Finite Group Databasebeta

← #217101

Subgroups — #217101

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9906C211yesmaximal, characteristic, complemented
5#9901C211yesmaximal, characteristic, complemented
6#9709C211yesmaximal, characteristic, complemented
7#9697C211yesmaximal, characteristic, complemented
8#9138C211yesmaximal, characteristic
9#9137C211yesmaximal, characteristic
10#9136C211yesmaximal, characteristic
11#9135C211yesmaximal, characteristic
12#8594C211yesmaximal, characteristic, complemented
13#8593C211yesmaximal, characteristic, complemented
14#8591C211yesmaximal, characteristic, complemented
15#8590C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2862C2211yescharacteristic
19#2859C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2645—21no—
23#2625—21no—
24#2514C412yes—
25#2509—21no—
26#2508—21no—
27#2495—21no—
28#2494—21no—
29#2489C412yes—
30#2453C412yes—
31#2421C412yes—
32#2323—21no—
33#2320—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1940—21no—
38#1940—21no—
39#1851—21no—
40#1851—21no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1840—21no—
44#1831C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1818—21no—
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1750—21no—
54#1746—21no—
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×#125C2×C412yes—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8C2×C412yes—
77C2×C4×Q8—21no—
78C2×C4×Q8C2×C412yes—
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C22×C42C2311yescharacteristic
82#432—41no—
83#432—41no—
84#423—41no—
85#423—41no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105#401—22no—
106#400—22no—
107#400—22no—
108#400—22no—
109#400—22no—
110#398—22no—
111#397—22no—
112#397—22no—
113#391—21no—
114#391—21no—
115#391—21no—
116#391—21no—
117#391C2×C412yes—
118#391C2×C412yes—
119#391C2×C412yes—
120#391C2×C412yes—
121#389—22no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389C2×C412yes—
133#389C2×C412yes—
134#388—22no—
135#388—22no—
136#387—22no—
137#387—22no—
138#387—22no—
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#383—22no—
142#382—21no—
143#382—21no—
144#382—22no—
145#382C2×C412yes—
146#380—21no—
147#380—21no—
148#380—21no—
149#380C2×C412yes—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2×C412yes—
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4D411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4C2×C411yescharacteristic
160C4×C22⋊C4D411yescharacteristic
161C4×C22⋊C4D411yescharacteristic
162#376D411yescharacteristic
163#376D411yescharacteristic
164C43C2311yescharacteristic
165C43C2311yescharacteristic
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#333—41no—
179#333—41no—
180#333—41no—
181#333—41no—
182#332—41no—
183#332—41no—
184#332—41no—
185#332—41no—
186C22×Q8—21no—
187C22×Q8C22×C412yes—
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C2×M16—42no—
192C2×M16—81no—
193C2×M16—81no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210#125—82no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C22⋊Q8—82no—
214C4×Q8—42no—
215C4×Q8—42no—
216C4×Q8—82no—
217#117—41no—
218#117—41no—
219#117—41no—
220#117—41no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4C22⋊C411yescharacteristic
244C2×C22⋊C4C22⋊C411yescharacteristic
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2411yescharacteristic
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—42no—
286C4⋊C8—81no—
287C4⋊C8—81no—
288C4⋊C8—81no—
289C4⋊C8—81no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293#97—42no—
294#97—42no—
295C4×C8—42no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95C22⋊C411yescharacteristic
311#95C22⋊C411yescharacteristic
312#95C22⋊C411yescharacteristic
313#95C22⋊C411yescharacteristic
314#95C22⋊C411yescharacteristic
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317#95C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363M16—84no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—44no—
421C22⋊C4—82no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—44no—
449C42—82no—
450C42H(5,𝔽2)11yescharacteristic
451C42H(5,𝔽2)11yescharacteristic
452C422−1+411yescharacteristic
453C422−1+411yescharacteristic
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23C22×C22⋊C411yescharacteristic
462Q8—82no—
463Q8—82no—
464Q8—81no—
465Q8—81no—
466Q8—81no—
467Q8—81no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4#52011yescharacteristic
526C2×C4#52011yescharacteristic
527C2×C4#53511yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53711yescharacteristic
530C2×C4#53711yescharacteristic
531C2×C4#57511yescharacteristic
532C2×C4#57511yescharacteristic
533C2×C4#57511yescharacteristic
534C2×C4#57511yescharacteristic
535C2×C4#57711yescharacteristic
536C2×C4#57711yescharacteristic
537C2×C4#57811yescharacteristic
538C2×C4#57811yescharacteristic
539C8—81no—
540C8—81no—
541C8—81no—
542C8—81no—
543C8—81no—
544C8—81no—
545C8—81no—
546C8—81no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22#231111yescharacteristic
555C22#288811yescharacteristic
556C22#288911yescharacteristic
557C22#289611yescharacteristic
558C22#289611yescharacteristic
559C22#289811yescharacteristic
560C22#289811yescharacteristic
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#322211yescharacteristic
579C4#322311yescharacteristic
580C4#324211yescharacteristic
581C4#324211yescharacteristic
582C4#325611yescharacteristic
583C4#325611yescharacteristic
584C4#325611yescharacteristic
585C4#325611yescharacteristic
586C2—81no—
587C2#1776711yescharacteristic
588C2#1776711yescharacteristic
589C2#1776811yescharacteristic
590C2#1776811yescharacteristic
591C2#2187311yescharacteristic
592C2#2188111yescharacteristic
593C2#2189511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9906C2semidirect
5. #9901C2semidirect
6. #9709C2semidirect
7. #9697C2semidirect
12. #8594C2semidirect
13. #8593C2semidirect
14. #8591C2semidirect
15. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#219782
#207862
#99062
#99012
#97092
#96972
#91382
#91372
#91362
#91352
#85942
#85932
#85912
#85902