Finite Group Databasebeta

← #274673

Subgroups — #274673

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32957C211yesmaximal, characteristic
2#27962C211yesmaximal, characteristic
3Q8×C22⋊C8C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5#10626C211yesmaximal, characteristic, complemented
6#10623C211yesmaximal, characteristic, complemented
7#10483C211yesmaximal, characteristic, complemented
8#10482C211yesmaximal, characteristic, complemented
9#9216C211yesmaximal, characteristic, complemented
10#9216C211yesmaximal, characteristic, complemented
11#9128C211yesmaximal, characteristic
12#8831C211yesmaximal, characteristic
13#8726C211yesmaximal, characteristic
14#8649C211yesmaximal, characteristic, complemented
15#8647C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857C2211yescharacteristic
23#2285C2211yescharacteristic
24#2101—21no—
25#2101—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1984—21no—
29#1984—21no—
30#1984C2211yescharacteristic
31#1984C2211yescharacteristic
32#1863—21no—
33#1863—21no—
34#1852—21no—
35#1834C2211yescharacteristic
36#1834C2211yescharacteristic
37#1825—21no—
38#1825—21no—
39#1824—21no—
40#1803C2211yescharacteristic
41#1803C2211yescharacteristic
42#1803C2211yescharacteristic
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1787—21no—
50#1787—21no—
51C4×#103—21no—
52C4×#103—21no—
53C4×C22⋊C8—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×Q8D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—41no—
75C2×C4×Q8—41no—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#477—22no—
86#477—22no—
87#477—41no—
88#477—41no—
89C8×Q8—22no—
90C8×Q8—22no—
91C8×Q8—41no—
92C8×Q8—41no—
93#431—41no—
94#431—41no—
95C2×#107—21no—
96C2×#107—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8D411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#389—21no—
125#389—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#382—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158#335—41no—
159#334—41no—
160#333—41no—
161#333—41no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—21no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8C2×D411yescharacteristic
171C22×Q8C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C22⋊Q8—82no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8#4112yes—
207C4×Q8#4112yes—
208#117—42no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
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—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—41no—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42C2411yescharacteristic
250#107—42no—
251#107—42no—
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—21no—
263C4⋊C8—21no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266#103—42no—
267#103—42no—
268#103—42no—
269#103—42no—
270#103—42no—
271#103—42no—
272#103—42no—
273#103—42no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—42no—
284C4×C8—42no—
285C4×C8—42no—
286#95—41no—
287#95—41no—
288#95—41no—
289#95—41no—
290C24#12111yescharacteristic
291C2×Q8—24no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—81no—
309C2×Q8—81no—
310C2×Q8#12312yes—
311C2×Q8#13512yes—
312C2×Q8#13512yes—
313C2×Q8#13712yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—21no—
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—21no—
330C22×C4—21no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—81no—
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#13512yes—
346C22×C4#13712yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
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—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
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—81no—
407C4⋊C4—81no—
408C4⋊C4—81no—
409C4⋊C4—82no—
410C4⋊C4#12312yes—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C42—21no—
414C42—21no—
415C42—22no—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—42no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42—82no—
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23#46412yes—
437C23C2×#12111yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—22no—
441Q8—22no—
442Q8—22no—
443Q8—48no—
444Q8—41no—
445Q8—41no—
446Q8—161no—
447Q8—161no—
448Q8#48514yes—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—48no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—84no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4#46412yes—
496C2×C4C2×#12311yescharacteristic
497C2×C4#53411yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#54611yescharacteristic
500C2×C4#55111yescharacteristic
501C2×C4#55111yescharacteristic
502C2×C4C2×#13511yescharacteristic
503C2×C4C2×#13511yescharacteristic
504C2×C4C2×#13511yescharacteristic
505C2×C4C2×#13711yescharacteristic
506C2×C4C2×#13711yescharacteristic
507C2×C4#57511yescharacteristic
508C2×C4#57811yescharacteristic
509C2×C4#57811yescharacteristic
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8—81no—
519C8—81no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22#241211yescharacteristic
528C22C2×#46411yescharacteristic
529C22#304411yescharacteristic
530C22#305011yescharacteristic
531C22#305611yescharacteristic
532C22#306711yescharacteristic
533C22#306811yescharacteristic
534C4—22no—
535C4—24no—
536C4—21no—
537C4—21no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4—42no—
542C4—81no—
543C4—161no—
544C4—161no—
545C4—161no—
546C4—161no—
547C4C2×#48511yescharacteristic
548C4#309111yescharacteristic
549C4#319011yescharacteristic
550C4#319111yescharacteristic
551C4#330011yescharacteristic
552C4#330011yescharacteristic
553C4#331211yescharacteristic
554C4#331211yescharacteristic
555C2—42no—
556C2#1912311yescharacteristic
557C2#1917611yescharacteristic
558C2#1940611yescharacteristic
559C2#1947811yescharacteristic
560C2#2452911yescharacteristic
561C2#2455411yescharacteristic
562C2#2457511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10626C2semidirect
6. #10623C2semidirect
7. #10483C2semidirect
8. #10482C2semidirect
9. #9216C2semidirect
10. #9216C2semidirect
14. #8649C2semidirect
15. #8647C2semidirect

Maximal subgroups

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

subgroupindex
#329572
#279622
Q8×C22⋊C82
#207342
#106262
#106232
#104832
#104822
#92162
#91282
#88312
#87262
#86492
#86472