Finite Group Databasebeta

← #229730

Subgroups — #229730

Order 512.

692 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23677C211yesmaximal, characteristic
2#23149C211yesmaximal, characteristic, complemented
3#22651C211yesmaximal, characteristic
4C2×#1588C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9643C211yesmaximal, characteristic, complemented
7#9601C211yesmaximal, characteristic, complemented
8#9559C211yesmaximal, characteristic, complemented
9#9555C211yesmaximal, characteristic, complemented
10#8949C211yesmaximal, characteristic
11#8945C211yesmaximal, characteristic
12#8844C211yesmaximal, characteristic, complemented
13#8844C211yesmaximal, characteristic, complemented
14#8265C211yesmaximal, characteristic, complemented
15#8183C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#424—21no—
23C4×C4⋊Q8C2211yescharacteristic
24#2293C2211yescharacteristic
25#2084C411yescharacteristic
26#2071C411yescharacteristic
27#2066C411yescharacteristic
28#2066C411yescharacteristic
29#2055C411yescharacteristic
30#2055C411yescharacteristic
31#2043C411yescharacteristic
32#2013C411yescharacteristic
33#1984C2211yescharacteristic
34#1984C2211yescharacteristic
35#1939—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1916—21no—
39#1916—21no—
40#1851—21no—
41#1850—21no—
42#1842C2211yescharacteristic
43#1798—21no—
44C8×C4⋊C4—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#103C2211yescharacteristic
50#1744C2211yescharacteristic
51#1588—24no—
52SD16⋊C8—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—41no—
81C8×Q8—41no—
82C4×SD16—22no—
83C4×SD16—22no—
84C4×SD16—22no—
85C4×SD16—22no—
86#436—22no—
87#436—41no—
88C8×D4—22no—
89C8×D4—41no—
90C2×#106C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92#424—44no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#389—21no—
131#389—21no—
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#388—21no—
135#388—21no—
136#384C2×C411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336C2×C411yescharacteristic
156#336C2×C411yescharacteristic
157#336D411yescharacteristic
158#336D411yescharacteristic
159#335—22no—
160#335—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—41no—
164Q8⋊C8—41no—
165D4⋊C8—22no—
166D4⋊C8D414yes—
167#322—22no—
168#322—22no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C2×SD16—22no—
174C2×SD16—22no—
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—22no—
178C2×SD16—41no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—81no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4C2×D412yes—
212C4×D4#4112yes—
213#117—42no—
214#117—42no—
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⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243#106—22no—
244#106—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—41no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—81no—
262C4⋊C8C2×D412yes—
263#103—22no—
264#103—22no—
265#103—22no—
266#103—22no—
267#103—22no—
268#103—41no—
269#102—22no—
270#102—22no—
271C22⋊C8—42no—
272C22⋊C8—42no—
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—42no—
285C4×C8—42no—
286C4×C8—42no—
287C4×C8C2×D412yes—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295C24—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8#13112yes—
307C2×Q8#13112yes—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4C4×D412yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334SD16—28no—
335SD16—44no—
336M16—84no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
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—84no—
379C2×C8#13112yes—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—81no—
399C4⋊C4C4×D412yes—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—81no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42AGL(1,ℤ/8)12yes—
421C42#13712yes—
422C42C22×D411yescharacteristic
423C42C2×#4111yescharacteristic
424C42H(5,𝔽2)11yescharacteristic
425C422−1+411yescharacteristic
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23C2×C4×D411yescharacteristic
434Q8—22no—
435Q8—22no—
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442D4—22no—
443D4—22no—
444D4—41no—
445D4—41no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
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—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4#44012yes—
484C2×C4#44012yes—
485C2×C4C2×C4×D411yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4D4211yescharacteristic
488C2×C4#54711yescharacteristic
489C2×C4#54811yescharacteristic
490C2×C4#55311yescharacteristic
491C2×C4C2×#13111yescharacteristic
492C2×C4C2×#13111yescharacteristic
493C2×C4C2×#13111yescharacteristic
494C2×C4#56811yescharacteristic
495C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
496C2×C4C2×#13711yescharacteristic
497C2×C4#57711yescharacteristic
498C2×C4#57811yescharacteristic
499C8—22no—
500C8—42no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—42no—
507C8—41no—
508C8—41no—
509C8—81no—
510C8—81no—
511C8—81no—
512C8—81no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22#236111yescharacteristic
521C22#292911yescharacteristic
522C22#293511yescharacteristic
523C22C2×#44011yescharacteristic
524C22#294711yescharacteristic
525C22#295511yescharacteristic
526C22#295711yescharacteristic
527C4—22no—
528C4—21no—
529C4—21no—
530C4—22no—
531C4—41no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—41no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4#292911yescharacteristic
541C4#293311yescharacteristic
542C4#299211yescharacteristic
543C4#299411yescharacteristic
544C4#327211yescharacteristic
545C4#327611yescharacteristic
546C4#336711yescharacteristic
547C4#337211yescharacteristic
548C2—82no—
549C2#1790811yescharacteristic
550C2#1791911yescharacteristic
551C2#1807111yescharacteristic
552C2#1808311yescharacteristic
553C2#2299111yescharacteristic
554C2#2300211yescharacteristic
555C2#2301711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23149C2semidirect
6. #9643C2semidirect
7. #9601C2semidirect
8. #9559C2semidirect
9. #9555C2semidirect
12. #8844C2semidirect
13. #8844C2semidirect
14. #8265C2semidirect

Maximal subgroups

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

subgroupindex
#236772
#231492
#226512
C2×#15882
C2×SD16⋊C82
#96432
#96012
#95592
#95552
#89492
#89452
#88442
#82652
#81832