Finite Group Databasebeta

← #243082

Subgroups — #243082

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21880C211yesmaximal, characteristic
2#21868C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic, complemented
4#17465C211yesmaximal, characteristic
5#10644C211yesmaximal, characteristic, complemented
6#10643C211yesmaximal, characteristic, complemented
7#10642C211yesmaximal, characteristic, complemented
8#10641C211yesmaximal, characteristic, complemented
9#10640C211yesmaximal, characteristic, complemented
10#10639C211yesmaximal, characteristic, complemented
11#10638C211yesmaximal, characteristic, complemented
12#10637C211yesmaximal, characteristic, complemented
13#8080C211yesmaximal, characteristic, complemented
14#7961C211yesmaximal, characteristic, complemented
15#7957C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2285C2211yescharacteristic
23#1998—21no—
24#1992—21no—
25#1967—21no—
26#1959—21no—
27#1901—21no—
28#1900—21no—
29#1875—21no—
30#1873—21no—
31#1805C2211yescharacteristic
32#1805C2211yescharacteristic
33#1804C2211yescharacteristic
34#1804C2211yescharacteristic
35#1801C2211yescharacteristic
36#1801C2211yescharacteristic
37#1800C2211yescharacteristic
38#1800C2211yescharacteristic
39#1573—22no—
40#1571C2211yescharacteristic
41#1571C2211yescharacteristic
42#1570C2211yescharacteristic
43#1570C2211yescharacteristic
44#1569—22no—
45#1483—22no—
46#1482—22no—
47#1479C2211yescharacteristic
48#1479C2211yescharacteristic
49#1478C2211yescharacteristic
50#1478C2211yescharacteristic
51#1301C2211yescharacteristic
52#1301C2211yescharacteristic
53#1301C2211yescharacteristic
54#1301C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1286C2211yescharacteristic
60#1286C2211yescharacteristic
61#1286C2211yescharacteristic
62#1286C2211yescharacteristic
63C22×M16—21no—
64#529—21no—
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72C2×#126—21no—
73C2×#125D411yescharacteristic
74C2×#124D411yescharacteristic
75C2×#123—21no—
76C2×C22⋊Q8D411yescharacteristic
77C2×#121D411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4C2311yescharacteristic
81C2×#117—21no—
82C2×#117—21no—
83C22×C42C2311yescharacteristic
84#432—21no—
85#432—21no—
86#432—21no—
87#432—21no—
88#432—21no—
89#432—21no—
90#432—21no—
91#432—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431D412yes—
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#423—21no—
103#423—21no—
104#423—21no—
105#423—21no—
106#423C2311yescharacteristic
107#423C2311yescharacteristic
108#423C2311yescharacteristic
109#423C2311yescharacteristic
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×#103—21no—
116C2×#103—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#101—21no—
120C2×#101—21no—
121C2×#100—21no—
122C2×#100—21no—
123#407—21no—
124#407—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C4×M16—21no—
128C4×M16D412yes—
129C2×#97—21no—
130C2×C4×C8—21no—
131#384—21no—
132#382C2311yescharacteristic
133#382C2311yescharacteristic
134#380—21no—
135#380—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138#343—21no—
139#343—21no—
140#343—21no—
141#343—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#335—22no—
147#335—22no—
148#334—22no—
149#334—22no—
150Q8⋊C8—22no—
151Q8⋊C8—22no—
152D4⋊C8—22no—
153D4⋊C8—22no—
154C22×Q8C2×D411yescharacteristic
155C22×D4—21no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C23×C4#4111yescharacteristic
161C2×M16—22no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—41no—
167C2×M16—81no—
168C2×M16—81no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—41no—
172C22×C8—41no—
173#126—41no—
174#126—41no—
175#126—41no—
176#126—41no—
177#125—21no—
178#125—21no—
179#125—21no—
180#125—21no—
181#124—21no—
182#124—21no—
183#124—21no—
184#124—21no—
185#123—41no—
186#123—41no—
187#123—41no—
188#123—41no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196#121—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—81no—
214#117—81no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—41no—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265#103—81no—
266#103—81no—
267#102—81no—
268#102—81no—
269#101—81no—
270#101—81no—
271#100—81no—
272#100—81no—
273C22⋊C8—22no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—41no—
279C22⋊C8—41no—
280C22⋊C8—41no—
281C22⋊C8—41no—
282C22⋊C8—41no—
283#97—22no—
284#97—41no—
285#97—41no—
286#97—41no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—41no—
290C4×C8—41no—
291C4×C8C2×D412yes—
292C4×C8C2×D412yes—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297C24—21no—
298C24#12311yescharacteristic
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—82no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—82no—
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4#12311yescharacteristic
332C22×C4#12311yescharacteristic
333C22×C4#12311yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4#12311yescharacteristic
337C22×C4#12411yescharacteristic
338C22×C4#12411yescharacteristic
339C22×C4#12611yescharacteristic
340C22×C4#12611yescharacteristic
341C22×C4#12611yescharacteristic
342C22×C4#12611yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—82no—
355M16—82no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—41no—
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—82no—
371C2×C8—82no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—82no—
401C42—41no—
402C42—41no—
403C42—82no—
404C42#12111yescharacteristic
405C42#12111yescharacteristic
406C42#12311yescharacteristic
407C42#12311yescharacteristic
408C42#12311yescharacteristic
409C42#12311yescharacteristic
410C42#12411yescharacteristic
411C42#12411yescharacteristic
412C42#12611yescharacteristic
413C42#12611yescharacteristic
414C42#12611yescharacteristic
415C42#12611yescharacteristic
416C42C22×D411yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C42C2×#4111yescharacteristic
420C23—21no—
421C23—21no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—81no—
428C23—81no—
429C23#39711yescharacteristic
430C23#39711yescharacteristic
431C23C2×#12311yescharacteristic
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—82no—
441D4—82no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4#39711yescharacteristic
480C2×C4#39711yescharacteristic
481C2×C4#39711yescharacteristic
482C2×C4#39711yescharacteristic
483C2×C4#39711yescharacteristic
484C2×C4#39711yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12411yescharacteristic
489C2×C4C2×#12611yescharacteristic
490C2×C4C2×#12611yescharacteristic
491C2×C4#57611yescharacteristic
492C2×C4#57711yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57811yescharacteristic
495C8—42no—
496C8—42no—
497C8—81no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—161no—
519C22—161no—
520C22C2×#39711yescharacteristic
521C22#309611yescharacteristic
522C22#309711yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—161no—
537C4—161no—
538C4#306211yescharacteristic
539C4#306311yescharacteristic
540C4#309811yescharacteristic
541C4#309911yescharacteristic
542C2—21no—
543C2—21no—
544C2—41no—
545C2—41no—
546C2—161no—
547C2#2364311yescharacteristic
548C2#2364911yescharacteristic
549C2#2365211yescharacteristic
subgroupG/H
center, socleC22C2×#397
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17822C2semidirect
5. #10644C2semidirect
6. #10643C2semidirect
7. #10642C2semidirect
8. #10641C2semidirect
9. #10640C2semidirect
10. #10639C2semidirect
11. #10638C2semidirect
12. #10637C2semidirect
13. #8080C2semidirect
14. #7961C2semidirect
15. #7957C2semidirect

Maximal subgroups

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

subgroupindex
#218802
#218682
#178222
#174652
#106442
#106432
#106422
#106412
#106402
#106392
#106382
#106372
#80802
#79612
#79572