Finite Group Databasebeta

← #360349

Subgroups — #360349

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1824C211yesmaximal, characteristic
2C2×C8×C22⋊C4C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#13924C211yesmaximal, characteristic
5#13907C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10820C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2212yes—
17C2×#389C2212yes—
18C2×#380—21no—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#1824—24no—
23C8×C22⋊C4—24no—
24C2×#340C418yescomplemented
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#336C2212yes—
36#1445—22no—
37#1441—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—24no—
50C23×C8—42no—
51C23×C8C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4C2×C414yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42C2312yes—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8D414yes—
82C2×C4×C8—24no—
83C2×C4×C8—24no—
84C2×C4×C8—42no—
85#389—24no—
86#389—42no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—44no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340C2×C414yes—
111#340C2×C414yes—
112#340C2×C414yes—
113#340C2×C414yes—
114#336—24no—
115C25C2×D411yescharacteristic
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—44no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C22×C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—28no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—28no—
161C22×C8—44no—
162C22×C8—48no—
163C22×C8—48no—
164C22×C8C22×C412yes—
165C22×C8C22×C412yes—
166C22×C8C2×D414yes—
167C22×C8#4114yes—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4SD1614yes—
226C2×C22⋊C4Q1614yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C22⋊C8—28no—
234C4×C8—48no—
235#95—24no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—44no—
240#95—82no—
241#95—82no—
242#95—82no—
243#95—82no—
244#95—82no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24C4×D412yes—
250C24#12112yes—
251C24#12412yes—
252C24C2×SD1612yes—
253C24C2×Q1612yes—
254C24#13512yes—
255C24#13512yes—
256C24C22×D411yescharacteristic
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
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—48no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—48no—
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—84no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4#11714yes—
330C22×C4C4×D414yes—
331C22×C4C4×D412yes—
332C22×C4#12112yes—
333C22×C4#12314yes—
334C22×C4#12412yes—
335C22×C4#12614yes—
336C22×C4C2×SD1614yes—
337C22×C4C2×Q1614yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4112yes—
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)12yes—
347C22×C42−1+412yes—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—416no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8#11714yes—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—48no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—48no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—88no—
390C4⋊C4#11714yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23#38814yes—
416C23C4×SD1614yes—
417C23C4×Q1614yes—
418C23#46514yes—
419C23#46714yes—
420C23#48714yes—
421C23#48814yes—
422C23C2×C4×D411yescharacteristic
423C23C2×#12111yescharacteristic
424C23C2×#12411yescharacteristic
425C23#54612yes—
426C23#54712yes—
427C23C22×SD1611yescharacteristic
428C23C22×Q1611yescharacteristic
429C23C2×#13511yescharacteristic
430C23C2×#13511yescharacteristic
431C23#57612yes—
432C23#57712yes—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—816no—
472C2×C4—816no—
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×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#38814yes—
494C2×C4C2×#11711yescharacteristic
495C2×C4C2×#11711yescharacteristic
496C2×C4C2×#11711yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4#51811yescharacteristic
499C2×C4#51911yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C2×C4C2×#12611yescharacteristic
502C2×C4#52912yes—
503C2×C4#52912yes—
504C2×C4#53912yes—
505C2×C4#53912yes—
506C2×C4#54612yes—
507C2×C4#54712yes—
508C2×C4#55112yes—
509C2×C4#55112yes—
510C2×C4#55212yes—
511C2×C4#55212yes—
512C8—24no—
513C8—42no—
514C8—44no—
515C22—24no—
516C22—24no—
517C22—28no—
518C22—24no—
519C22—28no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—28no—
524C22—28no—
525C22#193318yes—
526C22C2×#38811yescharacteristic
527C22#234211yescharacteristic
528C22#234411yescharacteristic
529C22#242512yes—
530C22#248812yes—
531C22C2×C4×SD1611yescharacteristic
532C22C2×C4×Q1611yescharacteristic
533C22#294811yescharacteristic
534C22#294911yescharacteristic
535C22C2×#46511yescharacteristic
536C22C2×#46711yescharacteristic
537C22#306212yes—
538C22C2×#48711yescharacteristic
539C22C2×#48811yescharacteristic
540C22#313612yes—
541C22#329512yes—
542C22#329712yes—
543C22#329912yes—
544C22#330112yes—
545C22#333512yes—
546C22#333712yes—
547C22#333812yes—
548C22#333912yes—
549C4—24no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—162no—
557C4—162no—
558C4#228311yescharacteristic
559C4#231311yescharacteristic
560C4#234211yescharacteristic
561C4#234411yescharacteristic
562C4#241912yes—
563C4#249312yes—
564C2—28no—
565C2#1387411yescharacteristic
566C2C2×#193311yescharacteristic
567C2#2108911yescharacteristic
568C2#2257411yescharacteristic
569C2#2257511yescharacteristic
570C2#2260911yescharacteristic
571C2#2261211yescharacteristic
572C2#2523412yes—
573C2#2523812yes—
574C2#2769712yes—
575C2#2770112yes—
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
6. #11316C2semidirect
7. #11293C2semidirect
8. #10880C2semidirect
9. #10820C2semidirect
24. C2×#340C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18242
C2×C8×C22⋊C42
C22×#3402
#139242
#139072
#113162
#112932
#108802
#108202
#107302
#107212