Finite Group Databasebeta

← #374620

Subgroups — #374620

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#424C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1987C211yesmaximal, characteristic, complemented
5#13777C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11227C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10722C212yesmaximal
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#424—24no—
17C2×#424—24no—
18C2×#424—24no—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23#2628—24no—
24#2628C418yescomplemented
25C2×#400—21no—
26C2×#399—21no—
27C2×#395—21no—
28C2×#393—21no—
29C2×#389C2212yes—
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1988—24no—
36#1987—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1437—21no—
46#1437—21no—
47#1436—21no—
48#1436—21no—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—42no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—42no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C414yes—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117C2×C414yes—
77C2×#117D414yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#424—232no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8D414yes—
106C2×C4⋊C8D414yes—
107#400—28no—
108#399—28no—
109#395—28no—
110#393—28no—
111#389—24no—
112#389C2×C418yes—
113#382—22no—
114#382C2×C414yes—
115#380—24no—
116#380—24no—
117#380C2×C418yes—
118#380Q818yescomplemented
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—42no—
137#340—42no—
138#340—42no—
139#340—42no—
140C25C2×D411yescharacteristic
141C22×Q8—24no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—82no—
145C22×Q8—82no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C22×C412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×Q814yes—
164C23×C4#4114yes—
165C23×C4C2411yescharacteristic
166C2×M16—28no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—44no—
170C2×M16—44no—
171C22×C8—28no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—48no—
189C22×C8—48no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C4⋊Q8—28no—
197C4⋊Q8—28no—
198C4⋊Q8—44no—
199C4⋊Q8—44no—
200#117—28no—
201#117—28no—
202#117—28no—
203#117—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C22⋊C414yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C2×Q814yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4C22⋊C414yes—
251C2×C22⋊C4C22⋊C414yes—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—82no—
259C2×C42—82no—
260C2×C42C22×C412yes—
261C2×C42C22×C412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C4⋊C8—28no—
269C4⋊C8—28no—
270C4⋊C8—28no—
271C4⋊C8—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—42no—
278#95—42no—
279#95—82no—
280#95—82no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C2×C22⋊C412yes—
289C24C22⋊Q814yes—
290C24C22×D411yescharacteristic
291C2×Q8—48no—
292C2×Q8—88no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—88no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
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×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C4×Q818yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4C22⋊Q814yes—
365C22×C4#12414yes—
366C22×C4C4⋊Q814yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q812yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+412yes—
377M16—416no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—416no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—88no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—84no—
428C42—84no—
429C42C4×D414yes—
430C42C4×D414yes—
431C42#12014yes—
432C42#12314yes—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23#39114yes—
454C23C22×C22⋊C411yescharacteristic
455C23C2×#12111yescharacteristic
456C23C2×#12111yescharacteristic
457C23C2×C22⋊Q812yes—
458C23#53712yes—
459C23#57612yes—
460C23#57712yes—
461C23#57712yes—
462C23#57812yes—
463Q8—816no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#38614yes—
521C2×C4#38614yes—
522C2×C4#39114yes—
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×C4×Q812yes—
526C2×C4#52012yes—
527C2×C4C2×#12011yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×C4⋊Q811yescharacteristic
531C2×C4#53511yescharacteristic
532C2×C4#53611yescharacteristic
533C2×C4#54612yes—
534C2×C4#54612yes—
535C2×C4D4×Q812yes—
536C2×C4D4×Q812yes—
537C2×C4#55212yes—
538C2×C4#55212yes—
539C2×C4#55412yes—
540C2×C4#55412yes—
541C8—42no—
542C8—42no—
543C8—42no—
544C8—42no—
545C22—24no—
546C22—28no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22C2×#38611yescharacteristic
555C22C2×#39111yescharacteristic
556C22#231111yescharacteristic
557C22#241712yes—
558C22#241712yes—
559C22#289712yes—
560C22#289812yes—
561C22#305711yescharacteristic
562C22#305711yescharacteristic
563C22#305811yescharacteristic
564C22#305911yescharacteristic
565C22#306212yes—
566C22#306312yes—
567C22#308112yes—
568C22#308212yes—
569C22#308312yes—
570C22#308412yes—
571C22#325712yes—
572C22#325812yes—
573C22#326212yes—
574C22#326312yes—
575C4—24no—
576C4—24no—
577C4—24no—
578C4—42no—
579C4—42no—
580C4—84no—
581C4—84no—
582C4—84no—
583C4—84no—
584C4#236012yes—
585C4#236012yes—
586C4#243411yescharacteristic
587C4#243511yescharacteristic
588C4#250211yescharacteristic
589C4#250311yescharacteristic
590C2—44no—
591C2#1413911yescharacteristic
592C2#2097411yescharacteristic
593C2#2097511yescharacteristic
594C2#2131512yes—
595C2#2189612yes—
596C2#2484712yes—
597C2#2484912yes—
598C2#2485112yes—
599C2#2485312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1988C2semidirect
4. C2×#1987C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect
8. #10850C2semidirect
9. #10850C2semidirect
24. #2628C4semidirect
118. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#26282
C2×#19882
C2×#19872
#137772
#112282
#112272
#108502
#107222
#107212