Finite Group Databasebeta

← #4100370

Subgroups — #4100370

Order 512.

2772 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2707C214yesmaximal, complemented
2#17135C211yesmaximal, characteristic, complemented
3#17121C212yesmaximal, complemented
4#15966C211yesmaximal, characteristic, complemented
5#15458C214yesmaximal, complemented
6#15417C212yesmaximal, complemented
7#15309C214yesmaximal, complemented
8#15261C212yesmaximal
9#15071C212yesmaximal, complemented
10#14976C214yesmaximal, complemented
11#14410C212yesmaximal, complemented
12#14226C212yesmaximal, complemented
13#14154C211yesmaximal, characteristic
14C22×#127C2211yescharacteristic
15C22×#125C2212yes—
16C22×#123C2212yes—
17#2707—232no—
18C2×#398C2212yes—
19C2×#397C2214yes—
20C2×#396C2214yes—
21C2×#396C2214yes—
22C2×#396C2214yes—
23C2×#396C2214yes—
24C2×#394C2214yes—
25C2×#390C2214yes—
26C2×#390C2214yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#380C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1445C2212yes—
40#1444C2212yes—
41#1444C2214yes—
42#1444C2212yes—
43#1442C2212yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2212yes—
47#1441C2212yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2211yescharacteristic
54#1438C2214yes—
55#1438C2211yescharacteristic
56#1438C2211yescharacteristic
57#1438C2211yescharacteristic
58#1438C2212yes—
59#1438C2212yes—
60#1438C2212yes—
61#1438C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1434C2212yes—
67C4×#95C2212yes—
68C23×D4C2312yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#127—24no—
74C2×#127—42no—
75C2×#127—42no—
76C2×#125—216no—
77C2×#125—44no—
78C2×#123—216no—
79C2×#123—44no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2314yes—
108#398D4116yescomplemented
109#397—216no—
110#396—216no—
111#396—216no—
112#396—216no—
113#396—48no—
114#394—216no—
115#390—216no—
116#390—48no—
117#389—216no—
118#389—44no—
119#388—216no—
120#388—216no—
121#388—48no—
122#387—216no—
123#386—28no—
124#386—44no—
125#385—216no—
126#380—216no—
127C4×C4⋊C4—42no—
128C4×C4⋊C4—42no—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2314yes—
149C25C2412yes—
150C25C2412yes—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—28no—
154C22×D4—22no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4#4114yes—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185#127—416no—
186#125—432no—
187#123—432no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4C2×D4116yes—
216C2×C4⋊C4C2×D418yes—
217C2×C4⋊C4C2×D418yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
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×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—44no—
282#95C2×D4116yes—
283#95C2×D418yes—
284C24—24no—
285C24—24no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—22no—
294C24—24no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24C2×#4114yes—
302C24C2511yescharacteristic
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—416no—
308C2×D4—416no—
309C2×D4—44no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—88no—
313C2×D4—88no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—84no—
320C2×D4—84no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
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—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—48no—
374C22×C4—416no—
375C22×C4—88no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4C22×D414yes—
379C22×C4C22×D418yes—
380C22×C4C22×D418yes—
381C22×C4C22×D414yes—
382C22×C4C22×D414yes—
383C22×C4C2×#4112yes—
384C22×C4H(5,𝔽2)14yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C42−1+414yes—
388C4⋊C4—232no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—416no—
393C4⋊C4—48no—
394C4⋊C4—416no—
395C4⋊C4—48no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—84no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—48no—
402C22⋊C4—416no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—28no—
413C42—48no—
414C42—84no—
415C42—88no—
416C23—28no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—28no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23C23×D412yes—
459C23C22×#4111yescharacteristic
460C23C2×H(5,𝔽2)11yescharacteristic
461C23C2×H(5,𝔽2)12yes—
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×2−1+412yes—
464C23#58512yes—
465C23#58514yes—
466D4—816no—
467D4—816no—
468D4—88no—
469D4—88no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—42no—
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—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—816no—
531C2×C4D4214yes—
532C2×C4#56314yes—
533C22—44no—
534C22—48no—
535C22—48no—
536C22—44no—
537C22—44no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—82no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—82no—
547C22—82no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22C2×D4211yescharacteristic
552C22D4×#4112yes—
553C22#348814yes—
554C22#349011yescharacteristic
555C22#349112yes—
556C22#349612yes—
557C22#349714yes—
558C22#352012yes—
559C22C2×#56311yescharacteristic
560C22#353612yes—
561C22#353914yes—
562C22#354311yescharacteristic
563C22#354511yescharacteristic
564C22#355112yes—
565C22#355112yes—
566C22#355714yes—
567C4—24no—
568C4—24no—
569C4—42no—
570C4—44no—
571C4—44no—
572C4—48no—
573C4—48no—
574C4—44no—
575C4—84no—
576C4—84no—
577C4—82no—
578C2—84no—
579C2—162no—
580C2#4438214yes—
581C2#4524412yes—
582C2#4619911yescharacteristic
583C2#4632912yes—
584C2#4719511yescharacteristic
585C2#4745712yes—
586C2#5835512yes—
587C2#5981411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2707C2semidirect
2. #17135C2semidirect
3. #17121C2semidirect
4. #15966C2semidirect
5. #15458C2semidirect
6. #15417C2semidirect
7. #15309C2semidirect
9. #15071C2semidirect
10. #14976C2semidirect
11. #14410C2semidirect
12. #14226C2semidirect
108. #398D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27072
#171352
#171212
#159662
#154582
#154172
#153092
#152612
#150712
#149762
#144102
#142262
#141542