Finite Group Databasebeta

← #849178

Subgroups — #849178

Order 512.

2442 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic
3#16688C212yesmaximal, complemented
4#16686C212yesmaximal, complemented
5#16060C214yesmaximal, complemented
6#15945C212yesmaximal, complemented
7#15944C212yesmaximal, complemented
8#15073C212yesmaximal
9#14935C212yesmaximal
10#14934C212yesmaximal
11#14695C212yesmaximal
12#14394C214yesmaximal, complemented
13#13992C212yesmaximal
14#13972C212yesmaximal
15#13923C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2583—28no—
21C2×#398C2211yescharacteristic
22C2×#398C2211yescharacteristic
23C2×#395C2211yescharacteristic
24C2×#395C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#382—28no—
32C2×#382C2212yes—
33C2×#382C2212yes—
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45#1446C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1436C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—42no—
85C2×C22⋊Q8—42no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42—24no—
111C22×C42—28no—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116#398—24no—
117#398—24no—
118#395—24no—
119#395—24no—
120#389—28no—
121#389—28no—
122#387—216no—
123#385—28no—
124#385—28no—
125#385—28no—
126#382—28no—
127#382—28no—
128#382—28no—
129#382—28no—
130#381—28no—
131#381—28no—
132#381—44no—
133#381—44no—
134#380—42no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—42no—
137C2×#95—28no—
138C2×#95—28no—
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C25C2411yescharacteristic
171C22×Q8—22no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—44no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C22⋊Q8—416no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—416no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—416no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4#4118yes—
266C2×C42—232no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42#4118yes—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24C22×D412yes—
311C24C2×#4114yes—
312C24H(5,𝔽2)12yes—
313C24H(5,𝔽2)12yes—
314C242−1+414yes—
315C24C2511yescharacteristic
316C2×Q8—44no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8—84no—
320C22×C4—216no—
321C22×C4—216no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
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—28no—
359C22×C4—28no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
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×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4118yes—
417C22×C4C2×#4114yes—
418C22×C42−1+414yes—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—416no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C22⋊C4—216no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C42—216no—
438C42—216no—
439C42—84no—
440C42—84no—
441C23—22no—
442C23—22no—
443C23—28no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—24no—
450C23—24no—
451C23—28no—
452C23#53414yes—
453C23#54218yes—
454C23#54818yes—
455C23#55514yes—
456C23#56314yes—
457C23C23×D411yescharacteristic
458C23C22×#4112yes—
459C23C22×#4112yes—
460C23C2×H(5,𝔽2)11yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×2−1+412yes—
463C23C2×2−1+412yes—
464C23#58514yes—
465Q8—88no—
466C2×C4—232no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—432no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—42no—
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—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—88no—
541C2×C4—88no—
542C2×C4—816no—
543C2×C4#52514yes—
544C2×C4#53614yes—
545C22—216no—
546C22—28no—
547C22—28no—
548C22—28no—
549C22—28no—
550C22—24no—
551C22—24no—
552C22#273118yes—
553C22C2×#52511yescharacteristic
554C22C2×#53411yescharacteristic
555C22C2×#53611yescharacteristic
556C22C2×#54212yes—
557C22C2×#54812yes—
558C22C2×#55511yescharacteristic
559C22#347514yes—
560C22#348512yes—
561C22#349212yes—
562C22#349212yes—
563C22#349514yes—
564C22#351514yes—
565C22C2×#56311yescharacteristic
566C22#354512yes—
567C22#354512yes—
568C22#356314yes—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—48no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—84no—
579C4—84no—
580C4—84no—
581C4—82no—
582C4—82no—
583C2—28no—
584C2#3505711yescharacteristic
585C2#3651612yes—
586C2#3711911yescharacteristic
587C2#3790714yes—
588C2#3912212yes—
589C2#4614512yes—
590C2C2×#273111yescharacteristic
591C2#5933912yes—
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
3. #16688C2semidirect
4. #16686C2semidirect
5. #16060C2semidirect
6. #15945C2semidirect
7. #15944C2semidirect
12. #14394C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C22×#3822
#166882
#166862
#160602
#159452
#159442
#150732
#149352
#149342
#146952
#143942
#139922
#139722
#139232