Finite Group Databasebeta

← #874898

Subgroups — #874898

Order 512.

2650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C212yesmaximal
2#17075C211yesmaximal, characteristic, complemented
3#17072C211yesmaximal, characteristic, complemented
4#17004C212yesmaximal, complemented
5#15676C214yesmaximal, complemented
6#15627C214yesmaximal, complemented
7#14821C212yesmaximal
8#14758C214yesmaximal
9#14697C212yesmaximal
10#14587C212yesmaximal, complemented
11#14441C212yesmaximal, complemented
12#14047C214yesmaximal
13C2×C4⋊#95C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2583—216no—
19C2×#398C2212yes—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#389C2214yes—
27C2×#389C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C22×#95C2214yes—
38#1446C2214yes—
39#1445C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2212yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2214yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64#1434C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C23×Q8C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2312yes—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8—44no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4D418yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42—24no—
103C22×C42—42no—
104C22×C42—42no—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109#398—28no—
110#398—28no—
111#398—28no—
112#398—44no—
113#395—28no—
114#395—28no—
115#395—48no—
116#389—216no—
117#389—216no—
118#389—28no—
119#389—44no—
120#387—48no—
121#385—216no—
122#385—216no—
123#380—44no—
124#380—44no—
125#380Q8116yescomplemented
126C4×C22⋊C4—28no—
127C2×#95—48no—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2411yescharacteristic
153C22×Q8—24no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—44no—
157C22×Q8—44no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—28no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C2×D414yes—
180C23×C4C2×D418yes—
181C23×C4C2×Q818yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C22⋊Q8—432no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D418yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×Q818yes—
248C2×C4⋊C4C2×Q818yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4C2×Q8116yes—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—42no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—416no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42#4114yes—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—44no—
292#95C2×Q8116yes—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—21no—
298C24—22no—
299C24C22×D412yes—
300C24C22×Q814yes—
301C24H(5,𝔽2)12yes—
302C24H(5,𝔽2)12yes—
303C242−1+414yes—
304C24C2511yescharacteristic
305C2×Q8—48no—
306C2×Q8—88no—
307C2×Q8—88no—
308C2×Q8—84no—
309C2×Q8—84no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—216no—
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—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
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—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—48no—
382C22×C4—416no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—416no—
389C22×C4—48no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4C22×D414yes—
396C22×C4C22×D414yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×Q814yes—
400C22×C4C22×Q818yes—
401C22×C4C22×Q818yes—
402C22×C4C22×Q814yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4114yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C42−1+412yes—
410C4⋊C4—216no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—416no—
419C4⋊C4—48no—
420C4⋊C4—416no—
421C4⋊C4—416no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C4⋊C4—84no—
428C22⋊C4—232no—
429C22⋊C4—416no—
430C42—44no—
431C42—48no—
432C42—44no—
433C42—88no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23#53414yes—
453C23#54418yes—
454C23D4×Q818yes—
455C23Q8214yes—
456C23#56314yes—
457C23C23×D411yescharacteristic
458C23C23×Q812yes—
459C23C22×#4111yescharacteristic
460C23C22×#4111yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)12yes—
465C23C2×2−1+411yescharacteristic
466C23C2×2−1+412yes—
467C23#58512yes—
468Q8—816no—
469C2×C4—232no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—432no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—816no—
545C2×C4—816no—
546C2×C4#53214yes—
547C2×C4#53714yes—
548C2×C4#55418yes—
549C22—24no—
550C22—24no—
551C22—28no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—28no—
556C22—28no—
557C22—24no—
558C22—28no—
559C22#273418yes—
560C22C2×#53211yescharacteristic
561C22C2×#53411yescharacteristic
562C22C2×#53711yescharacteristic
563C22C2×#54412yes—
564C22C2×D4×Q812yes—
565C22C2×#55412yes—
566C22C2×Q8211yescharacteristic
567C22Q8×#4112yes—
568C22#349011yescharacteristic
569C22#349011yescharacteristic
570C22#349211yescharacteristic
571C22#349211yescharacteristic
572C22#349512yes—
573C22#349612yes—
574C22#350112yes—
575C22#350312yes—
576C22#352012yes—
577C22C2×#56311yescharacteristic
578C22#353412yes—
579C22#353612yes—
580C22#354311yescharacteristic
581C22#354511yescharacteristic
582C22#356312yes—
583C4—24no—
584C4—24no—
585C4—28no—
586C4—44no—
587C4—44no—
588C4—44no—
589C4—48no—
590C4—48no—
591C4—44no—
592C4—84no—
593C4—82no—
594C4—84no—
595C2—28no—
596C2#3505111yescharacteristic
597C2#3656112yes—
598C2#3712911yescharacteristic
599C2#3787312yes—
600C2#3844712yes—
601C2#3961412yes—
602C2#4614511yescharacteristic
603C2#4619911yescharacteristic
604C2C2×#273411yescharacteristic
605C2#5932211yescharacteristic
606C2#5992311yescharacteristic
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
2. #17075C2semidirect
3. #17072C2semidirect
4. #17004C2semidirect
5. #15676C2semidirect
6. #15627C2semidirect
10. #14587C2semidirect
11. #14441C2semidirect
125. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
#170752
#170722
#170042
#156762
#156272
#148212
#147582
#146972
#145872
#144412
#140472
C2×C4⋊#952