Finite Group Databasebeta

← #895565

Subgroups — #895565

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic
2C2×#2734C212yesmaximal
3C2×C4×C4⋊Q8C211yesmaximal, characteristic
4#17078C212yesmaximal
5#16563C212yesmaximal
6#15859C214yesmaximal
7#15623C214yesmaximal
8#14880C212yesmaximal
9#14816C214yesmaximal
10#14699C212yesmaximal
11#14467C211yesmaximal, characteristic
12#14060C214yesmaximal
13#14024C212yesmaximal
14C22×C4⋊Q8C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#2734—28no—
19#2734—216no—
20C2×#398C2212yes—
21C2×#398C2214yes—
22C2×#398C2212yes—
23C2×#395C2214yes—
24C2×#395C2214yes—
25C2×#395C2214yes—
26C4×C4⋊Q8—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#387C2214yes—
40C2×#384C2211yescharacteristic
41C2×C4×C4⋊C4C2214yes—
42C2×C43C2211yescharacteristic
43#1445C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69C4×#95C2212yes—
70C23×Q8C2312yes—
71C23×Q8C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—28no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8D418yes—
79C2×C4×Q8—24no—
80C2×C4×Q8—24no—
81C2×C4×Q8—42no—
82C2×C4×Q8—42no—
83C2×C4×Q8—42no—
84C2×C4×Q8—42no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112#398—28no—
113#398—216no—
114#398—44no—
115#395—216no—
116#395—216no—
117#395—216no—
118#391—28no—
119#391—24no—
120#391—44no—
121#391—44no—
122#391D418yes—
123#389—216no—
124#389—28no—
125#389—216no—
126#389—44no—
127#389Q8116yescomplemented
128#387—216no—
129#387—216no—
130#387—216no—
131#384—24no—
132C4×C4⋊C4—216no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C43—24no—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C2×D414yes—
175C22×Q8C2×D414yes—
176C22×Q8C2×D414yes—
177C22×Q8#4114yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C4⋊Q8—232no—
192C4⋊Q8—432no—
193C4×Q8—416no—
194C4×Q8—416no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C2×D4116yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×Q818yes—
246C2×C4⋊C4C2×Q8116yes—
247C2×C4⋊C4C2×Q818yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42C2×D414yes—
273C2×C42C2×D414yes—
274C2×C42C2×Q818yes—
275#95—28no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95C2×D4116yes—
297#95C2×Q8116yes—
298C24C2511yescharacteristic
299C2×Q8—28no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—84no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8#121116yes—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
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—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4C22×D414yes—
385C22×C4C22×D414yes—
386C22×C4C22×D412yes—
387C22×C4C22×D414yes—
388C22×C4C22×D414yes—
389C22×C4C22×D412yes—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×Q818yes—
393C22×C4C22×Q818yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q814yes—
396C22×C4C22×Q814yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4114yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+414yes—
402C22×C42−1+414yes—
403C4⋊C4—232no—
404C4⋊C4—216no—
405C4⋊C4—232no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—416no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—416no—
414C4⋊C4—416no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C42—28no—
422C42—28no—
423C42—216no—
424C42—44no—
425C42—48no—
426C42—48no—
427C42—48no—
428C42—48no—
429C42—84no—
430C23C23×D411yescharacteristic
431C23C23×D411yescharacteristic
432C23C23×Q812yes—
433C23C22×#4111yescharacteristic
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×H(5,𝔽2)11yescharacteristic
436C23C2×2−1+412yes—
437C23C2×2−1+412yes—
438C23#58511yescharacteristic
439C23#58512yes—
440C23#58511yescharacteristic
441Q8—416no—
442Q8—416no—
443Q8—816no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
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—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4C2×#12114yes—
523C2×C4C2×#12118yes—
524C2×C4D4×Q818yes—
525C2×C4D4×Q818yes—
526C2×C4Q8214yes—
527C22C22×#12111yescharacteristic
528C22#345812yes—
529C22#345911yescharacteristic
530C22#345911yescharacteristic
531C22C2×D4×Q812yes—
532C22C2×D4×Q812yes—
533C22C2×Q8211yescharacteristic
534C22Q8×#4112yes—
535C22#348411yescharacteristic
536C22#348511yescharacteristic
537C22#348912yes—
538C22#349611yescharacteristic
539C22#349611yescharacteristic
540C22#349712yes—
541C22#350212yes—
542C22#350312yes—
543C22#352312yes—
544C22#352911yescharacteristic
545C22#354411yescharacteristic
546C22#355011yescharacteristic
547C22#355912yes—
548C22#356312yes—
549C22#356412yes—
550C4—24no—
551C4—28no—
552C4—24no—
553C4—28no—
554C4—28no—
555C4—42no—
556C4—44no—
557C4—44no—
558C4—48no—
559C4—48no—
560C4—82no—
561C4—84no—
562C4—84no—
563C2#3498411yescharacteristic
564C2Q8×#12112yes—
565C2#3726611yescharacteristic
566C2#3768912yes—
567C2#3936512yes—
568C2#3970312yes—
569C2#4622811yescharacteristic
570C2#4631211yescharacteristic
571C2#5228711yescharacteristic
572C2#5915111yescharacteristic
573C2#5983211yescharacteristic
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
127. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×C4×C4⋊Q82
#170782
#165632
#158592
#156232
#148802
#148162
#146992
#144672
#140602
#140242