Finite Group Databasebeta

← #385841

Subgroups — #385841

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1878C212yesmaximal, complemented
3#13969C212yesmaximal
4#11379C212yesmaximal, complemented
5#11190C212yesmaximal, complemented
6#10970C212yesmaximal
7#10789C212yesmaximal, complemented
8#10721C211yesmaximal, characteristic
9#10714C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2212yes—
14C2×#467—28no—
15C2×#467C418yescomplemented
16C22×#107C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C2×#391C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1878—28no—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1441—22no—
33#1439—22no—
34#1437—22no—
35#1436—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C23×Q8C2312yes—
39C24×C4C2311yescharacteristic
40C22×Q16—41no—
41C22×Q16—42no—
42C22×Q16—41no—
43C22×Q16C2×C414yes—
44C22×Q16D418yes—
45C23×C8—24no—
46C23×C8—42no—
47C23×C8—41no—
48C23×C8—41no—
49C23×C8C2×C414yes—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8C2×C418yes—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C42—22no—
74C22×C42C2312yes—
75#467—232no—
76C2×#107—24no—
77C2×#107C2×C414yes—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103C2×C418yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—41no—
89C2×C22⋊C8—41no—
90#391—24no—
91#391—42no—
92#380—24no—
93#380—42no—
94C4×C22⋊C4—41no—
95C4×C22⋊C4—41no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105#340—24no—
106#340—41no—
107#340—42no—
108#340—42no—
109#340—41no—
110#336—24no—
111#336—42no—
112C25C2×D411yescharacteristic
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8C22×C414yes—
125C22×Q8C2×D414yes—
126C22×Q8C2×D414yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4#4112yes—
144C23×C4C2411yescharacteristic
145C2×Q16—44no—
146C2×Q16—44no—
147C2×Q16—48no—
148C2×Q16—48no—
149C2×Q16C22⋊C4116yescomplemented
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—28no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8—48no—
168C22×C8—48no—
169C22×C8C22×C412yes—
170C22×C8C22×C412yes—
171C22×C8C2×D414yes—
172C22⋊Q8—216no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C22⋊Q8—48no—
176C4×Q8—48no—
177C4×Q8—48no—
178C4×Q8—48no—
179C4×Q8—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C22×C414yes—
203C2×C4⋊C4#4114yes—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—44no—
222#107—28no—
223#103—216no—
224#103—48no—
225#103—48no—
226C22⋊C8—44no—
227#95—41no—
228#95—41no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—82no—
235#95—82no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24C4×D412yes—
246C24#13714yes—
247C24C22×D411yescharacteristic
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8C2×C22⋊C418yes—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4C4×D412yes—
340C22×C4#13514yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C2×#4112yes—
344C22×C4C2×#4111yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)12yes—
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C42−1+411yescharacteristic
351Q16—416no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—416no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8C2×C22⋊C414yes—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—416no—
375C4⋊C4—416no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C22⋊C4—48no—
385C22⋊C4—416no—
386C22⋊C4—88no—
387C42—416no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23#44114yes—
415C23C2×C4×D411yescharacteristic
416C23#53411yescharacteristic
417C23#53711yescharacteristic
418C23D4212yes—
419C23#54612yes—
420C23C2×#13512yes—
421C23C2×#13712yes—
422C23#57611yescharacteristic
423C23#57811yescharacteristic
424C23#57811yescharacteristic
425C23#57811yescharacteristic
426Q8—28no—
427Q8—28no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—44no—
433Q8—44no—
434C2×C4—22no—
435C2×C4—28no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4C22×C22⋊C411yescharacteristic
488C2×C4C4×#4111yescharacteristic
489C2×C4#51811yescharacteristic
490C2×C4#51811yescharacteristic
491C2×C4#51811yescharacteristic
492C2×C4#52011yescharacteristic
493C2×C4#53412yes—
494C2×C4#53412yes—
495C2×C4#53812yes—
496C2×C4#53811yescharacteristic
497C2×C4#53911yescharacteristic
498C2×C4#54012yes—
499C2×C4#54612yes—
500C2×C4#54612yes—
501C2×C4#54712yes—
502C2×C4#55312yes—
503C2×C4#56012yes—
504C2×C4#56212yes—
505C8—24no—
506C8—42no—
507C8—82no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—44no—
519C22#232511yescharacteristic
520C22D4×C22⋊C411yescharacteristic
521C22#236411yescharacteristic
522C22#266812yes—
523C22#267112yes—
524C22C4×#13511yescharacteristic
525C22C2×#44111yescharacteristic
526C22#295011yescharacteristic
527C22#295111yescharacteristic
528C22#319012yes—
529C22#320211yescharacteristic
530C22#320311yescharacteristic
531C22#324812yes—
532C22#326111yescharacteristic
533C22#326211yescharacteristic
534C22#327012yes—
535C22#327012yes—
536C22#328512yes—
537C22#328712yes—
538C22#330012yes—
539C22#330212yes—
540C22#331212yes—
541C22#331412yes—
542C4—42no—
543C4—48no—
544C4—44no—
545C4—84no—
546C4—82no—
547C4—84no—
548C4—162no—
549C4#230811yescharacteristic
550C4#233311yescharacteristic
551C4#233911yescharacteristic
552C4#237711yescharacteristic
553C4#268812yes—
554C4#270812yes—
555C2—44no—
556C2#1441711yescharacteristic
557C2#2218311yescharacteristic
558C2#2219711yescharacteristic
559C2#2244011yescharacteristic
560C2#2245111yescharacteristic
561C2#2312911yescharacteristic
562C2#2313311yescharacteristic
563C2#3040912yes—
564C2#3041112yes—
565C2#3060212yes—
566C2#3060412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1878C2semidirect
4. #11379C2semidirect
5. #11190C2semidirect
7. #10789C2semidirect
15. C2×#467C4semidirect
149. C2×Q16C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#18782
#139692
#113792
#111902
#109702
#107892
#107212
#107142