Finite Group Databasebeta

← #387724

Subgroups — #387724

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic
3#13925C211yesmaximal, characteristic
4#11077C212yesmaximal
5#11052C212yesmaximal
6#11010C212yesmaximal
7#10932C212yesmaximal
8#10926C212yesmaximal
9#10806C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2568—24no—
19#2568C418yescomplemented
20C2×#400—21no—
21C2×#400—21no—
22C2×#391—21no—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#382—21no—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1984—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1446—21no—
37#1445—21no—
38#1436—21no—
39#1436—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8D411yescharacteristic
43C23×Q8C2312yes—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8C2×C414yes—
52C2×C4×Q8—41no—
53C2×C4×Q8—41no—
54C2×C4×Q8—41no—
55C2×C4×Q8—41no—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—41no—
73C2×C4⋊C8—41no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×C4×C8—22no—
87C2×C4×C8—41no—
88C2×C4×C8—41no—
89#400—28no—
90#400—28no—
91#391—28no—
92#391—24no—
93#391—24no—
94#391C2×C418yes—
95#391C2×C418yes—
96#382—28no—
97#382—41no—
98#382—41no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—41no—
101C4×C4⋊C4—41no—
102C4×C4⋊C4C2×C414yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#336—24no—
120#336—24no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8C22⋊C414yes—
139C22×Q8C22×C414yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8#4114yes—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22⋊C414yes—
202C2×C4⋊C4C22⋊C414yes—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4#4114yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—24no—
219C2×C42—44no—
220C2×C42—42no—
221C2×C42—82no—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C4⋊C8—44no—
229#103—48no—
230#103—48no—
231#103—48no—
232#103—48no—
233C4×C8—44no—
234#95—24no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—44no—
243#95—44no—
244#95—82no—
245#95—82no—
246C24C22×D411yescharacteristic
247C2×Q8—28no—
248C2×Q8—28no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—44no—
256C2×Q8—48no—
257C2×Q8—48no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
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—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4#13514yes—
337C22×C4#13714yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)12yes—
347C22×C42−1+412yes—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—48no—
392C42—84no—
393C23C22×C22⋊C411yescharacteristic
394C23#53411yescharacteristic
395C23#53411yescharacteristic
396C23#54812yes—
397C23#55012yes—
398C23C2×#13512yes—
399C23C2×#13712yes—
400C23#57712yes—
401C23#57812yes—
402Q8—416no—
403Q8—44no—
404Q8—44no—
405Q8—44no—
406Q8—44no—
407Q8—82no—
408Q8—82no—
409Q8—82no—
410Q8—82no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
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—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4#41614yes—
468C2×C4#41914yes—
469C2×C4C4×#4112yes—
470C2×C4#51811yescharacteristic
471C2×C4#51811yescharacteristic
472C2×C4#52012yes—
473C2×C4#53411yescharacteristic
474C2×C4#53411yescharacteristic
475C2×C4#53812yes—
476C2×C4#53812yes—
477C2×C4#54212yes—
478C2×C4#54212yes—
479C2×C4#54812yes—
480C2×C4#55012yes—
481C2×C4#55112yes—
482C2×C4#55511yescharacteristic
483C2×C4#55511yescharacteristic
484C2×C4H(3,𝔽4)11yescharacteristic
485C2×C4#56212yes—
486C2×C4#56311yescharacteristic
487C8—44no—
488C8—82no—
489C22#230811yescharacteristic
490C22#234112yes—
491C22#273012yes—
492C22#273111yescharacteristic
493C22#273311yescharacteristic
494C22C2×#41611yescharacteristic
495C22C2×#41911yescharacteristic
496C22#289812yes—
497C22#319012yes—
498C22#319012yes—
499C22#320011yescharacteristic
500C22#320112yes—
501C22#320211yescharacteristic
502C22#336012yes—
503C22#336412yes—
504C22#337212yes—
505C22#337412yes—
506C22#339312yes—
507C22#339412yes—
508C22#339812yes—
509C22#340012yes—
510C4—24no—
511C4—24no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—84no—
517C4—84no—
518C4—84no—
519C4—162no—
520C4#233712yes—
521C4#237211yescharacteristic
522C4#237911yescharacteristic
523C4#273012yes—
524C4#273111yescharacteristic
525C4#273311yescharacteristic
526C2#1449211yescharacteristic
527C2#2176211yescharacteristic
528C2#2177111yescharacteristic
529C2#2254112yes—
530C2#2254712yes—
531C2#3260412yes—
532C2#3260812yes—
533C2#3263412yes—
534C2#3267312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2568C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#19842
#139252
#110772
#110522
#110102
#109322
#109262
#108062