Finite Group Databasebeta

← #227559

Subgroups — #227559

Order 512.

779 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23003C211yesmaximal, characteristic
2#23003C211yesmaximal, characteristic
3#22542C211yesmaximal, characteristic, complemented
4#22524C211yesmaximal, characteristic, complemented
5C2×#1626C211yesmaximal, characteristic
6#9830C211yesmaximal, characteristic
7#9574C211yesmaximal, characteristic, complemented
8#9569C211yesmaximal, characteristic, complemented
9#9230C211yesmaximal, characteristic
10#9222C211yesmaximal, characteristic
11#9057C211yesmaximal, characteristic
12#8884C211yesmaximal, characteristic, complemented
13#8884C211yesmaximal, characteristic, complemented
14#8209C211yesmaximal, characteristic, complemented
15#8203C211yesmaximal, characteristic, complemented
16C2×#557C2211yescharacteristic
17C2×#486C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#460C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×#440—21no—
23C2×#440—21no—
24#2366C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26Q8×C4⋊C4C2211yescharacteristic
27#1992—21no—
28#1990C2211yescharacteristic
29#1989—21no—
30#1988C2211yescharacteristic
31#1981—21no—
32#1981C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956—21no—
36#1956C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1936—21no—
40#1936—21no—
41#1880—21no—
42#1880—21no—
43#1862—21no—
44#1862—21no—
45#1845C2211yescharacteristic
46#1803—21no—
47#1803—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52#1763—21no—
53C4×#103—21no—
54C4×#103C2211yescharacteristic
55C4×#103C2211yescharacteristic
56#1626C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70#557—22no—
71#557—22no—
72#557—41no—
73#557C2×C414yescomplemented
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2311yescharacteristic
77C2×#125—21no—
78C2×#125—41no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87#486—41no—
88#486C2×C414yes—
89#460—22no—
90#460—22no—
91#460C2×C414yes—
92#460C2×C414yes—
93#446—22no—
94#446—41no—
95C8×Q8—22no—
96C8×Q8—41no—
97#440—44no—
98#440—44no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—41no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103D411yescharacteristic
115C2×#103D411yescharacteristic
116C2×#103D411yescharacteristic
117C2×#103D411yescharacteristic
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102—21no—
122C2×#102—21no—
123C2×#102—21no—
124C2×#102—41no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#390—21no—
133#390—21no—
134#389—21no—
135#389—21no—
136#389—41no—
137#389—41no—
138#386—41no—
139#384—21no—
140#384—21no—
141#384C2311yescharacteristic
142#384C2311yescharacteristic
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2311yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#340D411yescharacteristic
159#336—21no—
160#336—21no—
161#332C2×C414yescomplemented
162#331—41no—
163Q8⋊C8C2×C414yescomplemented
164Q8⋊C8C2×C414yescomplemented
165#322—41no—
166C22×Q8—21no—
167C22×Q8C2×D411yescharacteristic
168C22×Q8C2×D411yescharacteristic
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C23×C4—41no—
173C2×SD16—24no—
174C2×SD16—24no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×SD16—42no—
178C2×SD16—42no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—24no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—42no—
193C4⋊Q8C22×C412yes—
194#127—21no—
195#127—21no—
196#127—21no—
197#127C22×C412yes—
198#125—24no—
199#125—42no—
200#125—44no—
201#125—44no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8—81no—
213C4×Q8—81no—
214C4×Q8C22×C412yes—
215C4×Q8C22×C412yes—
216C4×Q8#4112yes—
217C4×Q8#4112yes—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—81no—
221C4×D4—81no—
222C2×C4⋊C4—21no—
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—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42C2411yescharacteristic
256#107—42no—
257#107—42no—
258#107—82no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8C22×C412yes—
268C4⋊C8C22×C412yes—
269#103—22no—
270#103—22no—
271#103—22no—
272#103—41no—
273#103—42no—
274#103—42no—
275#103—42no—
276#103—81no—
277#103—81no—
278#103—81no—
279#102—24no—
280#102—24no—
281#102—44no—
282#102—42no—
283#102—42no—
284#102—81no—
285#102—81no—
286#97—42no—
287#97—42no—
288#97—42no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—42no—
295C4×C8C22×C412yes—
296#95—21no—
297#95—21no—
298#95—41no—
299#95—41no—
300C24—41no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—22no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8C4×D412yes—
316C2×Q8C4×D412yes—
317C2×Q8AGL(1,ℤ/8)12yes—
318C2×Q8#13712yes—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—81no—
334C2×D4—81no—
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—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—161no—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352SD16—48no—
353SD16—48no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
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×C8AGL(1,ℤ/8)12yes—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
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—82no—
411C4⋊C4—81no—
412C4⋊C4—81no—
413C4⋊C4—82no—
414C4⋊C4—161no—
415C4⋊C4—161no—
416C4⋊C4C4×D412yes—
417C4⋊C4C4×D412yes—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—161no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—44no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—82no—
431C42—82no—
432C42C23×C411yescharacteristic
433C42C2×#4111yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C23—42no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23D4211yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—22no—
447Q8—22no—
448Q8—22no—
449Q8—41no—
450Q8—41no—
451Q8—42no—
452Q8—42no—
453Q8—81no—
454Q8—81no—
455Q8#44014yescomplemented
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—82no—
461D4—81no—
462D4—81no—
463D4—81no—
464D4—81no—
465D4—161no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—164no—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4#51811yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#54611yescharacteristic
507C2×C4#55111yescharacteristic
508C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4C2×#13711yescharacteristic
511C2×C4#57511yescharacteristic
512C2×C4#57611yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57811yescharacteristic
515C2×C4#57811yescharacteristic
516C8—22no—
517C8—42no—
518C8—42no—
519C8—42no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—81no—
524C8—81no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—161no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22#235811yescharacteristic
533C22#326511yescharacteristic
534C22#327011yescharacteristic
535C22#327411yescharacteristic
536C22#327711yescharacteristic
537C22#328811yescharacteristic
538C22#328911yescharacteristic
539C4—24no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—161no—
551C4—161no—
552C4C2×#44011yescharacteristic
553C4C4×AGL(1,ℤ/8)11yescharacteristic
554C4#294711yescharacteristic
555C4#295111yescharacteristic
556C4#330811yescharacteristic
557C4#330811yescharacteristic
558C4#330911yescharacteristic
559C4#332011yescharacteristic
560C2—162no—
561C2#2020011yescharacteristic
562C2#2020211yescharacteristic
563C2#2021611yescharacteristic
564C2#2021811yescharacteristic
565C2#2285511yescharacteristic
566C2#2286311yescharacteristic
567C2#2289311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22542C2semidirect
4. #22524C2semidirect
7. #9574C2semidirect
8. #9569C2semidirect
12. #8884C2semidirect
13. #8884C2semidirect
14. #8209C2semidirect
15. #8203C2semidirect
56. #1626C4semidirect
73. #557C2×C4semidirect
161. #332C2×C4semidirect
163. Q8⋊C8C2×C4semidirect
164. Q8⋊C8C2×C4semidirect
455. Q8#440semidirect

Maximal subgroups

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

subgroupindex
#230032
#225422
#225242
C2×#16262
#98302
#95742
#95692
#92302
#92222
#90572
#88842
#82092
#82032