Finite Group Databasebeta

← #225103

Subgroups — #225103

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626C211yesmaximal, characteristic
2Q8×C22⋊C8C211yesmaximal, characteristic
3#20734C211yesmaximal, characteristic
4#9559C211yesmaximal, characteristic, complemented
5#9555C211yesmaximal, characteristic, complemented
6#9547C211yesmaximal, characteristic, complemented
7#9543C211yesmaximal, characteristic, complemented
8#8846C211yesmaximal, characteristic, complemented
9#8846C211yesmaximal, characteristic, complemented
10#8710C211yesmaximal, characteristic
11#8598C211yesmaximal, characteristic
12#8311C211yesmaximal, characteristic
13#8205C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2285C2211yescharacteristic
21#1984—21no—
22#1984—21no—
23#1984C2211yescharacteristic
24#1984C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1917—21no—
28#1916—21no—
29#1824C411yescharacteristic
30#1824C411yescharacteristic
31#1803C411yescharacteristic
32#1803C411yescharacteristic
33#1803C411yescharacteristic
34#1803C411yescharacteristic
35#1798—21no—
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1787C411yescharacteristic
40#1787C411yescharacteristic
41#1770C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1750—21no—
48C4×C22⋊C8—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×Q8C2×C411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—41no—
73C2×C4×Q8—41no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C8×Q8—22no—
84C8×Q8—22no—
85C8×Q8—41no—
86C8×Q8—41no—
87#431—41no—
88#431—41no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8C2×C411yescharacteristic
111C2×C22⋊C8C2×C411yescharacteristic
112C2×C22⋊C8D411yescharacteristic
113C2×C22⋊C8D411yescharacteristic
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#382—21no—
127#380—21no—
128#380C2×C411yescharacteristic
129#380C2×C411yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#334—41no—
155#334—41no—
156Q8⋊C8—22no—
157Q8⋊C8—22no—
158Q8⋊C8—41no—
159Q8⋊C8—41no—
160#322—41no—
161#322—41no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C22×C411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8#4112yes—
205C4×Q8#4112yes—
206#117—81no—
207#117—81no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
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—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—81no—
258C4⋊C8—81no—
259#103—22no—
260#103—22no—
261#103—22no—
262#103—22no—
263#103—22no—
264#103—22no—
265#103—41no—
266#103—41no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—41no—
270C22⋊C8—41no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274#97—81no—
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—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—42no—
288C4×C8—42no—
289C4×C8—81no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294C24C4×D411yescharacteristic
295C2×Q8—24no—
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—42no—
305C2×Q8—42no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8#11712yes—
311C2×Q8#13112yes—
312C2×Q8#13112yes—
313C2×Q8#13112yes—
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—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—81no—
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4#13712yes—
346C22×C4#13712yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C2×C8—22no—
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—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—81no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4#11712yes—
415C22⋊C4—81no—
416C22⋊C4—81no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—42no—
426C42—42no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42C2×#4111yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C42H(5,𝔽2)11yescharacteristic
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23#44112yes—
444C23C2×C4×D411yescharacteristic
445Q8—22no—
446Q8—22no—
447Q8—22no—
448Q8—22no—
449Q8—22no—
450Q8—48no—
451Q8—41no—
452Q8—41no—
453Q8#43314yes—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—48no—
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—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—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—84no—
500C2×C4#44112yes—
501C2×C4C2×#11711yescharacteristic
502C2×C4#51811yescharacteristic
503C2×C4#54611yescharacteristic
504C2×C4#54611yescharacteristic
505C2×C4#55111yescharacteristic
506C2×C4#55111yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4C2×#13111yescharacteristic
509C2×C4C2×#13111yescharacteristic
510C2×C4#56811yescharacteristic
511C2×C4C2×#13711yescharacteristic
512C2×C4C2×#13711yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57711yescharacteristic
515C8—42no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—81no—
525C8—81no—
526C8—81no—
527C8—81no—
528C8—81no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22#234211yescharacteristic
537C22#292911yescharacteristic
538C22#293311yescharacteristic
539C22C2×#44111yescharacteristic
540C22#294911yescharacteristic
541C22#295511yescharacteristic
542C22#295611yescharacteristic
543C4—24no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—42no—
549C4—41no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4C2×#43311yescharacteristic
557C4#292411yescharacteristic
558C4#297611yescharacteristic
559C4#297711yescharacteristic
560C4#331611yescharacteristic
561C4#331611yescharacteristic
562C4#332011yescharacteristic
563C4#332011yescharacteristic
564C2—42no—
565C2#1790611yescharacteristic
566C2#1792311yescharacteristic
567C2#1800311yescharacteristic
568C2#1803011yescharacteristic
569C2#2255811yescharacteristic
570C2#2256711yescharacteristic
571C2#2258811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9559C2semidirect
5. #9555C2semidirect
6. #9547C2semidirect
7. #9543C2semidirect
8. #8846C2semidirect
9. #8846C2semidirect
13. #8205C2semidirect
14. #8201C2semidirect

Maximal subgroups

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

subgroupindex
#226262
Q8×C22⋊C82
#207342
#95592
#95552
#95472
#95432
#88462
#87102
#85982
#83112
#82052
#82012
#81492