Finite Group Databasebeta

← #385946

Subgroups — #385946

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1918C212yesmaximal, complemented
3#13997C212yesmaximal
4#11067C212yesmaximal, complemented
5#11024C212yesmaximal, complemented
6#10970C212yesmaximal
7#10932C212yesmaximal, complemented
8#10721C211yesmaximal, characteristic
9#10714C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C2×#467—28no—
14C2×#467C418yescomplemented
15C22×#107C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1918—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1445—22no—
35#1439—22no—
36#1437—22no—
37C4⋊#95—22no—
38C23×Q8C2312yes—
39C24×C4C2311yescharacteristic
40C22×Q16—24no—
41C22×Q16—41no—
42C22×Q16—42no—
43C22×Q16—41no—
44C22×Q16C2×C414yes—
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—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D412yes—
67C22×C42—22no—
68C22×C42C2312yes—
69#467—232no—
70C2×#107C2×C414yes—
71C2×#107D418yes—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103C2×C418yes—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—41no—
83C2×C22⋊C8—41no—
84#391—24no—
85#391—42no—
86#391—42no—
87#391—42no—
88#389—24no—
89#389—42no—
90#380—24no—
91#380—42no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—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—24no—
107#340—24no—
108#340—41no—
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×Q8—22no—
125C22×Q8—22no—
126C22×Q8C22×C414yes—
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—28no—
146C2×Q16—44no—
147C2×Q16—44no—
148C2×Q16—48no—
149C2×Q16—48no—
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—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—82no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—82no—
218#107C22⋊C4116yescomplemented
219#103—216no—
220#103—48no—
221#103—48no—
222C22⋊C8—44no—
223#95—41no—
224#95—41no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24C4×D412yes—
242C24#13714yes—
243C24C22×D411yescharacteristic
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
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—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
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—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
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×C4C4×D412yes—
336C22×C4#13514yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C42−1+411yescharacteristic
347Q16—416no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—416no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8C2×C22⋊C414yes—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—416no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—88no—
379C4⋊C4C2×C22⋊C418yes—
380C22⋊C4—48no—
381C22⋊C4—416no—
382C22⋊C4—88no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—88no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23#44114yes—
411C23C2×C4×D411yescharacteristic
412C23#53411yescharacteristic
413C23#53711yescharacteristic
414C23D4212yes—
415C23#54612yes—
416C23C2×#13512yes—
417C23C2×#13712yes—
418C23#57611yescharacteristic
419C23#57811yescharacteristic
420C23#57811yescharacteristic
421C23#57811yescharacteristic
422Q8—44no—
423Q8—44no—
424Q8—44no—
425Q8—44no—
426Q8—82no—
427Q8—82no—
428Q8—82no—
429Q8—82no—
430C2×C4—22no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
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×C4C22×C22⋊C411yescharacteristic
484C2×C4C4×#4111yescharacteristic
485C2×C4#51811yescharacteristic
486C2×C4#51811yescharacteristic
487C2×C4#51811yescharacteristic
488C2×C4#52011yescharacteristic
489C2×C4#53512yes—
490C2×C4#53512yes—
491C2×C4#53812yes—
492C2×C4#53811yescharacteristic
493C2×C4#53911yescharacteristic
494C2×C4#54012yes—
495C2×C4#54612yes—
496C2×C4#54812yes—
497C2×C4#55012yes—
498C2×C4#55312yes—
499C2×C4#56012yes—
500C2×C4#56212yes—
501C8—24no—
502C8—42no—
503C8—82no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22#232511yescharacteristic
516C22D4×C22⋊C411yescharacteristic
517C22#236411yescharacteristic
518C22#267312yes—
519C22#267612yes—
520C22C4×#13511yescharacteristic
521C22C2×#44111yescharacteristic
522C22#295011yescharacteristic
523C22#295111yescharacteristic
524C22#319012yes—
525C22#320211yescharacteristic
526C22#320311yescharacteristic
527C22#324812yes—
528C22#326111yescharacteristic
529C22#326211yescharacteristic
530C22#327012yes—
531C22#327012yes—
532C22#328012yes—
533C22#328312yes—
534C22#330012yes—
535C22#330212yes—
536C22#331212yes—
537C22#331412yes—
538C4—42no—
539C4—48no—
540C4—44no—
541C4—84no—
542C4—84no—
543C4—82no—
544C4—162no—
545C4#230911yescharacteristic
546C4#233311yescharacteristic
547C4#234111yescharacteristic
548C4#237711yescharacteristic
549C4#271112yes—
550C4#271412yes—
551C2—44no—
552C2#1441811yescharacteristic
553C2#2218311yescharacteristic
554C2#2219711yescharacteristic
555C2#2244011yescharacteristic
556C2#2244811yescharacteristic
557C2#2312911yescharacteristic
558C2#2313311yescharacteristic
559C2#3070912yes—
560C2#3071212yes—
561C2#3091012yes—
562C2#3091212yes—
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×#1918C2semidirect
4. #11067C2semidirect
5. #11024C2semidirect
7. #10932C2semidirect
14. C2×#467C4semidirect
218. #107C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#19182
#139972
#110672
#110242
#109702
#109322
#107212
#107142