Finite Group Databasebeta

← #219116

Subgroups — #219116

Order 512.

585 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22307C211yesmaximal, characteristic, complemented
2#22214C211yesmaximal, characteristic
3C4×#489C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic, complemented
5#9639C211yesmaximal, characteristic
6#9544C211yesmaximal, characteristic, complemented
7#9543C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic
9#9541C211yesmaximal, characteristic
10#8850C211yesmaximal, characteristic, complemented
11#8849C211yesmaximal, characteristic
12#8848C211yesmaximal, characteristic, complemented
13#8847C211yesmaximal, characteristic
14#8808C211yesmaximal, characteristic, complemented
15#8132C211yesmaximal, characteristic, complemented
16C2×#489C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×#124C2211yescharacteristic
21#1993C411yescharacteristic
22#1984—21no—
23#1984—21no—
24#1984C2211yescharacteristic
25#1959C411yescharacteristic
26#1954C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1936C411yescharacteristic
30#1936C411yescharacteristic
31#1935C411yescharacteristic
32#1935C411yescharacteristic
33#1924—21no—
34#1924—21no—
35#1916—21no—
36#1916—21no—
37#1851—21no—
38#1851—21no—
39#1842C2211yescharacteristic
40#1836C411yescharacteristic
41#1831C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#124C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#489—24no—
74#446—41no—
75#446—41no—
76#436—41no—
77#436—41no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
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×#97—21no—
99C2×#97C2×C411yescharacteristic
100C2×#97C2×C411yescharacteristic
101C2×#97C2×C411yescharacteristic
102C2×#97C2×C411yescharacteristic
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#391C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#389—21no—
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#388—21no—
120#388—21no—
121#384C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—41no—
150#335—41no—
151#334—41no—
152#334—41no—
153Q8⋊C8—41no—
154Q8⋊C8—41no—
155D4⋊C8—41no—
156D4⋊C8—41no—
157C22×Q8—21no—
158C22×Q8C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C2×M16—42no—
162C2×M16—81no—
163C2×M16—81no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—22no—
179C4⋊Q8M1614yescomplemented
180#124—22no—
181#124—22no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—81no—
187C4×Q8—81no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—42no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8—81no—
231C4⋊C8—81no—
232#103—22no—
233#103—22no—
234#103—22no—
235#103—22no—
236#102—22no—
237#102—22no—
238#102—22no—
239#102—22no—
240C22⋊C8—42no—
241C22⋊C8—81no—
242C22⋊C8—81no—
243#97—22no—
244#97—22no—
245#97—22no—
246#97—22no—
247#97—42no—
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—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—42no—
261C4×C8—42no—
262C4×C8—81no—
263C4×C8—81no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24—21no—
273C2×Q8—24no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—81no—
281C2×Q8—81no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308M16—84no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
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—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
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#13112yes—
350C2×C8#13112yes—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—24no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4C2×M1612yes—
369C4⋊C4C2×M1612yes—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—81no—
375C22⋊C4—81no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42C2×M1612yes—
391C42#13112yes—
392C42H(5,𝔽2)11yescharacteristic
393C42H(5,𝔽2)11yescharacteristic
394C422−1+411yescharacteristic
395C422−1+411yescharacteristic
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C2×C4×D411yescharacteristic
404Q8—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—84no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—41no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
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—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4#43512yes—
451C2×C4#43512yes—
452C2×C4#52011yescharacteristic
453C2×C4#52011yescharacteristic
454C2×C4#53511yescharacteristic
455C2×C4#53611yescharacteristic
456C2×C4#54211yescharacteristic
457C2×C4#54211yescharacteristic
458C2×C4C22×M1611yescharacteristic
459C2×C4C2×#13111yescharacteristic
460C2×C4C2×#13111yescharacteristic
461C2×C4C2×#13111yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57811yescharacteristic
466C8—22no—
467C8—22no—
468C8—42no—
469C8—42no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22#232611yescharacteristic
488C22C2×#43511yescharacteristic
489C22#293111yescharacteristic
490C22#294611yescharacteristic
491C22#294711yescharacteristic
492C22#295611yescharacteristic
493C22#295611yescharacteristic
494C4—22no—
495C4—22no—
496C4—21no—
497C4—21no—
498C4—41no—
499C4—41no—
500C4—42no—
501C4—42no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4#299111yescharacteristic
508C4#299111yescharacteristic
509C4#299411yescharacteristic
510C4#299411yescharacteristic
511C4#322411yescharacteristic
512C4#322511yescharacteristic
513C4#324311yescharacteristic
514C4#324411yescharacteristic
515C2—161no—
516C2#1787311yescharacteristic
517C2#1787411yescharacteristic
518C2#1796811yescharacteristic
519C2#1796911yescharacteristic
520C2#2221011yescharacteristic
521C2#2223311yescharacteristic
522C2#2223711yescharacteristic
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 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22307C2semidirect
4. #9870C2semidirect
6. #9544C2semidirect
7. #9543C2semidirect
10. #8850C2semidirect
12. #8848C2semidirect
14. #8808C2semidirect
15. #8132C2semidirect
179. C4⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
#223072
#222142
C4×#4892
#98702
#96392
#95442
#95432
#95422
#95412
#88502
#88492
#88482
#88472
#88082
#81322