Finite Group Databasebeta

← #225491

Subgroups — #225491

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22735C211yesmaximal, characteristic, complemented
2#22701C211yesmaximal, characteristic
3#21274C211yesmaximal, characteristic
4C2×#1582C211yesmaximal, characteristic
5C2×#1583C211yesmaximal, characteristic
6#9597C211yesmaximal, characteristic, complemented
7#9597C211yesmaximal, characteristic, complemented
8#8848C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic
10#8844C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic
12#8807C211yesmaximal, characteristic, complemented
13#8570C211yesmaximal, characteristic, complemented
14#8569C211yesmaximal, characteristic
15#8261C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21#2295C2211yescharacteristic
22#2293C2211yescharacteristic
23#1968C411yescharacteristic
24#1968C411yescharacteristic
25#1941C411yescharacteristic
26#1941C411yescharacteristic
27#1940—21no—
28#1939—21no—
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1934C411yescharacteristic
34#1934C411yescharacteristic
35#1933C411yescharacteristic
36#1933C411yescharacteristic
37#1851—21no—
38#1850—21no—
39#1832—21no—
40#1831C2211yescharacteristic
41#1831C2211yescharacteristic
42#1798—21no—
43C4×#106C2211yescharacteristic
44#1768—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1729—21no—
50C42×C8C2211yescharacteristic
51#1583—24no—
52#1582—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×M16—21no—
71C23×C8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75#446—22no—
76#446—41no—
77C8×Q8—22no—
78C8×Q8—41no—
79C4×SD16D414yes—
80C4×SD16Q814yes—
81#436—22no—
82#436—41no—
83C8×D4—22no—
84C8×D4—41no—
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2×C411yescharacteristic
112C2×C4×C8C2×C411yescharacteristic
113C2×C4×C8C2×C411yescharacteristic
114C2×C4×C8C2×C411yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#388—21no—
130#388—21no—
131#387—21no—
132#387—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137#376—21no—
138C43C2311yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#336—21no—
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#336D411yescharacteristic
155#336D411yescharacteristic
156#335—41no—
157#335—41no—
158#334—22no—
159#334—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164#322—22no—
165#322—22no—
166C22×Q8C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C2×SD16—41no—
170C2×SD16—41no—
171C2×SD16C4⋊C414yes—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16—81no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—81no—
184C22×C8C22×C411yescharacteristic
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8C2×D412yes—
195C4×Q8C2×Q812yes—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4C2×D412yes—
199C4×D4C2×Q812yes—
200#117—81no—
201#117—81no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
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×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42C2411yescharacteristic
228#106—41no—
229#106—41no—
230#106C4⋊C414yes—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
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⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8—81no—
248#103—22no—
249#103—41no—
250#103C4⋊C414yes—
251#102—22no—
252#102—41no—
253#102C4⋊C414yes—
254C22⋊C8—42no—
255C22⋊C8—42no—
256#97—42no—
257C4×C8—22no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—41no—
269C4×C8—41no—
270C4×C8—41no—
271C4×C8—42no—
272C4×C8C2×D412yes—
273C4×C8C2×Q812yes—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284C24—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8C2×C4⋊C412yes—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4C2×C4⋊C412yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317SD16—44no—
318M16—84no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
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—84no—
357C2×C8C2×C4⋊C412yes—
358C2×C8C2×C4⋊C412yes—
359C2×C8#13112yes—
360C2×C8#13512yes—
361C2×C8#13512yes—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
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—42no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4C2×C4⋊C412yes—
379C4⋊C4C2×C4⋊C412yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—42no—
383C22⋊C4—81no—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C42—22no—
387C42—22no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42C22×D411yescharacteristic
403C42C22×Q811yescharacteristic
404C42H(5,𝔽2)11yescharacteristic
405C422−1+411yescharacteristic
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23C2×C4×D411yescharacteristic
414Q8—22no—
415Q8—22no—
416Q8—41no—
417Q8—41no—
418D4—22no—
419D4—22no—
420D4—41no—
421D4—41no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
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#44412yes—
461C2×C4C22×C4⋊C411yescharacteristic
462C2×C4#52011yescharacteristic
463C2×C4D4211yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4D4×Q811yescharacteristic
466C2×C4#55411yescharacteristic
467C2×C4C2×#13111yescharacteristic
468C2×C4#56811yescharacteristic
469C2×C4#56811yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57711yescharacteristic
475C8—22no—
476C8—22no—
477C8—22no—
478C8—22no—
479C8—22no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—81no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C8—81no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22D4×C4⋊C411yescharacteristic
497C22#293211yescharacteristic
498C22#293511yescharacteristic
499C22C4×#13511yescharacteristic
500C22#294911yescharacteristic
501C22C2×#44411yescharacteristic
502C22#295711yescharacteristic
503C4—22no—
504C4—21no—
505C4—21no—
506C4—41no—
507C4—42no—
508C4—41no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4#290511yescharacteristic
517C4#290611yescharacteristic
518C4#299211yescharacteristic
519C4#299511yescharacteristic
520C4#328211yescharacteristic
521C4#328211yescharacteristic
522C4#338111yescharacteristic
523C4#338111yescharacteristic
524C2—82no—
525C2#1797811yescharacteristic
526C2#1799011yescharacteristic
527C2#1806911yescharacteristic
528C2#1808911yescharacteristic
529C2#2265211yescharacteristic
530C2#2268311yescharacteristic
531C2#2268611yescharacteristic
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
1. #22735C2semidirect
6. #9597C2semidirect
7. #9597C2semidirect
8. #8848C2semidirect
10. #8844C2semidirect
12. #8807C2semidirect
13. #8570C2semidirect
15. #8261C2semidirect

Maximal subgroups

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

subgroupindex
#227352
#227012
#212742
C2×#15822
C2×#15832
#95972
#88482
#88472
#88442
#88432
#88072
#85702
#85692
#82612