Finite Group Databasebeta

← #207215

Subgroups — #207215

Order 512.

800 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
1#35294C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic
3#20550C211yesmaximal, characteristic
4#9907C211yesmaximal, characteristic, complemented
5#9906C211yesmaximal, characteristic, complemented
6#9903C211yesmaximal, characteristic, complemented
7#9902C211yesmaximal, characteristic, complemented
8#9901C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#9899C211yesmaximal, characteristic, complemented
11#9898C211yesmaximal, characteristic, complemented
12#8046C211yesmaximal, characteristic
13#8045C211yesmaximal, characteristic
14#8042C211yesmaximal, characteristic
15#8039C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2629C412yes—
25#2628C412yes—
26#2570—21no—
27#2570—21no—
28#2550—21no—
29#2550—21no—
30#2441—21no—
31#2441—21no—
32#2439—21no—
33#2439—21no—
34#2331C412yes—
35#2330C412yes—
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1940—21no—
40#1940—21no—
41#1850—21no—
42#1850—21no—
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1770—21no—
48C8×C4⋊C4—21no—
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1753—21no—
54#1753—21no—
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8C2×C412yes—
76C2×#125—21no—
77C2×#125—21no—
78C2×#125C2×C412yes—
79C2×#117—21no—
80C2×#117—21no—
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84#432—22no—
85#432—22no—
86#431—22no—
87#431Q814yescomplemented
88#423—22no—
89#423Q814yescomplemented
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#400—22no—
115#399—22no—
116#398—22no—
117#398—22no—
118#396—22no—
119#396—22no—
120#395—22no—
121#393—22no—
122#391—22no—
123#391—22no—
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#389—22no—
133#389—22no—
134#389C2×C412yes—
135#389C2×C412yes—
136#389C2×C412yes—
137#389C2×C412yes—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#384—21no—
143#384—21no—
144#384C2×C412yes—
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382—21no—
150#382—21no—
151#382—21no—
152#382—21no—
153#382C2×C412yes—
154#380—21no—
155#380—21no—
156#380C2×C412yes—
157#380C2×C412yes—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2×C412yes—
161C4×C4⋊C4C2×C412yes—
162C4×C4⋊C4D411yescharacteristic
163C4×C4⋊C4D411yescharacteristic
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C22⋊C4D411yescharacteristic
167C4×C22⋊C4D411yescharacteristic
168C4×C22⋊C4D411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170#376C2×C411yescharacteristic
171#376C2×C411yescharacteristic
172C43C2311yescharacteristic
173C43C2311yescharacteristic
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182C22×Q8—22no—
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—24no—
187C2×M16—81no—
188C2×M16—81no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—81no—
199C22×C8—81no—
200C4⋊Q8—44no—
201C4⋊Q8—81no—
202C4⋊Q8—81no—
203C4⋊Q8—81no—
204C4⋊Q8—81no—
205#125—44no—
206#125—81no—
207#125—81no—
208#125—81no—
209#125—81no—
210#117—42no—
211#117—42no—
212#117—81no—
213#117—81no—
214#117—81no—
215#117—81no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22×C412yes—
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—22no—
248C2×C42—22no—
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×Q812yes—
265C2×C42C2×Q812yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C2×C42C2411yescharacteristic
269C4⋊C8—24no—
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—42no—
283C4⋊C8—81no—
284C4⋊C8—81no—
285C4⋊C8—81no—
286C4⋊C8—81no—
287C4⋊C8C2×Q812yes—
288C4⋊C8C2×Q812yes—
289C4⋊C8C2×Q812yes—
290C4⋊C8C2×Q812yes—
291C22⋊C8—24no—
292C22⋊C8—42no—
293#97—42no—
294#97—81no—
295#97—81no—
296C4×C8—24no—
297C4×C8—81no—
298C4×C8—81no—
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#95C22⋊C411yescharacteristic
310#95C22⋊C411yescharacteristic
311#95C22⋊C411yescharacteristic
312#95C22⋊C411yescharacteristic
313#95C22⋊C411yescharacteristic
314#95C22⋊C411yescharacteristic
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C2×Q8—44no—
319C2×Q8—82no—
320C2×Q8—82no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C4×Q812yes—
347C22×C4C4×Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4C22⋊Q812yes—
350C22×C4#12412yes—
351C22×C4C4⋊Q812yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355M16—84no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
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—84no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
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—44no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C42—21no—
421C42—21no—
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—22no—
437C42—22no—
438C42—41no—
439C42—41no—
440C42—82no—
441C42—82no—
442C42C4×Q812yes—
443C42C4×Q812yes—
444C42C22⋊Q812yes—
445C42C22⋊Q812yes—
446C42#12412yes—
447C42C4⋊Q812yes—
448C42C22×Q811yescharacteristic
449C42C22×Q811yescharacteristic
450C42C2×#4111yescharacteristic
451C42C2×#4111yescharacteristic
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23#39112yes—
459C23C22×C22⋊C411yescharacteristic
460Q8—164no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
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—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4#39112yes—
514C2×C4#39112yes—
515C2×C4#39112yes—
516C2×C4#41312yes—
517C2×C4#41312yes—
518C2×C4C2×C4×Q811yescharacteristic
519C2×C4C2×C4×Q811yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×#12411yescharacteristic
523C2×C4C2×C4⋊Q811yescharacteristic
524C2×C4#57611yescharacteristic
525C2×C4#57611yescharacteristic
526C2×C4#57711yescharacteristic
527C2×C4#57711yescharacteristic
528C2×C4#57711yescharacteristic
529C2×C4#57711yescharacteristic
530C2×C4#57811yescharacteristic
531C2×C4#57811yescharacteristic
532C8—42no—
533C8—42no—
534C8—42no—
535C8—42no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8—81no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22C2×#39111yescharacteristic
548C22C2×#41311yescharacteristic
549C22C2×#41311yescharacteristic
550C22#289711yescharacteristic
551C22#289711yescharacteristic
552C22#289811yescharacteristic
553C22#289811yescharacteristic
554C4—22no—
555C4—22no—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4#308111yescharacteristic
572C4#308211yescharacteristic
573C4#308311yescharacteristic
574C4#308411yescharacteristic
575C4#313611yescharacteristic
576C4#313711yescharacteristic
577C4#316311yescharacteristic
578C4#316411yescharacteristic
579C2—42no—
580C2#1777111yescharacteristic
581C2#1777211yescharacteristic
582C2#1777511yescharacteristic
583C2#1777811yescharacteristic
584C2#2131511yescharacteristic
585C2#2132611yescharacteristic
586C2#2133711yescharacteristic
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. #9907C2semidirect
5. #9906C2semidirect
6. #9903C2semidirect
7. #9902C2semidirect
8. #9901C2semidirect
9. #9900C2semidirect
10. #9899C2semidirect
11. #9898C2semidirect
87. #431Q8semidirect
89. #423Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#219762
#205502
#99072
#99062
#99032
#99022
#99012
#99002
#98992
#98982
#80462
#80452
#80422
#80392