Finite Group Databasebeta

← #364797

Subgroups — #364797

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×#2962C211yesmaximal, characteristic
3#21323C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5C2×C4×#101C211yesmaximal, characteristic
6#11216C212yesmaximal
7#11062C212yesmaximal
8#10942C212yesmaximal
9#10896C212yesmaximal
10#10858C211yesmaximal, characteristic
11#10853C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078C418yescomplemented
16C2×#479—22no—
17C2×#477—21no—
18C2×#475—21no—
19#2962—24no—
20#2962—24no—
21C2×#446—21no—
22C2×#446—22no—
23C2×C8×Q8—21no—
24C2×#428C2212yes—
25C2×#424C2212yes—
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C2×#419C2212yes—
29C22×#103C2212yes—
30C22×#101C2211yescharacteristic
31C22×#101C2211yescharacteristic
32C2×#405C2212yes—
33C2×C4×M16C2211yescharacteristic
34C2×C4×M16C2211yescharacteristic
35C2×#382C2211yescharacteristic
36#1993—21no—
37#1993—21no—
38#1991—22no—
39#1805—21no—
40#1805—21no—
41#1805—22no—
42C4×#101—24no—
43C2×#343C2212yes—
44C2×#343C2211yescharacteristic
45C2×#343C2211yescharacteristic
46C2×#343C2212yes—
47C2×#341C2212yes—
48C2×#341C2212yes—
49#1736C2212yes—
50#1736C2212yes—
51C2×#340C2212yes—
52C23×Q8C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C23×C8C2312yes—
59#533—22no—
60#533—22no—
61#533C2×C414yes—
62#533D414yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×#125D412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—22no—
77C2×C4×Q8C2×C414yes—
78C2×#117C2312yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#479—216no—
85#477—28no—
86#475—28no—
87#446—216no—
88#446—28no—
89C8×Q8—28no—
90#428—24no—
91#428C2×C418yes—
92C2×#107—22no—
93C2×#106—22no—
94#424—24no—
95#424—24no—
96#423—22no—
97#423C2×C414yes—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D412yes—
107#419—24no—
108#419C2×C418yes—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—42no—
115C2×#101—22no—
116C2×#101—22no—
117C2×#101—22no—
118C2×#101—22no—
119C2×#101—22no—
120C2×#101Q814yescomplemented
121#405—24no—
122#405—24no—
123C4×M16—22no—
124C4×M16—22no—
125C4×M16—22no—
126C4×M16D414yes—
127C2×#97—22no—
128C2×#97—21no—
129C2×#97—21no—
130C2×C4×C8—22no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133#382—21no—
134#382—21no—
135#382—21no—
136#382—21no—
137C2×#95—21no—
138C2×#95—21no—
139#343—22no—
140#343—22no—
141#343—24no—
142#343—42no—
143#341—24no—
144#341—42no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8C22×C412yes—
165C22×Q8C2×Q812yes—
166C22×Q8#4112yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4#4111yescharacteristic
171C23×C4#4111yescharacteristic
172C23×C4C2411yescharacteristic
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—22no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—22no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—22no—
195C2×M16—21no—
196C2×M16C22×C414yes—
197C2×M16C2×D412yes—
198C2×M16C2×D412yes—
199C2×M16C2×Q812yes—
200C2×M16C2×Q812yes—
201C2×M16#4112yes—
202C2×M16#4112yes—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8C22⋊C414yes—
219C4⋊Q8C22⋊C414yes—
220#125—24no—
221#125C22⋊C418yes—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—42no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×Q8—42no—
238#117—22no—
239#117—22no—
240#117C22×C414yes—
241#117C2×D414yes—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C22⋊C4—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—42no—
270C2×C42C22×C412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42#4112yes—
278#107—28no—
279#106—28no—
280C4⋊C8—28no—
281C4⋊C8—28no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—24no—
287#103—28no—
288#103—28no—
289#103—44no—
290#103—44no—
291#101—28no—
292#101—28no—
293#97—28no—
294#97—24no—
295#97—42no—
296C4×C8—28no—
297C4×C8—24no—
298C4×C8—42no—
299#95—41no—
300#95—41no—
301#95—41no—
302#95—41no—
303C24C2×#4111yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—44no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—82no—
335C2×Q8C4×Q814yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12312yes—
363C22×C4#12312yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
378M16—22no—
379M16—22no—
380M16—22no—
381M16—24no—
382M16—24no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—41no—
390M16—41no—
391M16—42no—
392M16—41no—
393M16—42no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16C22⋊Q814yes—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—41no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4C2×C22⋊C414yes—
440C4⋊C4C2×C22⋊C414yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C42—24no—
444C42—24no—
445C42—44no—
446C42—44no—
447C42C2×C22⋊C412yes—
448C42C2×C22⋊C412yes—
449C42C4×D412yes—
450C42C4×D412yes—
451C42#12012yes—
452C42#12012yes—
453C42#12112yes—
454C42#12112yes—
455C23—22no—
456C23—22no—
457C23C2×C4×D411yescharacteristic
458C23C2×#12111yescharacteristic
459C23C2×#12311yescharacteristic
460C23#52911yescharacteristic
461C23#53311yescharacteristic
462C23#54111yescharacteristic
463C23#54211yescharacteristic
464Q8—24no—
465Q8—24no—
466Q8—48no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—42no—
472Q8—42no—
473Q8—84no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—162no—
501C2×C4—162no—
502C2×C4#38612yes—
503C2×C4#38612yes—
504C2×C4#38612yes—
505C2×C4#38612yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×Q811yescharacteristic
509C2×C4C4×#4111yescharacteristic
510C2×C4#51911yescharacteristic
511C2×C4#52011yescharacteristic
512C2×C4C2×#12011yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×C22⋊Q811yescharacteristic
515C2×C4#52511yescharacteristic
516C2×C4#53611yescharacteristic
517C2×C4#53711yescharacteristic
518C2×C4#54611yescharacteristic
519C2×C4#54711yescharacteristic
520C2×C4#54811yescharacteristic
521C2×C4#54811yescharacteristic
522C2×C4#54811yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4#54811yescharacteristic
525C2×C4D4×Q811yescharacteristic
526C2×C4D4×Q811yescharacteristic
527C2×C4D4×Q811yescharacteristic
528C2×C4#55011yescharacteristic
529C2×C4#55011yescharacteristic
530C2×C4#55211yescharacteristic
531C2×C4#55211yescharacteristic
532C2×C4#55411yescharacteristic
533C2×C4#55611yescharacteristic
534C8—24no—
535C8—44no—
536C8—42no—
537C8—42no—
538C8—42no—
539C8—42no—
540C8—42no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22C2×#38611yescharacteristic
550C22C4×C22⋊Q811yescharacteristic
551C22#232311yescharacteristic
552C22#242111yescharacteristic
553C22#242311yescharacteristic
554C22#242611yescharacteristic
555C22#242911yescharacteristic
556C22#306712yes—
557C22#306912yes—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—82no—
566C4—162no—
567C4Q8×C22⋊C411yescharacteristic
568C4#234111yescharacteristic
569C4#236011yescharacteristic
570C4#236211yescharacteristic
571C4#240711yescharacteristic
572C4#240911yescharacteristic
573C4#253611yescharacteristic
574C4#253811yescharacteristic
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2#1396911yescharacteristic
580C2#2100311yescharacteristic
581C2#2100611yescharacteristic
582C2#2531712yes—
583C2#2549412yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #3078C4semidirect
120. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29622
#213232
C2×#18052
C2×C4×#1012
#112162
#110622
#109422
#108962
#108582
#108532