Finite Group Databasebeta

← #229936

Subgroups — #229936

Order 512.

647 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23698C211yesmaximal, characteristic
2#23051C211yesmaximal, characteristic
3#23050C211yesmaximal, characteristic
4C2×#1579C211yesmaximal, characteristic, complemented
5C2×#1572C211yesmaximal, characteristic
6#9664C211yesmaximal, characteristic, complemented
7#9595C211yesmaximal, characteristic, complemented
8#9548C211yesmaximal, characteristic, complemented
9#9544C211yesmaximal, characteristic, complemented
10#8941C211yesmaximal, characteristic
11#8938C211yesmaximal, characteristic
12#8844C211yesmaximal, characteristic, complemented
13#8844C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24#2087C411yescharacteristic
25#2076C411yescharacteristic
26#2068C411yescharacteristic
27#2068C411yescharacteristic
28#2056C411yescharacteristic
29#2036C411yescharacteristic
30#2036C411yescharacteristic
31#2001C411yescharacteristic
32#1984C2211yescharacteristic
33#1984C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1924—21no—
37#1923—21no—
38#1923—21no—
39#1923—21no—
40#1917—21no—
41#1916—21no—
42C4×#107C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4—21no—
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1579—24no—
52#1572—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C8×Q8—22no—
76C8×Q8—22no—
77C8×Q8—41no—
78C8×Q8—41no—
79C4×Q16—22no—
80C4×Q16—22no—
81#432—22no—
82#432—41no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431—41no—
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8C2×C411yescharacteristic
110C2×C22⋊C8C2×C411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×C22⋊C8D411yescharacteristic
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#387—21no—
130#387—21no—
131#385—21no—
132#385—21no—
133#380C2×C411yescharacteristic
134#380C2×C411yescharacteristic
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
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#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158#335—22no—
159#335—22no—
160#334—22no—
161#334D414yes—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—41no—
165Q8⋊C8—41no—
166#322—22no—
167#322—22no—
168C22×Q8C22×C411yescharacteristic
169C22×Q8C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×Q16—22no—
174C2×Q16—22no—
175C2×Q16—22no—
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
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—41no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#107—22no—
231#107—22no—
232#107—41no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8C2×D412yes—
246C4⋊C8C2×D412yes—
247#103—22no—
248#103—22no—
249#103—22no—
250#103—22no—
251#103—22no—
252#103—41no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—41no—
256C22⋊C8—41no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—42no—
261#97—42no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—42no—
273C4×C8—42no—
274C4×C8—42no—
275C4×C8—42no—
276C4×C8C2×D412yes—
277C4×C8#4112yes—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288C24C4×D411yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8#13112yes—
300C2×Q8#13112yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331Q16—28no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
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—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
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×C8C4×D412yes—
373C2×C8C4×D412yes—
374C4⋊C4—24no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—81no—
393C4⋊C4#13112yes—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—81no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42C22×D411yescharacteristic
413C42C2×#4111yescharacteristic
414C42H(5,𝔽2)11yescharacteristic
415C422−1+411yescharacteristic
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23C2×C4×D411yescharacteristic
424Q8—22no—
425Q8—22no—
426Q8—22no—
427Q8—22no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—41no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
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—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4C2×C4×D411yescharacteristic
478C2×C4#52011yescharacteristic
479C2×C4D4211yescharacteristic
480C2×C4#54711yescharacteristic
481C2×C4#54811yescharacteristic
482C2×C4#55311yescharacteristic
483C2×C4C2×#13111yescharacteristic
484C2×C4C2×#13111yescharacteristic
485C2×C4C2×#13111yescharacteristic
486C2×C4#56811yescharacteristic
487C2×C4#57511yescharacteristic
488C2×C4#57511yescharacteristic
489C2×C4#57811yescharacteristic
490C2×C4#57811yescharacteristic
491C8—22no—
492C8—22no—
493C8—42no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—42no—
501C8—42no—
502C8—81no—
503C8—81no—
504C8—81no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22#236111yescharacteristic
513C22#292911yescharacteristic
514C22#293311yescharacteristic
515C22#294311yescharacteristic
516C22#295011yescharacteristic
517C22#295511yescharacteristic
518C22#295611yescharacteristic
519C4—21no—
520C4—21no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4#292911yescharacteristic
533C4#293511yescharacteristic
534C4#299211yescharacteristic
535C4#299411yescharacteristic
536C4#326911yescharacteristic
537C4#326911yescharacteristic
538C4#334911yescharacteristic
539C4#334911yescharacteristic
540C2—81no—
541C2#1790911yescharacteristic
542C2#1792511yescharacteristic
543C2#1800511yescharacteristic
544C2#1803211yescharacteristic
545C2#2299111yescharacteristic
546C2#2302111yescharacteristic
547C2#2303011yescharacteristic
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. C2×#1579C2semidirect
6. #9664C2semidirect
7. #9595C2semidirect
8. #9548C2semidirect
9. #9544C2semidirect
12. #8844C2semidirect
13. #8844C2semidirect
14. #8288C2semidirect

Maximal subgroups

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

subgroupindex
#236982
#230512
#230502
C2×#15792
C2×#15722
#96642
#95952
#95482
#95442
#89412
#89382
#88442
#82882
#81492