Finite Group Databasebeta

← #360813

Subgroups — #360813

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#1836C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#10849C211yesmaximal, characteristic
9#10835C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C212yesmaximal
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#97C2211yescharacteristic
19Q8×C4⋊C4—24no—
20Q8×C4⋊C4—24no—
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1836—24no—
30#1805—24no—
31#1763C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—22no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8C2×C414yes—
60C2×C4×Q8D414yes—
61C2×C4×Q8Q814yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—22no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103C2×C418yes—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97C2×C414yes—
92#389—216no—
93#389—24no—
94#389—24no—
95#384—28no—
96#384—22no—
97#384D414yes—
98#382—22no—
99#382—22no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D412yes—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—24no—
118#340—42no—
119#340—42no—
120#340C2×C418yes—
121#336—42no—
122#336—42no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8C22×C412yes—
129C22×Q8C2×D412yes—
130C22×Q8C2×Q812yes—
131C22×Q8#4114yes—
132C22×Q8#4114yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C22×C414yes—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8—48no—
165C4×Q8—48no—
166C4×Q8—48no—
167C4×Q8—48no—
168C4×Q8C4⋊C4116yes—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q812yes—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—28no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42—81no—
227C2×C42—81no—
228C2×C42C22×C412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×Q812yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234C4⋊C8—44no—
235#103—216no—
236#103—44no—
237#103—44no—
238#97—28no—
239#95—28no—
240#95—24no—
241#95—24no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—44no—
246#95—82no—
247#95—82no—
248C24C22×D411yescharacteristic
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8C2×C4⋊C414yes—
263C2×Q8#11718yes—
264C2×Q8#12414yes—
265C2×Q8#12414yes—
266C2×Q8#12518yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
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—24no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12012yes—
325C22×C4#12012yes—
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4#13714yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—28no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—216no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—416no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—84no—
379C4⋊C4—88no—
380C4⋊C4C2×C4⋊C414yes—
381C42—216no—
382C42—416no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—84no—
388C42C2×C4⋊C414yes—
389C42C4×D414yes—
390C42C22⋊Q814yes—
391C42#12314yes—
392C23C2×C4×D411yescharacteristic
393C23C2×#12011yescharacteristic
394C23C2×C22⋊Q811yescharacteristic
395C23#54612yes—
396C23#54812yes—
397C23C2×#13712yes—
398C23#57512yes—
399C23#57611yescharacteristic
400C23#57711yescharacteristic
401C23#57711yescharacteristic
402C23#57811yescharacteristic
403Q8—24no—
404Q8—28no—
405Q8—28no—
406Q8—24no—
407Q8—24no—
408Q8#38218yes—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#38514yes—
470C2×C4#38514yes—
471C2×C4C22×C4⋊C411yescharacteristic
472C2×C4C2×#11712yes—
473C2×C4C2×C4×D411yescharacteristic
474C2×C4#52012yes—
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#12411yescharacteristic
478C2×C4C2×#12411yescharacteristic
479C2×C4C2×#12512yes—
480C2×C4#53511yescharacteristic
481C2×C4#53611yescharacteristic
482C2×C4#54412yes—
483C2×C4#54612yes—
484C2×C4#54812yes—
485C2×C4#55212yes—
486C2×C4#55212yes—
487C2×C4#55312yes—
488C2×C4#55312yes—
489C8—44no—
490C8—81no—
491C8—81no—
492C22C2×#38511yescharacteristic
493C22#235212yes—
494C22#243511yescharacteristic
495C22#243611yescharacteristic
496C22#244212yes—
497C22C4×#13712yes—
498C22#294611yescharacteristic
499C22#294711yescharacteristic
500C22#300411yescharacteristic
501C22#300511yescharacteristic
502C22#301211yescharacteristic
503C22#301311yescharacteristic
504C22#307812yes—
505C22#308311yescharacteristic
506C22#308411yescharacteristic
507C22#330912yes—
508C22#331212yes—
509C22#331712yes—
510C22#332012yes—
511C22#336912yes—
512C22#337112yes—
513C22#337512yes—
514C22#337712yes—
515C4—28no—
516C4—24no—
517C4—24no—
518C4—44no—
519C4—44no—
520C4—82no—
521C4—84no—
522C4—84no—
523C4—161no—
524C4—161no—
525C4C2×#38211yescharacteristic
526C4#231711yescharacteristic
527C4#235212yes—
528C4#244212yes—
529C4#250311yescharacteristic
530C4#250411yescharacteristic
531C2#1388111yescharacteristic
532C2#2087411yescharacteristic
533C2#2087811yescharacteristic
534C2#2277512yes—
535C2#2277912yes—
536C2#2589311yescharacteristic
537C2#2589411yescharacteristic
538C2#2594111yescharacteristic
539C2#2594211yescharacteristic
540C2#2612012yes—
541C2#2612112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1763C4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#18362
C2×#18052
C2×#17632
#139552
#112932
#112922
#108492
#108352
#107892