Finite Group Databasebeta

← #364822

Subgroups — #364822

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×#2961C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4C2×#1801C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#11216C212yesmaximal
7#11062C212yesmaximal
8#10942C212yesmaximal
9#10896C212yesmaximal
10#10847C211yesmaximal, characteristic
11#10845C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078C418yescomplemented
16C2×#479—22no—
17C2×#477—21no—
18C2×#475—21no—
19#2961—24no—
20#2961—24no—
21C2×#446—22no—
22C2×#446—22no—
23C2×#428C2212yes—
24C2×#424C2212yes—
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419C2212yes—
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#101C2212yes—
32C2×#405C2212yes—
33C22×#97C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1989—22no—
36#1988—21no—
37#1987—21no—
38#1801—24no—
39#1763—22no—
40#1763—22no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#341C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C23×Q8C2311yescharacteristic
51C22×M16C2312yes—
52C22×M16C2312yes—
53C22×M16C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533D414yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125D412yes—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
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×Q8C2×C414yes—
75C2×#117C2312yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#479—216no—
84#477—28no—
85#475—28no—
86#446—216no—
87#446—216no—
88#428—24no—
89#428C2×C418yes—
90C2×#107—22no—
91C2×#106—22no—
92#424—24no—
93#424—24no—
94#423—22no—
95#423—22no—
96#423—22no—
97#423—22no—
98#423C2×C414yes—
99#423D414yes—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107#419—24no—
108#419C2×C418yes—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103Q814yescomplemented
117C2×#101—24no—
118C2×#101—24no—
119C2×#101—42no—
120#405—24no—
121#405—24no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—22no—
125C2×#97D412yes—
126C2×#97D412yes—
127C2×C4×C8—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C2×#95—21no—
133#343—24no—
134#343—24no—
135#343—42no—
136#341—24no—
137#341—42no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8C22×C412yes—
158C22×Q8C2×Q812yes—
159C22×Q8#4112yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4#4111yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16C22×C414yes—
184C2×M16C2×D414yes—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×Q812yes—
201C22×C8#4112yes—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8C22⋊C414yes—
205C4⋊Q8C22⋊C414yes—
206#125—24no—
207#125C22⋊C418yes—
208C22⋊Q8—24no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224#117—22no—
225#117—22no—
226#117C22×C414yes—
227#117C2×D414yes—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×Q812yes—
255C2×C4⋊C4#4112yes—
256C2×C22⋊C4—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—42no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—82no—
265C2×C42C22×C412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D412yes—
272C2×C42#4112yes—
273#107—28no—
274#106—28no—
275C4⋊C8—216no—
276C4⋊C8—24no—
277C4⋊C8—24no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282#103—28no—
283#103—28no—
284#103—28no—
285#103—28no—
286#101—48no—
287#97—28no—
288#97—24no—
289#97—24no—
290C4×C8—28no—
291#95—41no—
292#95—41no—
293C24C2×#4111yescharacteristic
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—81no—
324C2×Q8—81no—
325C2×Q8—81no—
326C2×Q8—81no—
327C2×Q8C4×Q814yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
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—24no—
379M16—24no—
380M16—24no—
381M16—24no—
382M16—42no—
383M16—42no—
384M16—42no—
385M16—42no—
386M16—42no—
387M16—42no—
388M16—42no—
389M16—42no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8C22⋊Q814yes—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4C2×C22⋊C414yes—
434C4⋊C4C2×C22⋊C414yes—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C42—24no—
438C42—24no—
439C42—44no—
440C42—44no—
441C42—82no—
442C42—82no—
443C42C2×C22⋊C412yes—
444C42C2×C22⋊C412yes—
445C42C4×D412yes—
446C42C4×D412yes—
447C42#12012yes—
448C42#12012yes—
449C42#12112yes—
450C42#12112yes—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23C2×C4×D411yescharacteristic
456C23C2×#12111yescharacteristic
457C23C2×#12311yescharacteristic
458C23#55211yescharacteristic
459C23#55211yescharacteristic
460C23#55411yescharacteristic
461C23#55611yescharacteristic
462Q8—24no—
463Q8—24no—
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—84no—
471Q8—84no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
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—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4#38612yes—
513C2×C4#38612yes—
514C2×C4#38612yes—
515C2×C4#38612yes—
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×C4×Q811yescharacteristic
519C2×C4C4×#4111yescharacteristic
520C2×C4#51911yescharacteristic
521C2×C4#52011yescharacteristic
522C2×C4C2×#12011yescharacteristic
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4#52511yescharacteristic
526C2×C4#52911yescharacteristic
527C2×C4#53311yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53711yescharacteristic
530C2×C4#54111yescharacteristic
531C2×C4#54211yescharacteristic
532C2×C4#54611yescharacteristic
533C2×C4#54711yescharacteristic
534C2×C4#54811yescharacteristic
535C2×C4#54811yescharacteristic
536C2×C4#54811yescharacteristic
537C2×C4#54811yescharacteristic
538C2×C4#54811yescharacteristic
539C2×C4D4×Q811yescharacteristic
540C2×C4D4×Q811yescharacteristic
541C2×C4D4×Q811yescharacteristic
542C2×C4#55011yescharacteristic
543C2×C4#55011yescharacteristic
544C8—24no—
545C8—44no—
546C8—42no—
547C8—44no—
548C8—82no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22C2×#38611yescharacteristic
562C22#236011yescharacteristic
563C22#236211yescharacteristic
564C22#242111yescharacteristic
565C22#242311yescharacteristic
566C22#253611yescharacteristic
567C22#253811yescharacteristic
568C22#306811yescharacteristic
569C22#306911yescharacteristic
570C22#306911yescharacteristic
571C22#306911yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—44no—
577C4—44no—
578C4—44no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4C4×C22⋊Q811yescharacteristic
583C4#232311yescharacteristic
584C4Q8×C22⋊C411yescharacteristic
585C4#234111yescharacteristic
586C4#240711yescharacteristic
587C4#240911yescharacteristic
588C4#242611yescharacteristic
589C4#242911yescharacteristic
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2#1396911yescharacteristic
595C2#2100411yescharacteristic
596C2#2100611yescharacteristic
597C2#2822111yescharacteristic
598C2#2822211yescharacteristic
599C2#2829811yescharacteristic
600C2#2830011yescharacteristic
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
116. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29612
#213122
C2×#18012
C2×#17632
#112162
#110622
#109422
#108962
#108472
#108452