Finite Group Databasebeta

← #203333

Subgroups — #203333

Order 512.

716 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#22012C211yesmaximal, characteristic, complemented
2#20558C211yesmaximal, characteristic, complemented
3C4×#442C211yesmaximal, characteristic
4#9546C211yesmaximal, characteristic, complemented
5#9545C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9538C211yesmaximal, characteristic, complemented
8#8947C211yesmaximal, characteristic
9#8947C211yesmaximal, characteristic
10#8845C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic, complemented
12#8276C211yesmaximal, characteristic, complemented
13#8174C211yesmaximal, characteristic, complemented
14#8174C211yesmaximal, characteristic, complemented
15#8134C211yesmaximal, characteristic, complemented
16C2×#442C411yescharacteristic
17C2×#442C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C42×D4C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1924—21no—
28#1924—21no—
29#1817—21no—
30#1797—21no—
31#1776C411yescharacteristic
32#1776C411yescharacteristic
33C4×#106C2211yescharacteristic
34#1770C2211yescharacteristic
35#1770C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1762C411yescharacteristic
39#1762C411yescharacteristic
40#1762C411yescharacteristic
41#1762C411yescharacteristic
42C4×#102C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1752—21no—
45#1752—21no—
46#1749—21no—
47#1744C2211yescharacteristic
48#1744C2211yescharacteristic
49C2×#336—21no—
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78#442—24no—
79#442—24no—
80C4×D8—22no—
81C4×D8—22no—
82C8×D4—22no—
83C8×D4—22no—
84C8×D4—22no—
85C8×D4—22no—
86C2×#107C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97C2×C411yescharacteristic
112C2×#97C2×C411yescharacteristic
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336D411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—22no—
153#334—22no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C2×D8—22no—
167C2×D8—22no—
168C2×D8M1614yes—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—24no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8—81no—
180C22×C8—81no—
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4#4112yes—
204C4×D4#4112yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—81no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42—81no—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42C2411yescharacteristic
234#107—22no—
235#106—22no—
236#106—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245C4⋊C8#4112yes—
246C4⋊C8#4112yes—
247#102—22no—
248#102—22no—
249#102—22no—
250#102—22no—
251#102—22no—
252#102M1614yes—
253C22⋊C8—24no—
254C22⋊C8—24no—
255C22⋊C8—81no—
256C22⋊C8—81no—
257C22⋊C8—81no—
258C22⋊C8—81no—
259#97—22no—
260#97—22no—
261#97—41no—
262#97—41no—
263#97—81no—
264#97—81no—
265#97—81no—
266#97—81no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285C24—21no—
286C24—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4#11712yes—
296C2×D4C2×M1612yes—
297C2×D4C2×M1612yes—
298C2×D4C2×M1612yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336D8—28no—
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—21no—
347C2×C8—22no—
348C2×C8—22no—
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—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—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—84no—
378C2×C8—84no—
379C2×C8C2×M1612yes—
380C2×C8C2×M1612yes—
381C4⋊C4—21no—
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#11712yes—
392C4⋊C4C2×M1612yes—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—81no—
416C42—81no—
417C42—81no—
418C42—81no—
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23C2×C4×D411yescharacteristic
438D4—22no—
439D4—22no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—41no—
444D4—41no—
445D4#43214yes—
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—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
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—41no—
481C2×C4—41no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4C2×#11711yescharacteristic
491C2×C4C4×#4111yescharacteristic
492C2×C4#52511yescharacteristic
493C2×C4#52511yescharacteristic
494C2×C4#52911yescharacteristic
495C2×C4#52911yescharacteristic
496C2×C4C22×M1611yescharacteristic
497C2×C4C22×M1611yescharacteristic
498C2×C4#56811yescharacteristic
499C2×C4#56811yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57611yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57711yescharacteristic
504C8—22no—
505C8—22no—
506C8—41no—
507C8—42no—
508C8—41no—
509C8—41no—
510C8—42no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8—81no—
518C22—24no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#229011yescharacteristic
533C22D4×M1611yescharacteristic
534C22D4×M1611yescharacteristic
535C22#294611yescharacteristic
536C22#294611yescharacteristic
537C22#295511yescharacteristic
538C22#295511yescharacteristic
539C4—21no—
540C4—21no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—42no—
545C4—41no—
546C4—42no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4C2×#43211yescharacteristic
552C4#292311yescharacteristic
553C4#297011yescharacteristic
554C4#297011yescharacteristic
555C4#311811yescharacteristic
556C4#311911yescharacteristic
557C4#312211yescharacteristic
558C4#312511yescharacteristic
559C2—44no—
560C2—82no—
561C2#1805011yescharacteristic
562C2#1805011yescharacteristic
563C2#1805011yescharacteristic
564C2#1805211yescharacteristic
565C2#2110711yescharacteristic
566C2#2112611yescharacteristic
567C2#2113811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22012C2semidirect
2. #20558C2semidirect
4. #9546C2semidirect
5. #9545C2semidirect
6. #9539C2semidirect
7. #9538C2semidirect
10. #8845C2semidirect
11. #8843C2semidirect
12. #8276C2semidirect
13. #8174C2semidirect
14. #8174C2semidirect
15. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#220122
#205582
C4×#4422
#95462
#95452
#95392
#95382
#89472
#88452
#88432
#82762
#81742
#81342