Finite Group Databasebeta

← #399615

Subgroups — #399615

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1865C211yesmaximal, characteristic
4C2×#1863C212yesmaximal
5#15149C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11346C211yesmaximal, characteristic
8#11328C212yesmaximal
9#11220C212yesmaximal
10#10789C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#107C2212yes—
15C22×#106C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C2×#398—21no—
20Q8×C4⋊C4—24no—
21Q8×C4⋊C4—24no—
22C2×#389—22no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2083—24no—
31#1865—24no—
32#1863—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1442—22no—
43#1441—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8D414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
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×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—42no—
81C2×#106—22no—
82C2×#106—24no—
83C2×#103—24no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103D414yes—
90C2×#103Q818yes—
91#398—44no—
92#389—216no—
93#389—24no—
94#389—24no—
95#389—24no—
96#389—22no—
97#389—22no—
98#389—42no—
99#384—28no—
100#384—22no—
101#384D414yes—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—24no—
117#340—42no—
118#336—24no—
119#336—22no—
120#336—22no—
121#336—42no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8C2×D412yes—
128C22×Q8C2×D412yes—
129C22×Q8C2×Q814yes—
130C22×Q8#4112yes—
131C22×Q8#4114yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8C2×D412yes—
159C22×C8C2×Q814yes—
160C4×Q8—216no—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—48no—
164C4×Q8—48no—
165C4×Q8—48no—
166C4×Q8—48no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×Q814yes—
217C2×C4⋊C4#4114yes—
218C2×C42—28no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—28no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233#107—48no—
234#106—44no—
235#103—28no—
236#103—216no—
237#103—44no—
238#95—28no—
239#95—24no—
240#95—44no—
241#95—44no—
242#95—42no—
243#95—42no—
244#95—44no—
245#95—81no—
246C24C22×D411yescharacteristic
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8#12114yes—
261C2×Q8C22⋊Q818yes—
262C2×Q8#12314yes—
263C2×Q8#12518yes—
264C2×Q8C4⋊Q814yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4#12012yes—
321C22×C4#12012yes—
322C22×C4#13512yes—
323C22×C4AGL(1,ℤ/8)12yes—
324C22×C4#13714yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q812yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C42−1+412yes—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—28no—
353C4⋊C4—216no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—416no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—84no—
375C4⋊C4—88no—
376C4⋊C4—88no—
377C4⋊C4—84no—
378C4⋊C4#12314yes—
379C42—216no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—416no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C23C2×#12011yescharacteristic
389C23#52512yes—
390C23D4211yescharacteristic
391C23#54611yescharacteristic
392C23D4×Q812yes—
393C23C2×#13511yescharacteristic
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×#13712yes—
396C23#57611yescharacteristic
397C23#57711yescharacteristic
398C23#57812yes—
399Q8—24no—
400Q8—28no—
401Q8—28no—
402Q8—24no—
403Q8—24no—
404Q8#39818yescomplemented
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4#44914yes—
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×C22⋊Q812yes—
467C2×C4C2×#12311yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4#52512yes—
470C2×C4C2×#12512yes—
471C2×C4C2×C4⋊Q811yescharacteristic
472C2×C4#53511yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#53912yes—
475C2×C4#53912yes—
476C2×C4D4211yescharacteristic
477C2×C4#54611yescharacteristic
478C2×C4#54712yes—
479C2×C4D4×Q812yes—
480C2×C4#55212yes—
481C2×C4#55412yes—
482C2×C4#55412yes—
483C8—44no—
484C8—42no—
485C8—81no—
486C22#240811yescharacteristic
487C22#243411yescharacteristic
488C22#244912yes—
489C22#247312yes—
490C22#284911yescharacteristic
491C22C2×#44911yescharacteristic
492C22#300411yescharacteristic
493C22#301111yescharacteristic
494C22#301211yescharacteristic
495C22#310812yes—
496C22#311212yes—
497C22#312012yes—
498C22#312412yes—
499C22#327012yes—
500C22#327411yescharacteristic
501C22#328411yescharacteristic
502C22#330511yescharacteristic
503C22#331212yes—
504C22#333211yescharacteristic
505C22#338312yes—
506C22#338512yes—
507C22#338612yes—
508C22#338712yes—
509C4—28no—
510C4—24no—
511C4—44no—
512C4—42no—
513C4—44no—
514C4—84no—
515C4—82no—
516C4—84no—
517C4—82no—
518C4—161no—
519C4C2×#39811yescharacteristic
520C4#244912yes—
521C4#247312yes—
522C4#248611yescharacteristic
523C4#250211yescharacteristic
524C4#262311yescharacteristic
525C2#1478611yescharacteristic
526C2#2423811yescharacteristic
527C2#2425511yescharacteristic
528C2#2584011yescharacteristic
529C2#2584711yescharacteristic
530C2#2637112yes—
531C2#2639712yes—
532C2#2729512yes—
533C2#2731112yes—
534C2#3300311yescharacteristic
535C2#3301711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
404. Q8#398semidirect

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×#20832
C2×#18652
C2×#18632
#151492
#113732
#113462
#113282
#112202
#107892