Finite Group Databasebeta

← #375149

Subgroups — #375149

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C2×#1875C211yesmaximal, characteristic
3C2×#1873C211yesmaximal, characteristic, complemented
4C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
5C2×#1735C211yesmaximal, characteristic, complemented
6#14138C211yesmaximal, characteristic, complemented
7#14136C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11023C212yesmaximal, complemented
10#10926C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C22×#106C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#102C2212yes—
25C22×#102C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C4×C8C2212yes—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1875—24no—
32#1873—24no—
33C8×C22⋊C4—24no—
34#1735—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1440—22no—
41#1438—22no—
42#1437—22no—
43#1436—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×SD16—22no—
51C22×SD16—22no—
52C22×SD16—42no—
53C22×SD16—42no—
54C22×SD16C2×C414yes—
55C23×C8—42no—
56C23×C8—42no—
57C23×C8C2×C414yes—
58C23×C8D414yes—
59C23×C8D414yes—
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8C2×C414yes—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121C2×C414yes—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—42no—
82C2×#117—42no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4D412yes—
94C22×C42—21no—
95C22×C42—21no—
96C22×C42C2312yes—
97#465—232no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106C2×C414yes—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×#103C2×C414yes—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—42no—
112C2×#102C2×C414yes—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8D414yes—
116C2×C4×C8—24no—
117C2×C4×C8—42no—
118C2×C4×C8—42no—
119#391—24no—
120#391—42no—
121#390—24no—
122#390—42no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95D412yes—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#336—24no—
134C25—21no—
135C25C2×D411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C22×C412yes—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—84no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4C22×C412yes—
170C23×C4C2×D414yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C2×SD16—28no—
179C2×SD16—48no—
180C2×SD16—44no—
181C2×SD16—44no—
182C2×SD16—48no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—48no—
194C22×C8—48no—
195C22×C8—44no—
196C22×C8—44no—
197C22×C8C22⋊C4116yescomplemented
198C22×C8C22×C412yes—
199C22×C8C22×C412yes—
200C22×C8C2×D414yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209#121—28no—
210#121—28no—
211#121—44no—
212#121—44no—
213#117—28no—
214#117—28no—
215#117—48no—
216#117—48no—
217#117—44no—
218#117—44no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4#4114yes—
245C2×C4⋊C4#4114yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—84no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42—42no—
270C2×C42—42no—
271#106—28no—
272#103—28no—
273#103—48no—
274#102—28no—
275#102—48no—
276C22⋊C8—28no—
277C4×C8—48no—
278#95—24no—
279#95—48no—
280#95—48no—
281#95—84no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—22no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—42no—
291C24—44no—
292C24—44no—
293C24C4×D412yes—
294C24#12112yes—
295C24#12712yes—
296C24C22×D411yescharacteristic
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—48no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—88no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4C2×C22⋊C414yes—
383C22×C4C4×D414yes—
384C22×C4C4×D412yes—
385C22×C4#12114yes—
386C22×C4#12112yes—
387C22×C4#12414yes—
388C22×C4#12712yes—
389C22×C4#13518yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4111yescharacteristic
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400SD16—416no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—416no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8C2×C22⋊C414yes—
418C2×C8C2×C22⋊C414yes—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C22⋊C4—216no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—816no—
442C42—48no—
443C42—48no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—42no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—42no—
466C23—82no—
467C23—82no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—82no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23—82no—
477C23#39014yes—
478C23C2×C4×D411yescharacteristic
479C23C2×#12111yescharacteristic
480C23C2×#12711yescharacteristic
481C23#54612yes—
482C23#54612yes—
483C23C2×#13514yes—
484C23#57714yes—
485Q8—42no—
486Q8—44no—
487Q8—42no—
488Q8—81no—
489Q8—81no—
490Q8—82no—
491D4—42no—
492D4—44no—
493D4—42no—
494D4—816no—
495D4—81no—
496D4—81no—
497D4—82no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#39014yes—
539C2×C4C22×C22⋊C411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C4×#4112yes—
542C2×C4#51812yes—
543C2×C4C2×#12111yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4C2×#12411yescharacteristic
547C2×C4#53212yes—
548C2×C4#53411yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53412yes—
551C2×C4#54612yes—
552C2×C4#54612yes—
553C2×C4#55312yes—
554C2×C4#55312yes—
555C2×C4#55312yes—
556C2×C4#55312yes—
557C8—24no—
558C8—24no—
559C8—44no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—48no—
564C22—48no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—88no—
569C22—82no—
570C22—82no—
571C22—164no—
572C22—162no—
573C22—162no—
574C22C2×#39011yescharacteristic
575C22#236412yes—
576C22#253911yescharacteristic
577C22#253911yescharacteristic
578C22#254312yes—
579C22C4×#13512yes—
580C22#294912yes—
581C22#305012yes—
582C22#305712yes—
583C22#314712yes—
584C22#315412yes—
585C22#329614yes—
586C22#330114yes—
587C22#330314yes—
588C22#330414yes—
589C4—24no—
590C4—48no—
591C4—82no—
592C4—84no—
593C4—84no—
594C4—82no—
595C4—82no—
596C4#229711yescharacteristic
597C4#230811yescharacteristic
598C4#236412yes—
599C4#254312yes—
600C4#259111yescharacteristic
601C4#259111yescharacteristic
602C2—44no—
603C2—162no—
604C2#1415211yescharacteristic
605C2#2122411yescharacteristic
606C2#2123111yescharacteristic
607C2#2313114yes—
608C2#2833912yes—
609C2#2834012yes—
610C2#2855614yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
3. C2×#1873C2semidirect
4. C2×C8×C22⋊C4C2semidirect
5. C2×#1735C2semidirect
6. #14138C2semidirect
8. #11378C2semidirect
9. #11023C2semidirect
10. #10926C2semidirect
11. #10925C2semidirect
20. C2×#465C4semidirect
197. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C2×#18752
C2×#18732
C2×C8×C22⋊C42
C2×#17352
#141382
#141362
#113782
#110232
#109262
#109252