Finite Group Databasebeta

← #368991

Subgroups — #368991

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic
6#11217C214yesmaximal
7#10899C214yesmaximal
8#10851C211yesmaximal, characteristic
9C4×#343C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165C418yescomplemented
14C2×#501—22no—
15C2×#500—21no—
16C2×#498—21no—
17C2×#428C2214yes—
18C2×C4×M16C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2370—22no—
21#2369—21no—
22#2365—21no—
23#2355—21no—
24#2355—22no—
25Q8×C4⋊C4—21no—
26#2329—22no—
27#2317C2212yes—
28C4×C22⋊Q8—22no—
29C2×#391—22no—
30#2285C2211yescharacteristic
31#2274C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1993—22no—
35#1990—21no—
36#1990—21no—
37#1986—24no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40#1736C2214yes—
41C2×#339C2214yes—
42C2×#339C2214yes—
43#1445—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C22×M16C2312yes—
48C22×M16C2312yes—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×#125D412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C42D412yes—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#501—216no—
80#500—28no—
81#498—28no—
82#428—28no—
83#428C2×C4116yes—
84C2×#107—24no—
85C2×#106—24no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16C2×C414yes—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#391—24no—
95#391—22no—
96#391—48no—
97#389—24no—
98#389—22no—
99#389—22no—
100#389—22no—
101#387—24no—
102#387—24no—
103#387—24no—
104#385—24no—
105#384—22no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#382—22no—
109#382D412yes—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C22⋊C4—24no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95—21no—
130C2×#95D412yes—
131#343—24no—
132#343Q818yescomplemented
133#340—24no—
134#340—24no—
135#339—28no—
136#339—28no—
137C22×Q8—24no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—44no—
141C22×Q8—44no—
142C22×Q8—41no—
143C22×Q8—41no—
144C22×Q8—42no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16C22×C414yes—
161C2×M16C2×Q814yes—
162C2×M16C2×Q814yes—
163C2×M16#4114yes—
164C22×C8—22no—
165C22×C8—22no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8C22⋊C414yes—
177C4⋊Q8C22⋊C414yes—
178#125—24no—
179#125C22⋊C418yes—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—84no—
189#117—22no—
190#117—22no—
191#117—24no—
192#117—24no—
193#117C22×C414yes—
194#117C2×D414yes—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×Q814yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42C22×C412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4112yes—
250C2×C42#4112yes—
251#107—216no—
252#106—216no—
253#97—24no—
254#97—42no—
255C4×C8—24no—
256C4×C8—42no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—48no—
266#95—48no—
267#95—42no—
268C24C22×D411yescharacteristic
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—416no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×Q8—82no—
279C2×Q8—82no—
280C2×Q8—88no—
281C2×Q8—82no—
282C2×Q8—82no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4C4⋊Q814yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q812yes—
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C42−1+412yes—
330C22×C42−1+412yes—
331M16—24no—
332M16—24no—
333M16—24no—
334M16—24no—
335M16—24no—
336M16—42no—
337M16—42no—
338M16C4×Q818yescomplemented
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—48no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C4⋊C4C2×C22⋊C414yes—
371C4⋊C4C2×C22⋊C414yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C42—24no—
377C42—24no—
378C42—44no—
379C42—44no—
380C42—44no—
381C42—44no—
382C42—82no—
383C42—84no—
384C42—84no—
385C42C2×C22⋊C412yes—
386C42C2×C22⋊C412yes—
387C42C4×D412yes—
388C42C4×D412yes—
389C42#12112yes—
390C42#12112yes—
391C42#12312yes—
392C42#12312yes—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23C2×C4×D411yescharacteristic
398C23C2×#12011yescharacteristic
399C23C2×#12111yescharacteristic
400C23C2×C4⋊Q811yescharacteristic
401C23#53211yescharacteristic
402C23#53611yescharacteristic
403C23#53611yescharacteristic
404Q8—48no—
405Q8—84no—
406Q8—88no—
407Q8—88no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4#38612yes—
453C2×C4#38612yes—
454C2×C4#38612yes—
455C2×C4#38612yes—
456C2×C4#45312yes—
457C2×C4#45412yes—
458C2×C4C22×C22⋊C411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C4×Q812yes—
461C2×C4#51912yes—
462C2×C4C2×#12111yescharacteristic
463C2×C4C2×#12311yescharacteristic
464C2×C4#53312yes—
465C2×C4#54112yes—
466C2×C4#54712yes—
467C2×C4#54812yes—
468C2×C4#54812yes—
469C2×C4D4×Q812yes—
470C2×C4D4×Q812yes—
471C2×C4D4×Q812yes—
472C2×C4#55612yes—
473C2×C4#55712yes—
474C8—28no—
475C8—44no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22C2×#38611yescharacteristic
489C22C4×C4⋊Q811yescharacteristic
490C22#232711yescharacteristic
491C22#243211yescharacteristic
492C22#243411yescharacteristic
493C22#254511yescharacteristic
494C22#254611yescharacteristic
495C22C2×#45311yescharacteristic
496C22C2×#45411yescharacteristic
497C22#302211yescharacteristic
498C22#302211yescharacteristic
499C4—22no—
500C4—22no—
501C4—22no—
502C4—44no—
503C4—44no—
504C4—44no—
505C4—84no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—84no—
510C4#188314yes—
511C4Q8×C22⋊C412yes—
512C4#236512yes—
513C4#242912yes—
514C4#244612yes—
515C2—21no—
516C2—21no—
517C2—21no—
518C2—21no—
519C2#1402411yescharacteristic
520C2C2×#188311yescharacteristic
521C2#2094411yescharacteristic
522C2#2569411yescharacteristic
523C2#2569611yescharacteristic
524C2#2582411yescharacteristic
525C2#2582611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3165C4semidirect
132. #343Q8semidirect
338. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342132
#213242
C2×#19862
#141552
#112172
#108992
#108512
C4×#3432