Finite Group Databasebeta

← #399143

Subgroups — #399143

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3C2×#1878C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5#15192C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11346C211yesmaximal, characteristic
8#11292C212yesmaximal
9#11067C212yesmaximal
10#10789C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19Q8×C4⋊C4—24no—
20Q8×C4⋊C4—24no—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2072—24no—
30#1878—28no—
31#1878—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1445—21no—
42#1443—22no—
43#1442—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×Q8C2311yescharacteristic
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16D414yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C2×C4×Q8D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—22no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—22no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—42no—
91#391—24no—
92#391—22no—
93#391—22no—
94#391—42no—
95#389—216no—
96#389—24no—
97#389—24no—
98#384—28no—
99#384—22no—
100#384D414yes—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#340—22no—
117#340—24no—
118#340—42no—
119#336—24no—
120#336—22no—
121#336—22no—
122#336—42no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8C2×D412yes—
138C22×Q8C2×D412yes—
139C22×Q8C2×D412yes—
140C22×Q8#4112yes—
141C22×Q8#4114yes—
142C22×Q8#4114yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×Q16—28no—
155C2×Q16—216no—
156C2×Q16—44no—
157C2×Q16—44no—
158C2×Q16—48no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8C2×D412yes—
176C4×Q8—216no—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—48no—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—48no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4#4114yes—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—28no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237#103—48no—
238#103—48no—
239#103—44no—
240#103—44no—
241#95—28no—
242#95—24no—
243#95—44no—
244#95—44no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—44no—
249#95—81no—
250#95—82no—
251C24C22×D411yescharacteristic
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8#12114yes—
284C2×Q8#12318yes—
285C2×Q8#12314yes—
286C2×Q8#12414yes—
287C2×Q8#12618yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4#12012yes—
342C22×C4#12012yes—
343C22×C4#13512yes—
344C22×C4#13714yes—
345C22×C4#13712yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C42−1+412yes—
356Q16—416no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—28no—
375C4⋊C4—216no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—416no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C4⋊C4#12314yes—
395C42—216no—
396C42—22no—
397C42—22no—
398C42—24no—
399C42—416no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C23C2×#12011yescharacteristic
405C23#52512yes—
406C23D4211yescharacteristic
407C23#54611yescharacteristic
408C23#54812yes—
409C23C2×#13511yescharacteristic
410C23C2×#13711yescharacteristic
411C23C2×#13712yes—
412C23#57611yescharacteristic
413C23#57711yescharacteristic
414C23#57812yes—
415Q8—24no—
416Q8—28no—
417Q8—28no—
418Q8—24no—
419Q8—24no—
420Q8—42no—
421Q8—44no—
422Q8—42no—
423Q8—81no—
424Q8—82no—
425Q8—81no—
426Q8#39718yes—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
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—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#45214yes—
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12312yes—
488C2×C4C2×#12311yescharacteristic
489C2×C4#52512yes—
490C2×C4C2×#12411yescharacteristic
491C2×C4C2×#12612yes—
492C2×C4#53511yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#53912yes—
495C2×C4#53912yes—
496C2×C4D4211yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#54712yes—
499C2×C4#54812yes—
500C2×C4#55212yes—
501C2×C4#55312yes—
502C2×C4#55312yes—
503C8—42no—
504C8—82no—
505C8—81no—
506C22#240811yescharacteristic
507C22#243611yescharacteristic
508C22#244312yes—
509C22#248112yes—
510C22#284911yescharacteristic
511C22C2×#45211yescharacteristic
512C22#300411yescharacteristic
513C22#301111yescharacteristic
514C22#301211yescharacteristic
515C22#310812yes—
516C22#311312yes—
517C22#312012yes—
518C22#312412yes—
519C22#327012yes—
520C22#327811yescharacteristic
521C22#328711yescharacteristic
522C22#330411yescharacteristic
523C22#331212yes—
524C22#333311yescharacteristic
525C22#336412yes—
526C22#336912yes—
527C22#337412yes—
528C22#337512yes—
529C4—28no—
530C4—24no—
531C4—44no—
532C4—42no—
533C4—44no—
534C4—82no—
535C4—84no—
536C4—82no—
537C4—162no—
538C4—161no—
539C4C2×#39711yescharacteristic
540C4#244312yes—
541C4#248112yes—
542C4#248611yescharacteristic
543C4#250411yescharacteristic
544C4#262311yescharacteristic
545C2#1477611yescharacteristic
546C2#2424211yescharacteristic
547C2#2425511yescharacteristic
548C2#2593111yescharacteristic
549C2#2594211yescharacteristic
550C2#2616912yes—
551C2#2618712yes—
552C2#2755112yes—
553C2#2756912yes—
554C2#3300811yescharacteristic
555C2#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

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×#20722
C2×#18782
#151922
#113732
#113462
#112922
#110672
#107892