Finite Group Databasebeta

← #230885

Subgroups — #230885

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23677C211yesmaximal, characteristic
2#23149C211yesmaximal, characteristic, complemented
3#22836C211yesmaximal, characteristic
4#9661C211yesmaximal, characteristic
5#9660C211yesmaximal, characteristic
6#9647C211yesmaximal, characteristic, complemented
7#9603C211yesmaximal, characteristic, complemented
8#9548C211yesmaximal, characteristic, complemented
9#9543C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic
11#8948C211yesmaximal, characteristic
12#8263C211yesmaximal, characteristic, complemented
13#8203C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15#8181C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21#2293C2211yescharacteristic
22#2084C411yescharacteristic
23#2071C411yescharacteristic
24#2066C411yescharacteristic
25#2066C411yescharacteristic
26#2055C411yescharacteristic
27#2055C411yescharacteristic
28#2043C411yescharacteristic
29#2013C411yescharacteristic
30#1984C2211yescharacteristic
31#1984C2211yescharacteristic
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1939—21no—
35#1924—21no—
36#1923—21no—
37#1916—21no—
38#1916—21no—
39#1831C2211yescharacteristic
40#1818—21no—
41#1770C2211yescharacteristic
42C8×C4⋊C4—21no—
43C8×C4⋊C4C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49#1750—21no—
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C8×Q8—22no—
75C8×Q8—22no—
76C8×Q8—41no—
77C8×Q8—41no—
78C4×SD16—41no—
79C4×SD16—41no—
80C4×SD16—41no—
81C4×SD16—41no—
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2×C411yescharacteristic
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#389—21no—
124#389—21no—
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#388—21no—
128#388—21no—
129#384C2×C411yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C43C2311yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#336—21no—
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#336D411yescharacteristic
151#336D411yescharacteristic
152#335—41no—
153#334—41no—
154Q8⋊C8—22no—
155Q8⋊C8—22no—
156Q8⋊C8—41no—
157Q8⋊C8—41no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160#322—41no—
161#322—41no—
162C22×Q8C22×C411yescharacteristic
163C22×Q8C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C2×SD16—22no—
167C2×SD16—22no—
168C2×SD16—22no—
169C2×SD16—41no—
170C2×SD16—41no—
171C2×SD16—41no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
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⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#106—41no—
231#106—41no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—41no—
242C4⋊C8—41no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8#4112yes—
246#103—22no—
247#103—22no—
248#103—22no—
249#103—41no—
250#103—41no—
251#103—41no—
252#102—41no—
253#102—41no—
254C22⋊C8—42no—
255C22⋊C8—42no—
256C22⋊C8—81no—
257C22⋊C8—81no—
258#97—42no—
259#97—42no—
260#97—81no—
261C4×C8—21no—
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—81no—
276C4×C8#4112yes—
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×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8#13112yes—
296C2×Q8#13112yes—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323SD16—44no—
324SD16—44no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
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—21no—
348C2×C8—21no—
349C2×C8—21no—
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—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8#13112yes—
366C2×C8#13712yes—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—81no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—42no—
390C22⋊C4—81no—
391C22⋊C4—81no—
392C22⋊C4—81no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—22no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C42H(5,𝔽2)11yescharacteristic
412C42H(5,𝔽2)11yescharacteristic
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23C2×C4×D411yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—22no—
424Q8—22no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—41no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
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—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4#44412yes—
472C2×C4C4×#4111yescharacteristic
473C2×C4#51811yescharacteristic
474C2×C4#54611yescharacteristic
475C2×C4#55011yescharacteristic
476C2×C4#55311yescharacteristic
477C2×C4#55511yescharacteristic
478C2×C4C2×#13111yescharacteristic
479C2×C4C2×#13111yescharacteristic
480C2×C4C2×#13111yescharacteristic
481C2×C4#56811yescharacteristic
482C2×C4C2×#13711yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57711yescharacteristic
486C8—22no—
487C8—22no—
488C8—42no—
489C8—42no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—42no—
496C8—41no—
497C8—81no—
498C8—81no—
499C8—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22#236311yescharacteristic
508C22#292911yescharacteristic
509C22#293511yescharacteristic
510C22C4×#13711yescharacteristic
511C22#294911yescharacteristic
512C22C2×#44411yescharacteristic
513C22#295611yescharacteristic
514C4—21no—
515C4—21no—
516C4—22no—
517C4—41no—
518C4—42no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4#296611yescharacteristic
528C4#296811yescharacteristic
529C4#297611yescharacteristic
530C4#297711yescharacteristic
531C4#330711yescharacteristic
532C4#332011yescharacteristic
533C4#340011yescharacteristic
534C4#340111yescharacteristic
535C2—161no—
536C2#1791511yescharacteristic
537C2#1792311yescharacteristic
538C2#1807711yescharacteristic
539C2#1808911yescharacteristic
540C2#2308911yescharacteristic
541C2#2311211yescharacteristic
542C2#2311411yescharacteristic
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
2. #23149C2semidirect
6. #9647C2semidirect
7. #9603C2semidirect
8. #9548C2semidirect
9. #9543C2semidirect
12. #8263C2semidirect
13. #8203C2semidirect
14. #8201C2semidirect

Maximal subgroups

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

subgroupindex
#236772
#231492
#228362
#96612
#96602
#96472
#96032
#95482
#95432
#89482
#82632
#82032
#82012
#81812