Finite Group Databasebeta

← #919297

Subgroups — #919297

Order 512.

2122 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2399C211yesmaximal, characteristic
3#17143C212yesmaximal
4#17085C212yesmaximal
5#16610C214yesmaximal
6#16553C214yesmaximal
7#15790C212yesmaximal
8#15625C212yesmaximal
9#15221C212yesmaximal
10#15158C212yesmaximal
11#15128C212yesmaximal
12#14266C212yesmaximal
13#14046C212yesmaximal
14#14041C212yesmaximal
15#13925C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2567—28no—
20#2399—28no—
21C2×#398C2212yes—
22C2×#398C2211yescharacteristic
23C2×#398C2211yescharacteristic
24C2×#395C2211yescharacteristic
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#395C2211yescharacteristic
28C2×#395C2212yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#384C2212yes—
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1446C2214yes—
46#1446C2212yes—
47#1445C2212yes—
48#1445C2214yes—
49#1445C2214yes—
50#1445C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2312yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4×Q8—24no—
84C2×C4×Q8—42no—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C2×C4×Q8—42no—
88C2×C4×Q8D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115#398—28no—
116#398—24no—
117#398—24no—
118#395—28no—
119#395—28no—
120#395—28no—
121#395—24no—
122#395—24no—
123#395—48no—
124#395—48no—
125#391—28no—
126#389—28no—
127#389—28no—
128#389—48no—
129#389—44no—
130#389—44no—
131#389Q8116yescomplemented
132#384—28no—
133#382—42no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—42no—
138#376—24no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C22×Q8—22no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8C2×D414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C4⋊Q8—416no—
200C4×Q8—416no—
201C4×Q8—416no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×Q818yes—
257C2×C4⋊C4C2×Q818yes—
258C2×C4⋊C4C2×Q8116yes—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—44no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42C2×D414yes—
275C2×C42C2×D414yes—
276C2×C42C2×D414yes—
277C2×C42C2×Q818yes—
278#95—24no—
279#95—22no—
280#95—22no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95C2×Q8116yes—
308C24C2511yescharacteristic
309C2×Q8—28no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—44no—
320C2×Q8—88no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—88no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×Q814yes—
406C22×C4C22×Q818yes—
407C22×C4C22×Q818yes—
408C22×C4C22×Q814yes—
409C22×C4C22×Q814yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+414yes—
414C22×C42−1+414yes—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—416no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C42—28no—
435C42—28no—
436C42—28no—
437C42—216no—
438C42—48no—
439C42—48no—
440C42—84no—
441C23C23×D411yescharacteristic
442C23C23×Q812yes—
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×H(5,𝔽2)12yes—
446C23C2×2−1+412yes—
447C23C2×2−1+412yes—
448C23#58512yes—
449C23#58512yes—
450Q8—416no—
451Q8—416no—
452Q8—816no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
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—44no—
538C2×C4#53414yes—
539C2×C4#53614yes—
540C2×C4#54418yes—
541C2×C4D4×Q818yes—
542C2×C4Q8214yes—
543C2×C4#56314yes—
544C22C2×#53411yescharacteristic
545C22C2×#53611yescharacteristic
546C22#344911yescharacteristic
547C22C2×#54412yes—
548C22C2×D4×Q812yes—
549C22C2×Q8211yescharacteristic
550C22#348812yes—
551C22#350112yes—
552C22#350212yes—
553C22#350312yes—
554C22#350412yes—
555C22C2×#56311yescharacteristic
556C22#353711yescharacteristic
557C22#354011yescharacteristic
558C22#354112yes—
559C22#354112yes—
560C22#354212yes—
561C22#355012yes—
562C22#355412yes—
563C22#355512yes—
564C22#355712yes—
565C4—24no—
566C4—48no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—44no—
572C4—42no—
573C4—44no—
574C4—48no—
575C4—44no—
576C4—82no—
577C4—84no—
578C4—84no—
579C4#273418yes—
580C2#3726911yescharacteristic
581C2#3743811yescharacteristic
582C2#3959012yes—
583C2#3959712yes—
584C2#4249612yes—
585C2#4282112yes—
586C2#4495012yes—
587C2C2×#273411yescharacteristic
588C2#5311312yes—
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

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#23992
#171432
#170852
#166102
#165532
#157902
#156252
#152212
#151582
#151282
#142662
#140462
#140412
#139252