Finite Group Databasebeta

← #313605

Subgroups — #313605

Order 512.

685 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27814C211yesmaximal, characteristic
2#26112C211yesmaximal, characteristic, complemented
3#25888C211yesmaximal, characteristic
4Q8×#102C211yesmaximal, characteristic
5#10622C211yesmaximal, characteristic
6#10541C211yesmaximal, characteristic, complemented
7#10413C211yesmaximal, characteristic
8#10357C211yesmaximal, characteristic, complemented
9#9821C211yesmaximal, characteristic, complemented
10#9566C211yesmaximal, characteristic, complemented
11#9554C211yesmaximal, characteristic
12#9216C211yesmaximal, characteristic, complemented
13#8975C211yesmaximal, characteristic, complemented
14#8447C211yesmaximal, characteristic
15C8×#103C211yesmaximal, characteristic, complemented
16C2×D4×Q8C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20#2563C2211yescharacteristic
21#2552C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23#2101—21no—
24#2100—21no—
25#2087—21no—
26#2036—21no—
27#2036—21no—
28#2025—21no—
29#1991—21no—
30#1990—21no—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1988—21no—
35#1988C2211yescharacteristic
36#1987C2211yescharacteristic
37#1981C2211yescharacteristic
38#1956C2211yescharacteristic
39#1954C2211yescharacteristic
40#1947C2211yescharacteristic
41#1939C2211yescharacteristic
42#1934—21no—
43#1863—21no—
44#1834C2211yescharacteristic
45#1803—21no—
46#1803C2211yescharacteristic
47#1802—21no—
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1787—21no—
51#1787—21no—
52C8×C4⋊C4C2211yescharacteristic
53C4×#103C2211yescharacteristic
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×Q8—21no—
68D4×Q8—22no—
69D4×Q8—22no—
70D4×Q8—22no—
71D4×Q8—41no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×#125—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—41no—
78C2×C4×Q8—21no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82#487—22no—
83#487—41no—
84#477—22no—
85#477—22no—
86C8×Q8—22no—
87C8×Q8—41no—
88C2×#107—21no—
89C2×#107—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—41no—
103C2×#103D411yescharacteristic
104C2×#103D411yescharacteristic
105C2×#102—21no—
106C2×#102—21no—
107C2×#102D411yescharacteristic
108C2×#102D411yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#398—41no—
117#393—41no—
118#390—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—41no—
123#384—21no—
124#384—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#382—21no—
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
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#340—21no—
147#340—41no—
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#340D411yescharacteristic
152#336—21no—
153#336—21no—
154#333—41no—
155#332—22no—
156Q8⋊C8—22no—
157D4⋊C8—41no—
158C82—41no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—41no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—81no—
167C22×Q8—81no—
168C22×Q8C2×D411yescharacteristic
169C22×D4C2×D411yescharacteristic
170C23×C4—21no—
171C23×C4—41no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8—24no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—42no—
190C4⋊Q8#4112yes—
191#125—42no—
192#125—81no—
193C22⋊Q8—44no—
194C22⋊Q8—42no—
195C22⋊Q8—44no—
196C22⋊Q8—42no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—42no—
200C4×Q8—81no—
201C4×Q8#4112yes—
202C4×Q8#4112yes—
203C4×D4—24no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—42no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—41no—
240C2×C42C2411yescharacteristic
241#107—42no—
242#107—81no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253#103—22no—
254#103—22no—
255#103—42no—
256#103—42no—
257#103—42no—
258#103—42no—
259#103—81no—
260#103—81no—
261#103—82no—
262#102—41no—
263#102—42no—
264#102—81no—
265#102Q1614yes—
266C22⋊C8—81no—
267C22⋊C8—81no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—42no—
275C4×C8—81no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—22no—
287C2×Q8—44no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—81no—
298C2×Q8—81no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8#12412yes—
304C2×Q8#13512yes—
305C2×Q8#13512yes—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4C2×Q1612yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—44no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335C2×C8—21no—
336C2×C8—21no—
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—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—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8C2×Q1612yes—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—44no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—82no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—161no—
397C4⋊C4—161no—
398C4⋊C4#12412yes—
399C4⋊C4C2×Q1612yes—
400C22⋊C4—44no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—161no—
404C22⋊C4—161no—
405C42—22no—
406C42—22no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—82no—
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C42H(5,𝔽2)11yescharacteristic
416C23—22no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23D4211yescharacteristic
424Q8—22no—
425Q8—22no—
426Q8—22no—
427Q8—42no—
428Q8—42no—
429Q8—88no—
430Q8—82no—
431Q8—81no—
432Q8—81no—
433Q8—161no—
434Q8#48814yes—
435D4—22no—
436D4—22no—
437D4—41no—
438D4—41no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
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—88no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4#52511yescharacteristic
480C2×C4C2×#12411yescharacteristic
481C2×C4#53811yescharacteristic
482C2×C4#54611yescharacteristic
483C2×C4#54611yescharacteristic
484C2×C4#55111yescharacteristic
485C2×C4C22×Q1611yescharacteristic
486C2×C4C2×#13511yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4#57511yescharacteristic
489C2×C4#57611yescharacteristic
490C2×C4#57711yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57811yescharacteristic
493C8—22no—
494C8—42no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22#247211yescharacteristic
510C22#327511yescharacteristic
511C22#328011yescharacteristic
512C22#328211yescharacteristic
513C22D4×Q1611yescharacteristic
514C22#329011yescharacteristic
515C22#329111yescharacteristic
516C4—24no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—82no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—161no—
527C4—161no—
528C4—161no—
529C4#310411yescharacteristic
530C4#312111yescharacteristic
531C4C2×#48811yescharacteristic
532C4#313611yescharacteristic
533C4#329611yescharacteristic
534C4#329711yescharacteristic
535C4#329911yescharacteristic
536C4#331311yescharacteristic
537C2—82no—
538C2#1990411yescharacteristic
539C2#1991811yescharacteristic
540C2#1999311yescharacteristic
541C2#2000111yescharacteristic
542C2#2722811yescharacteristic
543C2#2725611yescharacteristic
544C2#2728911yescharacteristic
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
2. #26112C2semidirect
6. #10541C2semidirect
8. #10357C2semidirect
9. #9821C2semidirect
10. #9566C2semidirect
12. #9216C2semidirect
13. #8975C2semidirect
15. C8×#103C2semidirect

Maximal subgroups

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

subgroupindex
#278142
#261122
#258882
Q8×#1022
#106222
#105412
#104132
#103572
#98212
#95662
#95542
#92162
#89752
#84472
C8×#1032