Finite Group Databasebeta

← #385839

Subgroups — #385839

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 571 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#11377C212yesmaximal, 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⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—22no—
75C22×C42C2312yes—
76#467—232no—
77C2×#107—24no—
78C2×#107C2×C414yes—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103C2×C418yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—41no—
90C2×C22⋊C8—41no—
91#391—24no—
92#391—42no—
93#380—24no—
94#380—42no—
95C4×C22⋊C4—41no—
96C4×C22⋊C4—41no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D412yes—
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—22no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4Q1614yescomplemented
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—44no—
225#107—28no—
226#103—216no—
227#103—48no—
228#103—48no—
229C22⋊C8—44no—
230#95—42no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—82no—
237#95—82no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—21no—
243C24—21no—
244C24—22no—
245C24—22no—
246C24C4×D412yes—
247C24C2×Q1612yes—
248C24#13512yes—
249C24C22×D411yescharacteristic
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—24no—
264C2×Q8—24no—
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×Q8—42no—
276C2×Q8—42no—
277C2×Q8C2×C22⋊C418yes—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
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—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
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×C4—42no—
340C22×C4—42no—
341C22×C4C4×D412yes—
342C22×C4C2×Q1614yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C2×#4112yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C42−1+411yescharacteristic
353Q16—416no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—416no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8C2×C22⋊C414yes—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—416no—
377C4⋊C4—416no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—416no—
388C22⋊C4—44no—
389C22⋊C4—88no—
390C42—416no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23C4×Q1614yes—
418C23C2×C4×D411yescharacteristic
419C23#53411yescharacteristic
420C23#53711yescharacteristic
421C23D4212yes—
422C23#54612yes—
423C23C22×Q1611yescharacteristic
424C23C2×#13511yescharacteristic
425C23#57511yescharacteristic
426C23#57511yescharacteristic
427C23#57712yes—
428C23#57812yes—
429Q8—28no—
430Q8—28no—
431Q8—44no—
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437C2×C4—22no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
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×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4C22×C22⋊C411yescharacteristic
491C2×C4C4×#4111yescharacteristic
492C2×C4#51811yescharacteristic
493C2×C4#51811yescharacteristic
494C2×C4#51811yescharacteristic
495C2×C4#52011yescharacteristic
496C2×C4#53412yes—
497C2×C4#53412yes—
498C2×C4#53812yes—
499C2×C4#53811yescharacteristic
500C2×C4#53911yescharacteristic
501C2×C4#54012yes—
502C2×C4#54612yes—
503C2×C4#54612yes—
504C2×C4#54712yes—
505C2×C4#55312yes—
506C2×C4#56012yes—
507C2×C4#56212yes—
508C8—24no—
509C8—42no—
510C8—82no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—44no—
517C22—44no—
518C22—42no—
519C22—42no—
520C22—44no—
521C22—44no—
522C22#232511yescharacteristic
523C22D4×C22⋊C411yescharacteristic
524C22#236411yescharacteristic
525C22#266812yes—
526C22#267112yes—
527C22C2×C4×Q1611yescharacteristic
528C22#294311yescharacteristic
529C22#295011yescharacteristic
530C22#295211yescharacteristic
531C22#318911yescharacteristic
532C22#319111yescharacteristic
533C22#320112yes—
534C22#324911yescharacteristic
535C22#325011yescharacteristic
536C22#325812yes—
537C22#327112yes—
538C22#327112yes—
539C22#328512yes—
540C22D4×Q1612yes—
541C22#329612yes—
542C22#329912yes—
543C22#331312yes—
544C22#331512yes—
545C4—42no—
546C4—48no—
547C4—44no—
548C4—84no—
549C4—82no—
550C4—84no—
551C4—162no—
552C4#230811yescharacteristic
553C4#233311yescharacteristic
554C4#233911yescharacteristic
555C4#237711yescharacteristic
556C4#268812yes—
557C4#270812yes—
558C2—44no—
559C2#1441711yescharacteristic
560C2#2218211yescharacteristic
561C2#2219211yescharacteristic
562C2#2244111yescharacteristic
563C2C22⋊C4×Q1611yescharacteristic
564C2#2312811yescharacteristic
565C2#2313611yescharacteristic
566C2#3040512yes—
567C2#3040712yes—
568C2#3059812yes—
569C2#3060012yes—
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. #11377C2semidirect
5. #11190C2semidirect
7. #10789C2semidirect
15. C2×#467C4semidirect
149. C2×Q16C22⋊C4semidirect
215. C2×C22⋊C4Q16semidirect

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
#113772
#111902
#109702
#107892
#107212
#107142