Finite Group Databasebeta

← #4607020

Subgroups — #4607020

Order 512.

2272 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2545C211yesmaximal, characteristic
3#17104C214yesmaximal, complemented
4#16618C214yesmaximal, complemented
5#16596C212yesmaximal
6#16510C212yesmaximal
7#16327C214yesmaximal, complemented
8#16027C214yesmaximal, complemented
9#15762C212yesmaximal, complemented
10#15761C212yesmaximal
11#15108C212yesmaximal, complemented
12#14880C212yesmaximal, complemented
13#14668C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×#124C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2734—28no—
20#2545—28no—
21C2×#399C2214yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#396C2214yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#394C2214yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1446C2212yes—
39#1444C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2212yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2212yes—
48#1441C2212yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2212yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2212yes—
62#1437C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1434C2212yes—
68#1434C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—48no—
76C2×C4⋊Q8—44no—
77C2×#124—24no—
78C2×#124—24no—
79C2×#124—42no—
80C2×#124—42no—
81C2×#124—44no—
82C2×C4×Q8—24no—
83C2×C4×Q8—42no—
84C2×C4×Q8—42no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110#399—48no—
111#398—28no—
112#398—44no—
113#396—216no—
114#395—216no—
115#395—48no—
116#395—48no—
117#394—48no—
118#393—44no—
119#393—44no—
120#391D418yes—
121#390—24no—
122#389—28no—
123#389—44no—
124#388—216no—
125#387—216no—
126C4×C4⋊C4—24no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2412yes—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—44no—
166C22×Q8C2×D414yes—
167C22×D4—22no—
168C22×D4—24no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—42no—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C4⋊Q8—416no—
185C4⋊Q8—432no—
186#124—416no—
187#124—416no—
188C4×Q8—416no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—44no—
269#95—42no—
270#95—42no—
271#95—44no—
272#95—42no—
273#95—42no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—44no—
281#95C2×D4116yes—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24C2511yescharacteristic
292C2×Q8—28no—
293C2×Q8—28no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—48no—
297C2×Q8—48no—
298C2×Q8—44no—
299C2×Q8—48no—
300C2×Q8—44no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—88no—
306C2×Q8—88no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—84no—
311C2×D4—84no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
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—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)14yes—
378C22×C4H(5,𝔽2)14yes—
379C22×C42−1+418yes—
380C22×C42−1+414yes—
381C4⋊C4—216no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—416no—
386C4⋊C4—416no—
387C4⋊C4—48no—
388C4⋊C4—416no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—88no—
392C4⋊C4—88no—
393C22⋊C4—416no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C42—28no—
402C42—44no—
403C42—48no—
404C42—44no—
405C42—48no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23C23×D411yescharacteristic
428C23C2×H(5,𝔽2)12yes—
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×H(5,𝔽2)12yes—
431C23C2×H(5,𝔽2)11yescharacteristic
432C23C2×2−1+414yes—
433C23C2×2−1+412yes—
434C23#58512yes—
435Q8—416no—
436Q8—88no—
437Q8—816no—
438D4—88no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
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—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4#53514yes—
508C22—84no—
509C22—84no—
510C22—84no—
511C22—82no—
512C22—82no—
513C22—84no—
514C22—82no—
515C22—84no—
516C22—84no—
517C22C2×#53511yescharacteristic
518C22#348411yescharacteristic
519C22#348411yescharacteristic
520C22#348612yes—
521C22#348912yes—
522C22#352911yescharacteristic
523C22#353712yes—
524C22#353712yes—
525C22#354412yes—
526C22#354714yes—
527C22#355012yes—
528C22#355011yescharacteristic
529C22#355214yes—
530C22#355214yes—
531C22#355912yes—
532C22#356114yes—
533C4—24no—
534C4—48no—
535C4—42no—
536C4—44no—
537C4—44no—
538C4—44no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—84no—
543C4—84no—
544C4—84no—
545C2—162no—
546C2#3671711yescharacteristic
547C2#4281212yes—
548C2#4622811yescharacteristic
549C2#4954912yes—
550C2#5207211yescharacteristic
551C2#5295312yes—
552C2#5351514yes—
553C2#5372612yes—
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
1. C2×#2734C2semidirect
3. #17104C2semidirect
4. #16618C2semidirect
7. #16327C2semidirect
8. #16027C2semidirect
9. #15762C2semidirect
11. #15108C2semidirect
12. #14880C2semidirect
13. #14668C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#25452
#171042
#166182
#165962
#165102
#163272
#160272
#157622
#157612
#151082
#148802
#146682