Finite Group Databasebeta

← #222956

Subgroups — #222956

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic
2#22399C212yesmaximal
3#9883C212yesmaximal, complemented
4#9085C212yesmaximal, complemented
5#9081C212yesmaximal, complemented
6#8704C212yesmaximal, complemented
7#8560C211yesmaximal, characteristic, complemented
8#8153C211yesmaximal, characteristic, complemented
9#7915C212yesmaximal, complemented
10C22×#125C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#435—22no—
13C2×#407C2212yes—
14C22×C22⋊C8C2212yes—
15#2454—22no—
16#2454—22no—
17#2454—22no—
18#2454—22no—
19C2×#399C412yes—
20C2×#396C412yes—
21C2×#390C411yescharacteristic
22C2×#390C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#385C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1955—22no—
29#1870—22no—
30#1869—22no—
31#1797C2212yes—
32#1779C2212yes—
33#1779C2212yes—
34#1749C2212yes—
35C2×#324C2212yes—
36C2×#324C2211yescharacteristic
37C2×#324C2211yescharacteristic
38#1280C2211yescharacteristic
39#1280C2212yes—
40#1280C2212yes—
41#1280C2212yes—
42#1280C2212yes—
43#1280C2212yes—
44#1280C2211yescharacteristic
45C23×D4C2×C411yescharacteristic
46C24×C4C2312yes—
47C22×M16—22no—
48C23×C8—22no—
49C2×#125—22no—
50C2×#125—21no—
51C2×#125—21no—
52C2×#125—44no—
53C2×C4×D4—44no—
54C2×C4×D4D412yes—
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4D412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2311yescharacteristic
75#435—48no—
76C2×C4⋊C8—22no—
77#407—24no—
78#407—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×#97—22no—
92#399—22no—
93#399—22no—
94#399—22no—
95#399—22no—
96#396—22no—
97#396—22no—
98#396—22no—
99#396—22no—
100#390—22no—
101#390—21no—
102#390—21no—
103#390—22no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—22no—
108#389—21no—
109#389—22no—
110#389—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C412yes—
123C2×#95C2311yescharacteristic
124C2×#95C2312yes—
125#336—22no—
126#336—22no—
127#324—24no—
128#324—41no—
129#324—41no—
130C25C22×C412yes—
131C22×D4—24no—
132C22×D4—42no—
133C22×D4—41no—
134C22×D4—41no—
135C22×D4C22⋊C412yes—
136C22×D4C22⋊C411yescharacteristic
137C22×D4C22⋊C411yescharacteristic
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—44no—
143C23×C4—82no—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2411yescharacteristic
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—82no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8—82no—
168#125—48no—
169#125—44no—
170#125—44no—
171C4×D4—28no—
172C4×D4—44no—
173C4×D4—44no—
174C4×D4—44no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—42no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—42no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C4⋊C8—44no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—44no—
244C22⋊C8—44no—
245C22⋊C8—82no—
246C22⋊C8—82no—
247C22⋊C8#4114yes—
248#97—44no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255C24—24no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24C2×C22⋊C412yes—
266C24C2×C22⋊C412yes—
267C24C23×C411yescharacteristic
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4#10114yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4#13112yes—
350C22×C4C22×D412yes—
351C22×C4C2×#4112yes—
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C42−1+411yescharacteristic
354M16—88no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—88no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4#9914yes—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C42—24no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23C2×#10114yes—
438C23C22×C22⋊C411yescharacteristic
439C23C4×#4111yescharacteristic
440C23#52011yescharacteristic
441C23C2×#13111yescharacteristic
442C23#56811yescharacteristic
443C23#56811yescharacteristic
444C23#56811yescharacteristic
445D4—44no—
446D4—48no—
447D4—44no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4C2×#9912yes—
512C2×C4C2×#9914yes—
513C2×C4C2×#10112yes—
514C2×C4#54612yes—
515C2×C4#54712yes—
516C8—44no—
517C8—82no—
518C8—82no—
519C8—82no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22#233911yescharacteristic
551C22#287911yescharacteristic
552C22#288011yescharacteristic
553C22C22×#9911yescharacteristic
554C22#288311yescharacteristic
555C22C22×#10111yescharacteristic
556C22#288811yescharacteristic
557C22#296711yescharacteristic
558C22#296811yescharacteristic
559C22#299211yescharacteristic
560C22#299511yescharacteristic
561C4—22no—
562C4—22no—
563C4—24no—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—82no—
571C4—82no—
572C4—81no—
573C4—81no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—44no—
579C2—82no—
580C2#1739711yescharacteristic
581C2#1740811yescharacteristic
582C2#1741911yescharacteristic
583C2#1742611yescharacteristic
584C2#2247511yescharacteristic
585C2#2247611yescharacteristic
586C2#2248111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9883C2semidirect
4. #9085C2semidirect
5. #9081C2semidirect
6. #8704C2semidirect
7. #8560C2semidirect
8. #8153C2semidirect
9. #7915C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
#223992
#98832
#90852
#90812
#87042
#85602
#81532
#79152