Finite Group Databasebeta

← #622267

Subgroups — #622267

Order 512.

2302 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×#2629C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3#15809C214yesmaximal, complemented
4#15767C214yesmaximal, complemented
5#15470C214yesmaximal
6#15385C214yesmaximal
7#15200C212yesmaximal, complemented
8#15156C212yesmaximal, complemented
9#14685C212yesmaximal
10#14343C212yesmaximal
11#14160C212yesmaximal, complemented
12#13911C212yesmaximal, complemented
13#13777C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2629—28no—
19C2×#400C2214yes—
20C2×#398C2214yes—
21C2×#396C2214yes—
22#2296—28no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#384C2211yescharacteristic
29C2×#382C2214yes—
30C2×#381C2214yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36#1445C2214yes—
37#1445C2214yes—
38#1445C2214yes—
39#1443C2214yes—
40#1443C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2211yescharacteristic
66#1434C2214yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C2×#125—24no—
72C2×#125—28no—
73C2×#117—24no—
74C2×#117—28no—
75C2×#117—28no—
76C2×#117D418yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4C2311yescharacteristic
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C42—24no—
96C22×C42—28no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2314yes—
103C22×C42C2311yescharacteristic
104#400—48no—
105#398—216no—
106#396—216no—
107#389—28no—
108#389—28no—
109#389—28no—
110#385—28no—
111#385—28no—
112#384—24no—
113#382—48no—
114#381—48no—
115#380—28no—
116#380—28no—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—28no—
119C4×C4⋊C4—28no—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C25C2411yescharacteristic
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4C2×D414yes—
158C23×C4#4118yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2412yes—
162C23×C4C2412yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171#125—416no—
172#117—232no—
173#117—416no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C2×D418yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4C2×D418yes—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—28no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—416no—
241C2×C42C2×D414yes—
242C2×C42C2×D414yes—
243C2×C42#4118yes—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24C22×D412yes—
278C24C2×#4114yes—
279C24C2511yescharacteristic
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—216no—
290C22×C4—216no—
291C22×C4—216no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4#124116yes—
359C22×C4C22×D414yes—
360C22×C4C22×D414yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C2×#4114yes—
364C22×C4C2×#4114yes—
365C22×C4C2×#4114yes—
366C22×C4C2×#4114yes—
367C22×C4C2×#4114yes—
368C22×C4H(5,𝔽2)14yes—
369C22×C42−1+414yes—
370C22×C42−1+412yes—
371C22×C42−1+412yes—
372C4⋊C4—216no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—416no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—88no—
384C4⋊C4—88no—
385C22⋊C4—216no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C42—28no—
391C42—416no—
392C42—44no—
393C42—44no—
394C42—48no—
395C42—48no—
396C42#123116yes—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23C2×#12414yes—
418C23C23×D411yescharacteristic
419C23C22×#4112yes—
420C23C22×#4112yes—
421C23C2×H(5,𝔽2)12yes—
422C23C2×2−1+411yescharacteristic
423C23C2×2−1+411yescharacteristic
424C23C2×2−1+412yes—
425C23#58512yes—
426C23#58512yes—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—232no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—432no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
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×C4C2×#12318yes—
503C2×C4C2×#12314yes—
504C2×C4C2×#12414yes—
505C2×C4C2×#12414yes—
506C2×C4#53614yes—
507C2×C4#55218yes—
508C2×C4#55218yes—
509C22—24no—
510C22—28no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—42no—
517C22—44no—
518C22C22×#12311yescharacteristic
519C22C22×#12411yescharacteristic
520C22C2×#53611yescharacteristic
521C22#346012yes—
522C22#346112yes—
523C22C2×#55212yes—
524C22C2×#55212yes—
525C22#348812yes—
526C22#348912yes—
527C22#349212yes—
528C22#349312yes—
529C22#349412yes—
530C22#349512yes—
531C22#351112yes—
532C22#351212yes—
533C22#354811yescharacteristic
534C22#355412yes—
535C22#355512yes—
536C22#355611yescharacteristic
537C22#355711yescharacteristic
538C22#355711yescharacteristic
539C4—28no—
540C4—24no—
541C4—44no—
542C4—44no—
543C4—44no—
544C4—84no—
545C4—84no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—82no—
550C4#250318yes—
551C2—44no—
552C2C2×#250311yescharacteristic
553C2#3525512yes—
554C2#3554212yes—
555C2#3727411yescharacteristic
556C2#3737711yescharacteristic
557C2#3798212yes—
558C2#3798312yes—
559C2#3798612yes—
560C2#3798712yes—
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. #15809C2semidirect
4. #15767C2semidirect
7. #15200C2semidirect
8. #15156C2semidirect
11. #14160C2semidirect
12. #13911C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#22962
#158092
#157672
#154702
#153852
#152002
#151562
#146852
#143432
#141602
#139112
#137772