Finite Group Databasebeta

← #305764

Subgroups — #305764

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26092C211yesmaximal, characteristic, complemented
2#26025C211yesmaximal, characteristic
3#22519C211yesmaximal, characteristic
4#22502C211yesmaximal, characteristic
5#10675C211yesmaximal, characteristic, complemented
6#10589C211yesmaximal, characteristic, complemented
7#10589C211yesmaximal, characteristic, complemented
8#10355C211yesmaximal, characteristic, complemented
9#8889C211yesmaximal, characteristic
10#8887C211yesmaximal, characteristic, complemented
11#8849C211yesmaximal, characteristic
12#8846C211yesmaximal, characteristic, complemented
13#8652C211yesmaximal, characteristic
14#8647C211yesmaximal, characteristic, complemented
15#8450C211yesmaximal, characteristic
16C2×#554C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#436C2211yescharacteristic
19#2552C2211yescharacteristic
20#2355C2211yescharacteristic
21#2354C2211yescharacteristic
22#2095—21no—
23#2068—21no—
24#2067—21no—
25#2025—21no—
26#1991—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1989—21no—
30#1989—21no—
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1956C2211yescharacteristic
34#1947—21no—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1921—21no—
40#1921—21no—
41#1894—21no—
42#1873—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1834C2211yescharacteristic
46#1823—21no—
47#1804—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#103C2211yescharacteristic
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55#1736—21no—
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68#554—22no—
69#554—22no—
70#554—41no—
71#554Q814yes—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#125—21no—
74C2×#125—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—41no—
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C8×Q8—22no—
83C8×Q8—41no—
84#436—22no—
85#436—41no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—41no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—41no—
101C2×#103D411yescharacteristic
102C2×#103D411yescharacteristic
103C2×#102—21no—
104C2×#102—21no—
105C2×#102D411yescharacteristic
106C2×#102D411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#398—41no—
113#391—21no—
114#390—21no—
115#389—21no—
116#389—21no—
117#388—41no—
118#387—41no—
119#384—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#334—41no—
153#334—41no—
154#333—41no—
155Q8⋊C8—22no—
156D4⋊C8—22no—
157C22×Q8—21no—
158C22×Q8C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—41no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8C2×Q812yes—
178#125—24no—
179#125—42no—
180#125—42no—
181#125—81no—
182C22⋊Q8—24no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—42no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8C2×Q812yes—
189C4×Q8#4112yes—
190C4×D4—24no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—42no—
194C4×D4C2×Q812yes—
195C4×D4#4112yes—
196#117—42no—
197#117—81no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
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⋊C4—21no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42C2411yescharacteristic
233#107—42no—
234#107—42no—
235#107—81no—
236#107—81no—
237#106—82no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—81no—
249#103—22no—
250#103—41no—
251#103—81no—
252#103—81no—
253#103—82no—
254#102—22no—
255#102—41no—
256#102—42no—
257#102—81no—
258C22⋊C8—42no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266#95—21no—
267#95—21no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—81no—
282C2×Q8—81no—
283C2×Q8C22⋊Q812yes—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4C22⋊Q812yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—44no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—161no—
309C22×C4—161no—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313M16—84no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
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—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8#13512yes—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—161no—
376C4⋊C4C22⋊Q812yes—
377C4⋊C4C22⋊Q812yes—
378C4⋊C4#13712yes—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—161no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—82no—
391C42—161no—
392C42C22×Q811yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23D4211yescharacteristic
404Q8—22no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408Q8—82no—
409Q8—161no—
410D4—22no—
411D4—22no—
412D4—41no—
413D4—41no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
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—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—164no—
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×C22⋊Q811yescharacteristic
456C2×C4#54311yescharacteristic
457C2×C4#54611yescharacteristic
458C2×C4#54611yescharacteristic
459C2×C4#55111yescharacteristic
460C2×C4C2×#13511yescharacteristic
461C2×C4C2×#13711yescharacteristic
462C2×C4#57511yescharacteristic
463C2×C4#57511yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—22no—
469C8—42no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22#243711yescharacteristic
485C22#326711yescharacteristic
486C22#327011yescharacteristic
487C22#327511yescharacteristic
488C22#328411yescharacteristic
489C22#328911yescharacteristic
490C22#329211yescharacteristic
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—82no—
500C4—81no—
501C4—161no—
502C4—161no—
503C4—161no—
504C4#307611yescharacteristic
505C4#307811yescharacteristic
506C4#308211yescharacteristic
507C4#308411yescharacteristic
508C4#329611yescharacteristic
509C4#330711yescharacteristic
510C4#331311yescharacteristic
511C4#332911yescharacteristic
512C2—82no—
513C2#1988111yescharacteristic
514C2#1994911yescharacteristic
515C2#2011411yescharacteristic
516C2#2025111yescharacteristic
517C2#2596411yescharacteristic
518C2#2597911yescharacteristic
519C2#2598211yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26092C2semidirect
5. #10675C2semidirect
6. #10589C2semidirect
7. #10589C2semidirect
8. #10355C2semidirect
10. #8887C2semidirect
12. #8846C2semidirect
14. #8647C2semidirect

Maximal subgroups

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

subgroupindex
#260922
#260252
#225192
#225022
#106752
#105892
#103552
#88892
#88872
#88492
#88462
#86522
#86472
#84502