39#1436—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—24no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8Q818yes—
79#389—22no—
80#389—22no—
81#389D414yes—
82#389D414yes—
83#384—22no—
84#384—22no—
85#384—22no—
86#384—22no—
87#382—24no—
88#382—24no—
89#382—24no—
90#382—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—44no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C43—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—22no—
115#336—22no—
116#336—24no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C2×Q814yes—
143C22×C8C2×Q814yes—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×Q814yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4114yes—
221C2×C42#4112yes—
222#107—48no—
223#106—48no—
224C4⋊C8—216no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—42no—
229#95—42no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—42no—
237C24C22×D411yescharacteristic
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#12312yes—
315C22×C4#12312yes—
316C22×C4C4⋊Q812yes—
317C22×C4C4⋊Q812yes—
318C22×C4#13512yes—
319C22×C4#13512yes—
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4#13712yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q812yes—
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C42−1+412yes—
331C22×C42−1+412yes—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—48no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4#12714yes—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—44no—
375C42—44no—
376C42—44no—
377C42—44no—
378C42—48no—
379C42—84no—
380C42C22⋊Q818yes—
381C42#12518yes—
382C23C2×#12111yescharacteristic
383C23C2×#12111yescharacteristic
384C23C2×#12111yescharacteristic
385C23C2×#12311yescharacteristic
386C23C2×C4⋊Q811yescharacteristic
387C23#53611yescharacteristic
388C23#53611yescharacteristic
389C23C2×#13511yescharacteristic
390C23C2×#13511yescharacteristic
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23C2×#13711yescharacteristic
393C23#57511yescharacteristic
394C23#57511yescharacteristic
395C23#57511yescharacteristic
396C23#57511yescharacteristic
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#39814yes—
478C2×C4#39814yes—
479C2×C4#46314yes—
480C2×C4#46414yes—
481C2×C4C2×C22⋊Q812yes—
482C2×C4C2×#12512yes—
483C2×C4C2×#12711yescharacteristic
484C2×C4#53211yescharacteristic
485C2×C4#53211yescharacteristic
486C2×C4#53211yescharacteristic
487C2×C4#53612yes—
488C2×C4#54112yes—
489C2×C4#54712yes—
490C2×C4#54712yes—
491C2×C4D4×Q812yes—
492C2×C4D4×Q812yes—
493C2×C4#55212yes—
494C2×C4#55212yes—
495C2×C4#55712yes—
496C2×C4#55712yes—
497C8—44no—
498C8—44no—
499C22C2×#39811yescharacteristic
500C22#248611yescharacteristic
501C22#248611yescharacteristic
502C22#254611yescharacteristic
503C22#254611yescharacteristic
504C22#262111yescharacteristic
505C22#285611yescharacteristic
506C22C2×#46311yescharacteristic
507C22C2×#46411yescharacteristic
508C22#303511yescharacteristic
509C22#303511yescharacteristic
510C22#303511yescharacteristic
511C22#303511yescharacteristic
512C22#303611yescharacteristic
513C22#303611yescharacteristic
514C22#303711yescharacteristic
515C22#303711yescharacteristic
516C22#304011yescharacteristic
517C22#304011yescharacteristic
518C22#309111yescharacteristic
519C22#309111yescharacteristic
520C22#309411yescharacteristic
521C22#309511yescharacteristic
522C22#316011yescharacteristic
523C22#316211yescharacteristic
524C22#316511yescharacteristic
525C22#316611yescharacteristic
526C22#322811yescharacteristic
527C22#323111yescharacteristic
528C22#323211yescharacteristic
529C22#323211yescharacteristic
530C22#323411yescharacteristic
531C22#323611yescharacteristic
532C22#324011yescharacteristic
533C22#324011yescharacteristic
534C4—24no—
535C4—24no—
536C4—24no—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—44no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—162no—
545C4—162no—
546C4#250612yes—
547C4#250612yes—
548C4#254212yes—
549C4#256512yes—
550C2#1515111yescharacteristic
551C2#2365611yescharacteristic
552C2#2365711yescharacteristic
553C2#2765011yescharacteristic
554C2#2765111yescharacteristic
555C2#2765211yescharacteristic
556C2#2765311yescharacteristic
557C2#2868811yescharacteristic
558C2#2869011yescharacteristic
559C2#2869211yescharacteristic
560C2#2869511yescharacteristic
561C2#2987011yescharacteristic
562C2#2987111yescharacteristic
563C2#3316411yescharacteristic
564C2#3316511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20842
C2×#20852
C2×#20832
C2×#19492
#146872
C4×#3892
#113582
#113572
#108502
#108322