Finite Group Databasebeta

← #535406

Subgroups — #535406

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2356C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3C2×#2327C211yesmaximal, characteristic
4#14411C214yesmaximal
5#14390C214yesmaximal
6#14142C212yesmaximal
7#14120C214yesmaximal
8#14087C214yesmaximal
9#14030C212yesmaximal
10#14024C212yesmaximal
11#13926C211yesmaximal, characteristic
12#13926C211yesmaximal, characteristic
13#13854C211yesmaximal, characteristic
14#13817C212yesmaximal
15#13779C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#2356—28no—
21#2351—28no—
22#2327C4116yescomplemented
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#384C2211yescharacteristic
34C2×#383C2212yes—
35C2×#382C2212yes—
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×#376C2212yes—
44C2×#376C2211yescharacteristic
45#1443C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C4×#95C2214yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2×C418yes—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—44no—
90C2×C4×Q8C2×C418yes—
91C2×C4×Q8C2×C418yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2314yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115#391—28no—
116#391—28no—
117#391—44no—
118#391C2×C418yes—
119#391C2×C418yes—
120#389—28no—
121#389—216no—
122#387—216no—
123#387C2×C4132yes—
124#387C2×C4132yes—
125#384—24no—
126#383—28no—
127#382—28no—
128#382C2×C418yes—
129#382D418yes—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—44no—
135#376—28no—
136#376C2×C418yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8C22×C414yes—
182C22×Q8C22×C414yes—
183C22×Q8#4114yes—
184C22×Q8#4114yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C4⋊Q8—232no—
201C4×Q8—232no—
202C4×Q8—232no—
203C4×Q8—432no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4C22×C418yes—
239C2×C4⋊C4C22×C4116yes—
240C2×C4⋊C4C2×D418yes—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—28no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C22×C418yes—
265C2×C42C22×C418yes—
266C2×C42C22×C414yes—
267C2×C42C22×C414yes—
268C2×C42C22×C414yes—
269C2×C42C2×D414yes—
270C2×C42#4114yes—
271#95—22no—
272#95—28no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—42no—
286#95—44no—
287#95—44no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95C22×C418yes—
299#95C22×C418yes—
300#95C22×C4116yes—
301#95C22×C4116yes—
302#95C22×C4116yes—
303#95C2×D418yes—
304#95C2×D418yes—
305C24C2511yescharacteristic
306C2×Q8—28no—
307C2×Q8—28no—
308C2×Q8—28no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—28no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8#117116yes—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
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—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4C23×C414yes—
394C22×C4C23×C414yes—
395C22×C4C23×C414yes—
396C22×C4C23×C414yes—
397C22×C4C23×C414yes—
398C22×C4C23×C414yes—
399C22×C4C23×C412yes—
400C22×C4C23×C412yes—
401C22×C4C23×C412yes—
402C22×C4C22×D418yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C2×#4114yes—
407C22×C4C2×#4114yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+414yes—
415C22×C42−1+414yes—
416C22×C42−1+412yes—
417C4⋊C4—232no—
418C4⋊C4—216no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—216no—
422C4⋊C4—216no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C42—28no—
431C42—216no—
432C42—216no—
433C42—28no—
434C42—416no—
435C42—416no—
436C42—48no—
437C42—48no—
438C42—88no—
439C42C4×D4116yes—
440C23C24×C411yescharacteristic
441C23C23×D411yescharacteristic
442C23C22×#4111yescharacteristic
443C23C22×#4111yescharacteristic
444C23C22×#4111yescharacteristic
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×2−1+411yescharacteristic
447C23C2×2−1+412yes—
448C23C2×2−1+412yes—
449C23#58512yes—
450C23#58512yes—
451Q8—232no—
452Q8—416no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4C2×#11718yes—
541C2×C4C2×#11714yes—
542C2×C4C2×C4×D414yes—
543C2×C4C2×C4×D414yes—
544C2×C4C2×C4×D414yes—
545C2×C4#52014yes—
546C2×C4#54614yes—
547C2×C4#54814yes—
548C2×C4#55214yes—
549C2×C4#55314yes—
550C22C22×#11711yescharacteristic
551C22C22×C4×D411yescharacteristic
552C22C2×#52011yescharacteristic
553C22C4×2−1+412yes—
554C22C4×2−1+412yes—
555C22C2×#54611yescharacteristic
556C22C2×#54811yescharacteristic
557C22C2×#55211yescharacteristic
558C22C2×#55311yescharacteristic
559C22#348912yes—
560C22#348912yes—
561C22#350911yescharacteristic
562C22#350911yescharacteristic
563C22#351011yescharacteristic
564C22#351011yescharacteristic
565C22#351612yes—
566C22#351812yes—
567C22#352312yes—
568C22#352412yes—
569C22#355212yes—
570C22#355512yes—
571C22#356212yes—
572C22#356312yes—
573C4—28no—
574C4—24no—
575C4—24no—
576C4—44no—
577C4—44no—
578C4—44no—
579C4—44no—
580C4—44no—
581C4—44no—
582C4—48no—
583C4—44no—
584C4—84no—
585C4—82no—
586C4—82no—
587C4#235218yes—
588C2C2×#235211yescharacteristic
589C2#3426212yes—
590C2#3426811yescharacteristic
591C2#3426811yescharacteristic
592C2#3443012yes—
593C2#4529911yescharacteristic
594C2#4530012yes—
595C2#4530611yescharacteristic
596C2#4530712yes—
597C2#4534411yescharacteristic
598C2#4534711yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #2327C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23562
C2×#23512
C2×#23272
#144112
#143902
#141422
#141202
#140872
#140302
#140242
#139262
#138542
#138172
#137792