Finite Group Databasebeta

← #877702

Subgroups — #877702

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17076C211yesmaximal, characteristic, complemented
2#17070C211yesmaximal, characteristic, complemented
3#16933C212yesmaximal, complemented
4#15542C214yesmaximal, complemented
5#15533C214yesmaximal, complemented
6#14822C212yesmaximal
7#14766C214yesmaximal
8#14697C212yesmaximal
9#14603C212yesmaximal, complemented
10#14446C212yesmaximal, complemented
11#13980C214yesmaximal
12#13923C212yesmaximal
13C2×C4⋊#95C211yesmaximal, characteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#400C2212yes—
18C2×#400C2212yes—
19C2×#398C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#387C2214yes—
23C2×#385C2214yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C22×#95C2214yes—
30#1445C2214yes—
31#1445C2214yes—
32#1444C2214yes—
33#1443C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1441C2214yes—
40#1441C2212yes—
41#1441C2212yes—
42#1441C2212yes—
43#1441C2214yes—
44#1441C2212yes—
45#1441C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2212yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2214yes—
61C4⋊#95C2211yescharacteristic
62#1434C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2312yes—
70C24×C4C2312yes—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4Q818yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C42—24no—
96C22×C42—42no—
97C22×C42—42no—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2314yes—
101C22×C42C2311yescharacteristic
102#400—28no—
103#400—44no—
104#398—48no—
105#389—28no—
106#389—44no—
107#387—48no—
108#385—216no—
109#380—28no—
110#380—44no—
111#380—44no—
112#380—44no—
113C4×C22⋊C4—44no—
114C2×#95—48no—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C25C2411yescharacteristic
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—28no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4C2×Q814yes—
165C23×C4C2×Q818yes—
166C23×C4#4118yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2412yes—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—416no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—416no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×Q814yes—
223C2×C4⋊C4C2×Q814yes—
224C2×C4⋊C4C2×Q818yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—42no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—416no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42#4114yes—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—24no—
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—42no—
261#95—42no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—44no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—22no—
278C24C22×Q812yes—
279C24C2×#4114yes—
280C24H(5,𝔽2)12yes—
281C24H(5,𝔽2)12yes—
282C242−1+414yes—
283C24C2511yescharacteristic
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—216no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—216no—
323C22×C4—216no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
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—48no—
358C22×C4—416no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—416no—
365C22×C4—48no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4C22×Q814yes—
372C22×C4C22×Q814yes—
373C22×C4C22×Q812yes—
374C22×C4C22×Q812yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4114yes—
380C22×C4C2×#4114yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C42−1+412yes—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—416no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—88no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C22⋊C4—416no—
401C22⋊C4—416no—
402C42—44no—
403C42—48no—
404C42—44no—
405C42—88no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23#53918yes—
425C23#54314yes—
426C23#55514yes—
427C23#55718yes—
428C23#56314yes—
429C23C23×Q811yescharacteristic
430C23C22×#4111yescharacteristic
431C23C22×#4111yescharacteristic
432C23C22×#4112yes—
433C23C2×H(5,𝔽2)11yescharacteristic
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×H(5,𝔽2)11yescharacteristic
436C23C2×H(5,𝔽2)12yes—
437C23C2×2−1+412yes—
438C23C2×2−1+411yescharacteristic
439C23#58512yes—
440C2×C4—232no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—432no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
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—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—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—816no—
516C2×C4—816no—
517C2×C4#53314yes—
518C2×C4#54414yes—
519C2×C4#55318yes—
520C22—28no—
521C22—24no—
522C22—24no—
523C22—28no—
524C22—28no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—28no—
530C22#280918yes—
531C22C2×#53311yescharacteristic
532C22C2×#53912yes—
533C22C2×#54311yescharacteristic
534C22C2×#54411yescharacteristic
535C22C2×#55312yes—
536C22C2×#55511yescharacteristic
537C22C2×#55712yes—
538C22#348012yes—
539C22#349011yescharacteristic
540C22#349011yescharacteristic
541C22#349211yescharacteristic
542C22#349211yescharacteristic
543C22#349512yes—
544C22#350112yes—
545C22#350512yes—
546C22#351112yes—
547C22#352012yes—
548C22C2×#56311yescharacteristic
549C22#353412yes—
550C22#353612yes—
551C22#354311yescharacteristic
552C22#354511yescharacteristic
553C22#356312yes—
554C4—28no—
555C4—24no—
556C4—24no—
557C4—44no—
558C4—44no—
559C4—44no—
560C4—44no—
561C4—84no—
562C4—84no—
563C4—82no—
564C4—84no—
565C4—84no—
566C2—28no—
567C2#3505211yescharacteristic
568C2#3659912yes—
569C2#3713411yescharacteristic
570C2#3789012yes—
571C2#3848512yes—
572C2#3975112yes—
573C2#4614511yescharacteristic
574C2#4619911yescharacteristic
575C2C2×#280911yescharacteristic
576C2#5930311yescharacteristic
577C2#5988511yescharacteristic
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
1. #17076C2semidirect
2. #17070C2semidirect
3. #16933C2semidirect
4. #15542C2semidirect
5. #15533C2semidirect
9. #14603C2semidirect
10. #14446C2semidirect

Maximal subgroups

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

subgroupindex
#170762
#170702
#169332
#155422
#155332
#148222
#147662
#146972
#146032
#144462
#139802
#139232
C2×C4⋊#952