Finite Group Databasebeta

← #822376

Subgroups — #822376

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C212yesmaximal
2C2×#2550C212yesmaximal
3#17072C212yesmaximal
4#15779C212yesmaximal
5#15698C214yesmaximal
6#15363C212yesmaximal
7#15228C212yesmaximal
8#15137C214yesmaximal
9#15128C211yesmaximal, characteristic
10#14236C212yesmaximal
11#14046C212yesmaximal
12#14006C212yesmaximal
13#13910C211yesmaximal, characteristic
14C4×#395C211yesmaximal, characteristic
15#13818C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2212yes—
18C22×C4×Q8C2211yescharacteristic
19#2584—216no—
20#2550—216no—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#395C2212yes—
24C2×#395C2211yescharacteristic
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#391C2214yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#382C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#1446C2212yes—
47#1446C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1444C2214yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2211yescharacteristic
62#1441C2214yes—
63#1441C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4×#95C2214yes—
75C4×#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—42no—
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—44no—
84C2×C4×Q8—28no—
85C2×C4×Q8—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2314yes—
113C22×C42C2314yes—
114C22×C42C2312yes—
115#398—216no—
116#398—216no—
117#395—216no—
118#395—216no—
119#395—24no—
120#395—28no—
121#391—216no—
122#389—28no—
123#389—216no—
124#389—28no—
125#389—216no—
126#389—28no—
127#389—28no—
128#389—216no—
129#389—48no—
130#389D4116yes—
131#389Q8116yescomplemented
132#384—42no—
133#384—44no—
134#382—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—42no—
139C4×C4⋊C4Q818yescomplemented
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C22×Q8—24no—
169C22×Q8—28no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—42no—
174C22×Q8—41no—
175C22×Q8—42no—
176C22×Q8—44no—
177C22×Q8—42no—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C4⋊Q8—416no—
194#125—432no—
195C4×Q8—416no—
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—24no—
203C2×C4⋊C4—24no—
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—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D4116yes—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×Q818yes—
252C2×C4⋊C4C2×Q8116yes—
253C2×C4⋊C4C2×Q818yes—
254C2×C4⋊C4C2×Q818yes—
255C2×C4⋊C4#4114yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C2×D418yes—
275C2×C42C2×Q814yes—
276C2×C42C2×Q818yes—
277C2×C42C2×Q818yes—
278#95—24no—
279#95—24no—
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—24no—
289#95—22no—
290#95—24no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—42no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—41no—
300#95—41no—
301#95—41no—
302#95—44no—
303#95C2×D4116yes—
304#95C2×Q818yes—
305#95C2×Q8116yes—
306C24C2511yescharacteristic
307C2×Q8—44no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8—416no—
311C2×Q8—48no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—88no—
315C2×Q8—84no—
316C2×Q8—84no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4C22×D418yes—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D418yes—
393C22×C4C22×D414yes—
394C22×C4C22×Q812yes—
395C22×C4C22×Q814yes—
396C22×C4C22×Q818yes—
397C22×C4C22×Q814yes—
398C22×C4C22×Q818yes—
399C22×C4C22×Q818yes—
400C22×C4C22×Q812yes—
401C22×C4C22×Q812yes—
402C22×C4C22×Q814yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+412yes—
409C22×C42−1+412yes—
410C4⋊C4—216no—
411C4⋊C4—216no—
412C4⋊C4—416no—
413C4⋊C4—416no—
414C4⋊C4—44no—
415C4⋊C4—416no—
416C4⋊C4—44no—
417C4⋊C4—416no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—416no—
423C4⋊C4—416no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—88no—
427C4⋊C4—84no—
428C4⋊C4—88no—
429C4⋊C4—84no—
430C4⋊C4C4⋊Q8132yes—
431C42—28no—
432C42—216no—
433C42—216no—
434C42—48no—
435C42—44no—
436C42—44no—
437C42—88no—
438C42—88no—
439C23C23×D412yes—
440C23C23×Q811yescharacteristic
441C23C23×Q812yes—
442C23C22×#4111yescharacteristic
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×2−1+411yescharacteristic
446C23C2×2−1+411yescharacteristic
447C23#58512yes—
448C23#58512yes—
449C23#58511yescharacteristic
450Q8—416no—
451Q8—816no—
452C2×C4—22no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4C2×C4⋊Q8116yes—
539C2×C4C2×C4⋊Q818yes—
540C2×C4#54414yes—
541C2×C4D4214yes—
542C2×C4D4×Q818yes—
543C2×C4#55714yes—
544C2×C4Q8214yes—
545C22C22×C4⋊Q812yes—
546C22C2×#54411yescharacteristic
547C22C2×D4211yescharacteristic
548C22C2×D4×Q812yes—
549C22D4×#4112yes—
550C22C2×#55711yescharacteristic
551C22C2×Q8211yescharacteristic
552C22Q8×#4112yes—
553C22#348512yes—
554C22#348812yes—
555C22#349712yes—
556C22#350112yes—
557C22#350111yescharacteristic
558C22#350212yes—
559C22#350412yes—
560C22#350412yes—
561C22#350511yescharacteristic
562C22#351111yescharacteristic
563C22#353611yescharacteristic
564C22#354011yescharacteristic
565C22#355111yescharacteristic
566C22#356311yescharacteristic
567C22#356512yes—
568C4—24no—
569C4—216no—
570C4—44no—
571C4—48no—
572C4—44no—
573C4—44no—
574C4—42no—
575C4—48no—
576C4—42no—
577C4—44no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—84no—
582C4#255118yes—
583C2C2×#255111yescharacteristic
584C2#3595012yes—
585C2#3646612yes—
586C2#3975111yescharacteristic
587C2#4298111yescharacteristic
588C2#4509912yes—
589C2#4524411yescharacteristic
590C2#4536411yescharacteristic
591C2#5900011yescharacteristic
592C2#5915312yes—
593C2#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
131. #389Q8semidirect
139. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
C2×#25502
#170722
#157792
#156982
#153632
#152282
#151372
#151282
#142362
#140462
#140062
#139102
C4×#3952
#138182