Finite Group Databasebeta

← #379007

Subgroups — #379007

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1971C211yesmaximal, characteristic, complemented
3C2×#1935C212yesmaximal, complemented
4C2×#1892C212yesmaximal
5C2×#1752C211yesmaximal, characteristic
6#13997C212yesmaximal
7#11327C211yesmaximal, characteristic, complemented
8#11095C211yesmaximal, characteristic
9#11024C211yesmaximal, characteristic, complemented
10#10932C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×#106C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#2001C418yescomplemented
29#1971—24no—
30#1935—28no—
31#1892—28no—
32#1752—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1445—22no—
41#1439—22no—
42#1437—22no—
43C4⋊#95—22no—
44C23×Q8C2312yes—
45C24×C4C2311yescharacteristic
46C22×Q16—22no—
47C22×Q16—22no—
48C22×Q16—42no—
49C22×Q16—42no—
50C22×Q16C2×C414yes—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C2×C22⋊Q8C2×C418yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—22no—
78C2×#106—42no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103C2×C418yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8C2×C414yes—
91C2×#97—24no—
92C2×#97—22no—
93C2×#97—42no—
94C2×#97D414yes—
95#391—24no—
96#391—24no—
97#391—24no—
98#391—42no—
99#389—24no—
100#389—42no—
101#380—24no—
102#380—42no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—42no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D412yes—
115#340—22no—
116#340—24no—
117#340—24no—
118#340C2×C414yes—
119#340D418yes—
120#336—22no—
121#336—22no—
122C25C2×D411yescharacteristic
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8C22×C414yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C22×C412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C2×Q16—28no—
156C2×Q16—48no—
157C2×Q16—44no—
158C2×Q16—44no—
159C2×Q16—48no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—28no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—48no—
175C22×C8—48no—
176C22×C8—44no—
177C22×C8—44no—
178C22×C8C22×C412yes—
179C22×C8C22×C412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D414yes—
182C22×C8C2×D412yes—
183C22×C8#4112yes—
184C22×C8#4112yes—
185C22⋊Q8—216no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
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—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—82no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—82no—
231C2×C42C2×D412yes—
232C2×C42#4112yes—
233#106—44no—
234#103—216no—
235#103—48no—
236#103—48no—
237C22⋊C8—28no—
238#97—44no—
239#97—44no—
240#95—24no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24C4×D412yes—
256C24#13714yes—
257C24C22×D411yescharacteristic
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
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×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
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×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4C4×D412yes—
350C22×C4#12114yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)12yes—
361Q16—416no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—416no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8C4×D414yes—
382C2×C8#12714yes—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—416no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—88no—
396C4⋊C4C4×D418yes—
397C22⋊C4—48no—
398C22⋊C4—416no—
399C22⋊C4—88no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—41no—
404C42—41no—
405C42—42no—
406C42—88no—
407C23—24no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23#44114yes—
428C23C2×C4×D411yescharacteristic
429C23C2×#12111yescharacteristic
430C23#53211yescharacteristic
431C23D4212yes—
432C23#54612yes—
433C23C2×#13712yes—
434C23#57512yes—
435C23#57712yes—
436C23#57812yes—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—82no—
442Q8—82no—
443Q8—82no—
444Q8—82no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D412yes—
500C2×C4C4×#4111yescharacteristic
501C2×C4#51812yes—
502C2×C4#52511yescharacteristic
503C2×C4#52911yescharacteristic
504C2×C4C2×#12711yescharacteristic
505C2×C4#53211yescharacteristic
506C2×C4#53212yes—
507C2×C4#53411yescharacteristic
508C2×C4#53511yescharacteristic
509C2×C4#53812yes—
510C2×C4#54612yes—
511C2×C4#54612yes—
512C2×C4#55012yes—
513C2×C4#55112yes—
514C2×C4#55312yes—
515C2×C4#55512yes—
516C8—24no—
517C8—24no—
518C8—42no—
519C8—42no—
520C22—24no—
521C22—28no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22C4×#12111yescharacteristic
530C22#235812yes—
531C22#254111yescharacteristic
532C22#254311yescharacteristic
533C22#255512yes—
534C22C2×#44111yescharacteristic
535C22#294311yescharacteristic
536C22#294712yes—
537C22#305612yes—
538C22#306312yes—
539C22#317311yescharacteristic
540C22#317711yescharacteristic
541C22#318112yes—
542C22#326612yes—
543C22#326712yes—
544C22#326912yes—
545C22#327012yes—
546C22#331112yes—
547C22#331212yes—
548C22#331512yes—
549C22#331612yes—
550C4—48no—
551C4—44no—
552C4—42no—
553C4—84no—
554C4—82no—
555C4—84no—
556C4—162no—
557C4C4×#12711yescharacteristic
558C4#232111yescharacteristic
559C4#236312yes—
560C4#253911yescharacteristic
561C4#255412yes—
562C4#261011yescharacteristic
563C2—44no—
564C2#1422811yescharacteristic
565C2#2152711yescharacteristic
566C2#2154211yescharacteristic
567C2#2285612yes—
568C2#2285912yes—
569C2#2849212yes—
570C2#2855812yes—
571C2#2892912yes—
572C2#2893412yes—
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×#1971C2semidirect
3. C2×#1935C2semidirect
7. #11327C2semidirect
9. #11024C2semidirect
10. #10932C2semidirect
11. #10703C2semidirect
28. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#19712
C2×#19352
C2×#18922
C2×#17522
#139972
#113272
#110952
#110242
#109322
#107032