Finite Group Databasebeta

← #2736657

Subgroups — #2736657

Order 512.

2122 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C212yesmaximal
2C2×#2506C212yesmaximal
3#17134C212yesmaximal
4#17128C214yesmaximal
5#17120C212yesmaximal
6#16688C212yesmaximal
7#16605C214yesmaximal
8#15151C211yesmaximal, characteristic
9#15146C212yesmaximal
10#15139C212yesmaximal
11#15137C212yesmaximal
12#15133C212yesmaximal
13#15133C212yesmaximal
14#13911C211yesmaximal, characteristic
15C4×#398C211yesmaximal, characteristic
16C22×#125C2214yes—
17#2822—216no—
18#2506—216no—
19C2×#400C2214yes—
20C2×#400C2214yes—
21C2×#400C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#398C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#387C2214yes—
35C2×#384C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#382C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43#1445C2211yescharacteristic
44#1445C2212yes—
45#1445C2212yes—
46#1444C2214yes—
47#1444C2211yescharacteristic
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1441C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2211yescharacteristic
77C2×#125—216no—
78C2×#125—216no—
79C2×#125—48no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107#400—216no—
108#400—216no—
109#400—216no—
110#398—216no—
111#398—28no—
112#398—216no—
113#398—216no—
114#398—216no—
115#398—44no—
116#398D418yes—
117#389—28no—
118#389—28no—
119#389—28no—
120#389—28no—
121#389—44no—
122#387—216no—
123#384—28no—
124#384—42no—
125#384—42no—
126#382—28no—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4D418yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2411yescharacteristic
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178#125—464no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
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—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C2×D418yes—
237C2×C4⋊C4C2×D418yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42C2×D418yes—
255C2×C42C2×D414yes—
256#95—24no—
257#95—24no—
258#95—28no—
259#95—28no—
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—24no—
269#95—24no—
270#95—24no—
271#95—42no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—41no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—41no—
281#95—41no—
282#95—44no—
283#95—42no—
284#95—44no—
285#95C2×D418yes—
286#95C2×D418yes—
287#95C2×D414yes—
288C24C2511yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
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—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D414yes—
370C22×C4C22×D418yes—
371C22×C4C22×D414yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C42−1+412yes—
378C22×C42−1+414yes—
379C22×C42−1+414yes—
380C22×C42−1+414yes—
381C22×C42−1+412yes—
382C4⋊C4—216no—
383C4⋊C4—216no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—416no—
389C4⋊C4—44no—
390C4⋊C4—416no—
391C4⋊C4—416no—
392C4⋊C4—416no—
393C4⋊C4—416no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—416no—
397C4⋊C4—48no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C42—28no—
404C42—216no—
405C42—48no—
406C42—48no—
407C42—44no—
408C42—44no—
409C23C23×D411yescharacteristic
410C23C23×D411yescharacteristic
411C23C2×H(5,𝔽2)11yescharacteristic
412C23C2×H(5,𝔽2)11yescharacteristic
413C23C2×2−1+412yes—
414C23C2×2−1+412yes—
415C23C2×2−1+411yescharacteristic
416C23C2×2−1+412yes—
417C23C2×2−1+411yescharacteristic
418C23#58512yes—
419C23#58511yescharacteristic
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4#53618yes—
508C2×C4#53714yes—
509C2×C4D4214yes—
510C2×C4#56314yes—
511C2×C4#56314yes—
512C2×C4#56418yes—
513C22C2×#53612yes—
514C22C2×#53711yescharacteristic
515C22C2×D4211yescharacteristic
516C22#348912yes—
517C22#349611yescharacteristic
518C22#349711yescharacteristic
519C22#349712yes—
520C22#349911yescharacteristic
521C22#349912yes—
522C22C2×#56311yescharacteristic
523C22C2×#56311yescharacteristic
524C22C2×#56412yes—
525C22#353611yescharacteristic
526C22#353811yescharacteristic
527C22#354712yes—
528C22#355012yes—
529C22#355111yescharacteristic
530C22#355512yes—
531C22#355612yes—
532C22#355712yes—
533C22#355911yescharacteristic
534C22#356312yes—
535C22#356312yes—
536C4—24no—
537C4—42no—
538C4—48no—
539C4—44no—
540C4—42no—
541C4—42no—
542C4—44no—
543C4—42no—
544C4—44no—
545C4—81no—
546C4—82no—
547C4—84no—
548C4—84no—
549C4—84no—
550C4#282218yescomplemented
551C2C2×#282211yescharacteristic
552C2#3934312yes—
553C2#3935911yescharacteristic
554C2#4280112yes—
555C2#4298011yescharacteristic
556C2#4548811yescharacteristic
557C2C4⋊#56412yes—
558C2#4632911yescharacteristic
559C2#5301711yescharacteristic
560C2#5310011yescharacteristic
561C2#5313112yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
550. C4#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#25062
#171342
#171282
#171202
#166882
#166052
#151512
#151462
#151392
#151372
#151332
#139112
C4×#3982