Finite Group Databasebeta

← #212192

Subgroups — #212192

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758C211yesmaximal, characteristic
2#22133C211yesmaximal, characteristic
3#20548C211yesmaximal, characteristic, complemented
4#9542C211yesmaximal, characteristic, complemented
5#9542C211yesmaximal, characteristic, complemented
6#9535C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8944C211yesmaximal, characteristic
9#8833C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic, complemented
11#8290C211yesmaximal, characteristic, complemented
12#8288C211yesmaximal, characteristic, complemented
13#8204C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C2×C8×D4C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C4×#121C2211yescharacteristic
20#2283C2211yescharacteristic
21#2101C411yescharacteristic
22#2092C411yescharacteristic
23#2085C411yescharacteristic
24#2077C411yescharacteristic
25#2076C411yescharacteristic
26#2073C411yescharacteristic
27#2034C411yescharacteristic
28#2033C411yescharacteristic
29#1954—21no—
30#1954C2211yescharacteristic
31#1939C2211yescharacteristic
32#1923—21no—
33#1869—21no—
34#1850—21no—
35#1797C2211yescharacteristic
36C4×#107C2211yescharacteristic
37C4×#106C2211yescharacteristic
38#1770C2211yescharacteristic
39#1770C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43#1768—21no—
44#1768—21no—
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1753—21no—
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#121C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C8×D4—22no—
74C8×D4—22no—
75#431—22no—
76#431—22no—
77C2×#107C2×C411yescharacteristic
78C2×#107C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81#423—22no—
82#423—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8C2×C411yescharacteristic
101C2×C22⋊C8C2×C411yescharacteristic
102C2×C22⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#390C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#388—21no—
118#386—21no—
119#383—21no—
120#382C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122#381—21no—
123#380C2×C411yescharacteristic
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
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#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#334—22no—
151#334—22no—
152D4⋊C8—22no—
153D4⋊C8—22no—
154#322—22no—
155#322—22no—
156C22×D4—21no—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C2×M16—24no—
163C2×M16—81no—
164C2×M16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—81no—
173C22×C8—81no—
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182#121—22no—
183#121—22no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—81no—
189C4×D4—81no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#107—22no—
217#107—41no—
218#106—22no—
219#106—41no—
220C4⋊C8—24no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—81no—
230C4⋊C8#4112yes—
231C4⋊C8#4112yes—
232#102—22no—
233#102—22no—
234#102—41no—
235#102—41no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—81no—
243C22⋊C8—81no—
244C22⋊C8—81no—
245#97—81no—
246#97—81no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—81no—
258C4×C8—81no—
259C4×C8—81no—
260C4×C8#4112yes—
261C4×C8#4112yes—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273C24—21no—
274C24C4×D411yescharacteristic
275C2×D4—24no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4#13112yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319M16—84no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
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—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—84no—
361C2×C8#12312yes—
362C2×C8#12312yes—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—81no—
377C4⋊C4#13112yes—
378C22⋊C4—24no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—81no—
382C22⋊C4—81no—
383C22⋊C4—81no—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—22no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C2×C4×D411yescharacteristic
417D4—22no—
418D4—22no—
419D4—41no—
420D4—41no—
421D4—84no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4C4×#4111yescharacteristic
469C2×C4C4×#4111yescharacteristic
470C2×C4C2×#12311yescharacteristic
471C2×C4#52511yescharacteristic
472C2×C4#52911yescharacteristic
473C2×C4#52911yescharacteristic
474C2×C4C2×#13111yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4#56811yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C8—22no—
483C8—22no—
484C8—42no—
485C8—41no—
486C8—41no—
487C8—42no—
488C8—42no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—81no—
494C8—81no—
495C8—81no—
496C22—41no—
497C22—42no—
498C22—41no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22C4×#12311yescharacteristic
511C22#292911yescharacteristic
512C22#293211yescharacteristic
513C22#294611yescharacteristic
514C22#295211yescharacteristic
515C22#295611yescharacteristic
516C22#295711yescharacteristic
517C4—21no—
518C4—21no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4#296811yescharacteristic
530C4#296811yescharacteristic
531C4#296911yescharacteristic
532C4#296911yescharacteristic
533C4#309611yescharacteristic
534C4#309911yescharacteristic
535C4#311811yescharacteristic
536C4#312611yescharacteristic
537C2—81no—
538C2—82no—
539C2#1791811yescharacteristic
540C2#1792711yescharacteristic
541C2#1798711yescharacteristic
542C2#1799311yescharacteristic
543C2#2156711yescharacteristic
544C2#2158811yescharacteristic
545C2#2159411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20548C2semidirect
4. #9542C2semidirect
5. #9542C2semidirect
6. #9535C2semidirect
7. #9533C2semidirect
9. #8833C2semidirect
10. #8711C2semidirect
11. #8290C2semidirect
12. #8288C2semidirect
13. #8204C2semidirect
14. #8202C2semidirect
15. #8149C2semidirect

Maximal subgroups

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

subgroupindex
#237582
#221332
#205482
#95422
#95352
#95332
#89442
#88332
#87112
#82902
#82882
#82042
#82022
#81492