Finite Group Databasebeta

← #4320302

Subgroups — #4320302

Order 512.

2162 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16824C211yesmaximal, characteristic
2#16671C214yesmaximal, complemented
3#16358C212yesmaximal
4#16254C211yesmaximal, characteristic
5#16166C214yesmaximal, complemented
6#15623C214yesmaximal, complemented
7#14893C212yesmaximal
8#14816C214yesmaximal, complemented
9#14720C212yesmaximal
10#14551C212yesmaximal
11#14449C212yesmaximal
12#13938C212yesmaximal
13#13795C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#400C2212yes—
19C2×#399C2212yes—
20C2×#398C2214yes—
21C2×#395C2212yes—
22C2×#395C2214yes—
23C2×#395C2212yes—
24C2×#391C2214yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#385C2214yes—
28C2×#383C2212yes—
29C2×C4×C4⋊C4C2212yes—
30#1445C2214yes—
31#1445C2214yes—
32#1444C2214yes—
33#1444C2214yes—
34#1444C2214yes—
35#1443C2214yes—
36#1441C2212yes—
37#1441C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2214yes—
41#1441C2212yes—
42#1441C2214yes—
43#1441C2212yes—
44#1441C2212yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2212yes—
48#1439C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2212yes—
52#1437C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2214yes—
58C4⋊#95C2214yes—
59#1434C2214yes—
60#1434C2212yes—
61C22⋊#95C2214yes—
62C22⋊#95C2211yescharacteristic
63C22⋊#95C2212yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C23×Q8C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—48no—
73C2×#117—24no—
74C2×#117—42no—
75C2×#117—42no—
76C2×#117—42no—
77C2×#117—42no—
78C2×#117Q818yescomplemented
79C22×C4⋊C4—24no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2314yes—
105#400—44no—
106#399—28no—
107#398—48no—
108#395—44no—
109#395—44no—
110#395—48no—
111#391—48no—
112#389—44no—
113#389—48no—
114#385—48no—
115#383—44no—
116C4×C4⋊C4—44no—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C25C2411yescharacteristic
144C22×Q8—24no—
145C22×Q8—28no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—44no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4C2×Q814yes—
163C23×C4#4114yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C22⋊Q8—432no—
177#117—416no—
178#117—416no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—416no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×Q818yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4C2×Q814yes—
234C2×C22⋊C4C2×Q814yes—
235C2×C22⋊C4#4114yes—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42C2×Q818yes—
245#95—24no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—42no—
255#95—44no—
256#95—42no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—44no—
271C24—24no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—24no—
276C24—21no—
277C24C22×Q812yes—
278C24C2×#4112yes—
279C24C2×#4112yes—
280C24C2511yescharacteristic
281C2×Q8—216no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—88no—
285C2×Q8—88no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—44no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4C22×Q814yes—
362C22×C4C22×Q814yes—
363C22×C4C22×Q812yes—
364C22×C4C22×Q812yes—
365C22×C4C2×#4114yes—
366C22×C4C2×#4114yes—
367C22×C4H(5,𝔽2)18yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+418yes—
372C22×C42−1+412yes—
373C4⋊C4—48no—
374C4⋊C4—416no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—416no—
382C4⋊C4—48no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—416no—
391C22⋊C4—416no—
392C22⋊C4—44no—
393C42—216no—
394C42—48no—
395C42—88no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—28no—
404C23—24no—
405C23—28no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—42no—
411C23—44no—
412C23#53314yes—
413C23C23×Q811yescharacteristic
414C23C22×#4111yescharacteristic
415C23C22×#4111yescharacteristic
416C23C2×H(5,𝔽2)11yescharacteristic
417C23C2×H(5,𝔽2)11yescharacteristic
418C23C2×H(5,𝔽2)11yescharacteristic
419C23C2×H(5,𝔽2)14yes—
420C23C2×2−1+411yescharacteristic
421C23C2×2−1+414yes—
422Q8—816no—
423C2×C4—232no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—88no—
495C2×C4—88no—
496C2×C4#54314yes—
497C2×C4#54414yes—
498C22—28no—
499C22—24no—
500C22—44no—
501C22—42no—
502C22—44no—
503C22—44no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22C2×#53311yescharacteristic
508C22C2×#54311yescharacteristic
509C22C2×#54411yescharacteristic
510C22#349011yescharacteristic
511C22#349012yes—
512C22#349011yescharacteristic
513C22#349211yescharacteristic
514C22#349211yescharacteristic
515C22#350314yes—
516C22#350912yes—
517C22#352014yes—
518C22#353712yes—
519C22#353712yes—
520C22#353812yes—
521C22#354814yes—
522C22#355012yes—
523C22#355214yes—
524C4—24no—
525C4—24no—
526C4—44no—
527C4—44no—
528C4—48no—
529C4—48no—
530C4—44no—
531C4—44no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4—84no—
536C2—44no—
537C2#3505211yescharacteristic
538C2#3574312yes—
539C2#3730712yes—
540C2#4293312yes—
541C2#4638411yescharacteristic
542C2#4675812yes—
543C2#4703411yescharacteristic
544C2#4959412yes—
545C2#4970312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #16671C2semidirect
5. #16166C2semidirect
6. #15623C2semidirect
8. #14816C2semidirect
78. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#168242
#166712
#163582
#162542
#161662
#156232
#148932
#148162
#147202
#145512
#144492
#139382
#137952