Finite Group Databasebeta

← #852331

Subgroups — #852331

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2792C212yesmaximal
2#17004C212yesmaximal
3#16563C212yesmaximal
4#15862C214yesmaximal
5#15683C214yesmaximal
6#15151C211yesmaximal, characteristic
7#15133C212yesmaximal
8#15128C211yesmaximal, characteristic
9#15128C212yesmaximal
10#14699C212yesmaximal
11#14587C212yesmaximal
12#14046C212yesmaximal
13#14046C212yesmaximal
14#14042C212yesmaximal
15C4×#384C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17#2792—216no—
18C2×#398C2214yes—
19C2×#398C2212yes—
20C2×#398C2212yes—
21C2×#395C2212yes—
22C2×#395C2214yes—
23C2×#389C2214yes—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C43C2211yescharacteristic
41#1446C2214yes—
42#1446C2212yes—
43#1445C2212yes—
44#1445C2212yes—
45#1444C2212yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74#1434C2214yes—
75#1434C2214yes—
76C23×Q8C2312yes—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—44no—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108#398—216no—
109#398—28no—
110#398—28no—
111#395—28no—
112#395—216no—
113#389—216no—
114#389—216no—
115#389—28no—
116#389—44no—
117#389—44no—
118#389—44no—
119#389Q8116yescomplemented
120#387—216no—
121#387—216no—
122#384—24no—
123#384—42no—
124#384D418yes—
125C4×C4⋊C4—44no—
126C4×C4⋊C4—42no—
127C4×C4⋊C4—44no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—44no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C43—24no—
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—44no—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C4⋊Q8—416no—
181C2×C4⋊C4—28no—
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—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
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—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D418yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×Q818yes—
237C2×C4⋊C4C2×Q8116yes—
238C2×C4⋊C4C2×Q818yes—
239C2×C4⋊C4#4114yes—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—42no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42C2×D414yes—
263C2×C42C2×Q818yes—
264C2×C42#4114yes—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—28no—
274#95—24no—
275#95—28no—
276#95—42no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—42no—
284#95—41no—
285#95—41no—
286#95—42no—
287#95—42no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95C2×Q8116yes—
294C24C2511yescharacteristic
295C2×Q8—48no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×Q8—84no—
299C2×Q8—88no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4C22×D412yes—
374C22×C4C22×D414yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D414yes—
378C22×C4C22×Q814yes—
379C22×C4C22×Q818yes—
380C22×C4C22×Q818yes—
381C22×C4C22×Q814yes—
382C22×C4C22×Q814yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)14yes—
391C22×C42−1+414yes—
392C22×C42−1+412yes—
393C4⋊C4—216no—
394C4⋊C4—232no—
395C4⋊C4—216no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—416no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—88no—
409C4⋊C4—88no—
410C4⋊C4—84no—
411C4⋊C4—88no—
412C4⋊C4—84no—
413C42—28no—
414C42—28no—
415C42—216no—
416C42—48no—
417C42—44no—
418C42—44no—
419C42—48no—
420C42—44no—
421C42—88no—
422C23C23×D411yescharacteristic
423C23C23×Q812yes—
424C23C22×#4111yescharacteristic
425C23C22×#4111yescharacteristic
426C23C2×H(5,𝔽2)11yescharacteristic
427C23C2×H(5,𝔽2)12yes—
428C23C2×2−1+411yescharacteristic
429C23C2×2−1+412yes—
430C23#58511yescharacteristic
431C23#58511yescharacteristic
432C23#58512yes—
433Q8—816no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
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—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4#53214yes—
521C2×C4#53914yes—
522C2×C4#54418yes—
523C2×C4D4×Q818yes—
524C2×C4#55418yes—
525C2×C4Q8214yes—
526C2×C4#56014yes—
527C22C2×#53211yescharacteristic
528C22C2×#53911yescharacteristic
529C22#345511yescharacteristic
530C22C2×#54412yes—
531C22C2×D4×Q812yes—
532C22C2×#55412yes—
533C22C2×Q8211yescharacteristic
534C22Q8×#4112yes—
535C22#349512yes—
536C22#349611yescharacteristic
537C22#349612yes—
538C22#349711yescharacteristic
539C22#350212yes—
540C22#350412yes—
541C22#350511yescharacteristic
542C22#350711yescharacteristic
543C22#352012yes—
544C22C2×#56011yescharacteristic
545C22#354712yes—
546C22#355011yescharacteristic
547C22#355111yescharacteristic
548C22#355112yes—
549C22#355512yes—
550C4—24no—
551C4—28no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—44no—
556C4—48no—
557C4—48no—
558C4—44no—
559C4—42no—
560C4—42no—
561C4—82no—
562C4—84no—
563C4—84no—
564C4#279218yescomplemented
565C2#3500611yescharacteristic
566C2#3656112yes—
567C2#3738311yescharacteristic
568C2#3937012yes—
569C2#3960312yes—
570C2#3997412yes—
571C2#4632811yescharacteristic
572C2#4634211yescharacteristic
573C2C2×#279211yescharacteristic
574C2#5983211yescharacteristic
575C2#5992511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
119. #389Q8semidirect
564. C4#2792semidirect

Maximal subgroups

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

subgroupindex
C2×#27922
#170042
#165632
#158622
#156832
#151512
#151332
#151282
#146992
#145872
#140462
#140422
C4×#3842