Finite Group Databasebeta

← #2923736

Subgroups — #2923736

Order 512.

2438 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2545C211yesmaximal, characteristic
3#16629C212yesmaximal, complemented
4#16466C214yesmaximal, complemented
5#15713C214yesmaximal, complemented
6#15392C214yesmaximal, complemented
7#15262C212yesmaximal
8#15262C212yesmaximal
9#15143C212yesmaximal, complemented
10#14880C212yesmaximal, complemented
11#14405C214yesmaximal, complemented
12#14242C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×Q8C2211yescharacteristic
19#2724—28no—
20#2545—28no—
21C2×#400C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#396C2214yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#384C2214yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40#1445C2212yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2212yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1440C2212yes—
53#1440C2214yes—
54#1440C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66#1434C2212yes—
67#1434C2214yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—44no—
75C2×#124—24no—
76C2×#124—24no—
77C2×#124—42no—
78C2×#124—42no—
79C2×#124—44no—
80C2×#123—28no—
81C2×#123—48no—
82C2×C4×Q8—24no—
83C2×C4×Q8—42no—
84C2×C4×Q8—42no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#400—28no—
113#398—28no—
114#398—44no—
115#397—216no—
116#396—216no—
117#395—48no—
118#395—48no—
119#391—24no—
120#390D418yes—
121#389—28no—
122#389—44no—
123#388—216no—
124#388—216no—
125#385—216no—
126#385—44no—
127#385—44no—
128#385—48no—
129#384—48no—
130C4×C4⋊C4—24no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C25C2412yes—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—42no—
164C22×D4—24no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—42no—
168C22×D4C2×D414yes—
169C23×C4—24no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C4⋊Q8—416no—
187#124—416no—
188#124—416no—
189#123—432no—
190C4×Q8—416no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×D4116yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—44no—
264C2×C42C2×D414yes—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95#4118yes—
287C24—24no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24C22×D414yes—
296C24C2511yescharacteristic
297C2×Q8—28no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×D4—28no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—84no—
311C2×D4—84no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—48no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4C22×D414yes—
376C22×C4C22×D412yes—
377C22×C4C22×D414yes—
378C22×C4C2×#4118yes—
379C22×C4C2×#4114yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C4H(5,𝔽2)14yes—
384C22×C42−1+418yes—
385C22×C42−1+414yes—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—416no—
390C4⋊C4—416no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C22⋊C4—232no—
400C22⋊C4—48no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—416no—
406C22⋊C4—48no—
407C22⋊C4—416no—
408C42—28no—
409C42—44no—
410C42—416no—
411C42—48no—
412C42—44no—
413C23—24no—
414C23—28no—
415C23—24no—
416C23—24no—
417C23—28no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23C23×D411yescharacteristic
435C23C22×#4112yes—
436C23C2×H(5,𝔽2)12yes—
437C23C2×H(5,𝔽2)11yescharacteristic
438C23C2×H(5,𝔽2)12yes—
439C23C2×H(5,𝔽2)11yescharacteristic
440C23C2×2−1+412yes—
441C23C2×2−1+414yes—
442Q8—416no—
443Q8—88no—
444D4—88no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4#53514yes—
514C2×C4#54718yes—
515C22—48no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—48no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22C2×#53511yescharacteristic
525C22C2×#54712yes—
526C22#348111yescharacteristic
527C22#348411yescharacteristic
528C22#348411yescharacteristic
529C22#349912yes—
530C22#351114yes—
531C22#352014yes—
532C22#352212yes—
533C22#353712yes—
534C22#353712yes—
535C22#353812yes—
536C22#354412yes—
537C22#355011yescharacteristic
538C22#355214yes—
539C22#355214yes—
540C4—24no—
541C4—28no—
542C4—48no—
543C4—42no—
544C4—44no—
545C4—48no—
546C4—44no—
547C4—42no—
548C4—84no—
549C4—84no—
550C4—82no—
551C4—82no—
552C2—84no—
553C2#3671711yescharacteristic
554C2#3948812yes—
555C2#4300412yes—
556C2#4486012yes—
557C2#5082314yes—
558C2#5154211yescharacteristic
559C2#5372612yes—
560C2#5907011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16629C2semidirect
4. #16466C2semidirect
5. #15713C2semidirect
6. #15392C2semidirect
9. #15143C2semidirect
10. #14880C2semidirect
11. #14405C2semidirect
12. #14242C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25452
#166292
#164662
#157132
#153922
#152622
#151432
#148802
#144052
#142422
#141572