Finite Group Databasebeta

← #368944

Subgroups — #368944

Order 512.

1255 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#34953C211yesmaximal, characteristic
3#21337C211yesmaximal, characteristic
4#21303C211yesmaximal, characteristic
5#21303C211yesmaximal, characteristic
6#11223C214yesmaximal
7#10919C214yesmaximal
8#10856C211yesmaximal, characteristic
9C4×#344C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12Q8×M16—22no—
13Q8×M16—22no—
14#2962—22no—
15#2962—22no—
16#2961—22no—
17#2960—22no—
18#2905C2212yes—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2873C2212yes—
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2584—22no—
25#2583C414yes—
26#2567—21no—
27#2546—21no—
28#2502C412yes—
29#2434C412yes—
30#2399—22no—
31C4×C4⋊Q8—22no—
32#2296C2214yes—
33C4×#117C2211yescharacteristic
34#1998—22no—
35#1997—21no—
36#1997—21no—
37#1985—22no—
38#1985—22no—
39#1984—22no—
40#1984—22no—
41C2×#344C2214yes—
42C2×#344C2212yes—
43C2×#344C2212yes—
44#1742C2214yes—
45#1728C2212yes—
46#1728C2214yes—
47#1728C2212yes—
48C23×Q8C2×C411yescharacteristic
49C22×M16C2312yes—
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×#125D412yes—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8C2×C412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8D412yes—
81C2×C4×Q8D412yes—
82C2×#117C2312yes—
83C2×#117C2312yes—
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#446—24no—
89#446—24no—
90#446—28no—
91#446—28no—
92C8×Q8—24no—
93C8×Q8—24no—
94#424—24no—
95#424—24no—
96#423—21no—
97#423—21no—
98#423—21no—
99#423—21no—
100#423—24no—
101#423—21no—
102#423—21no—
103#423—21no—
104#423—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D412yes—
108C2×C4⋊C8D412yes—
109#405—24no—
110#405—24no—
111C4×M16—21no—
112C4×M16—21no—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16—24no—
116C4×M16—21no—
117C4×M16—21no—
118C4×M16—21no—
119C4×M16—21no—
120C2×#97—21no—
121C2×#97D412yes—
122C2×C4×C8—21no—
123C2×C4×C8D412yes—
124#398—24no—
125#398—24no—
126#395—24no—
127#395—24no—
128#391—22no—
129#391—22no—
130#389—28no—
131#389—24no—
132#389—24no—
133#385C2×C418yes—
134#385C2×C418yes—
135#384—22no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C22⋊C4C2×C412yes—
140C4×C22⋊C4C2×C412yes—
141#376—21no—
142C43—21no—
143#344—24no—
144#344—28no—
145#344Q818yescomplemented
146#343—24no—
147#343—24no—
148#336—28no—
149#336—24no—
150#336—24no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—44no—
158C22×Q8C22⋊C412yes—
159C22×Q8C22⋊C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2411yescharacteristic
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—22no—
168C2×M16—21no—
169C2×M16—22no—
170C2×M16—21no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—24no—
174C2×M16—22no—
175C2×M16—24no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—24no—
183C2×M16—22no—
184C2×M16—24no—
185C2×M16C2×Q814yes—
186C2×M16C2×Q814yes—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C2×D412yes—
190C22×C8C2×D414yes—
191C22×C8C2×D412yes—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—48no—
201#125—22no—
202#125—22no—
203#125—22no—
204#125—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237#117—22no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117C2×Q814yes—
244#117#4114yes—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C414yes—
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C22⋊C412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4C22×C414yes—
267C2×C22⋊C4C22×C414yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×D414yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42#4111yescharacteristic
289C2×C42#4111yescharacteristic
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298C4⋊C8—41no—
299C4⋊C8—41no—
300#97—28no—
301#97—22no—
302#97—22no—
303#97—41no—
304#97—41no—
305C4×C8—28no—
306C4×C8—22no—
307C4×C8—22no—
308C4×C8—41no—
309C4×C8—41no—
310#95—22no—
311#95—22no—
312#95—28no—
313#95—28no—
314C24C23×C411yescharacteristic
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—82no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C414yes—
347C22×C4C2×C22⋊C414yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q812yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4112yes—
367C22×C42−1+412yes—
368C22×C42−1+412yes—
369M16—24no—
370M16—24no—
371M16—24no—
372M16—24no—
373M16—42no—
374M16—42no—
375M16—42no—
376M16—42no—
377M16—42no—
378M16—42no—
379M16—44no—
380M16—44no—
381M16—44no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—28no—
389C2×C8—24no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8C4⋊Q814yescomplemented
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—416no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—88no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4C4×Q818yes—
424C42—24no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—44no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—82no—
434C42#12011yescharacteristic
435C42#12011yescharacteristic
436C42#12011yescharacteristic
437C42#12011yescharacteristic
438C42#12111yescharacteristic
439C42#12111yescharacteristic
440C42#12111yescharacteristic
441C42#12111yescharacteristic
442C42#12111yescharacteristic
443C42#12111yescharacteristic
444C42#12111yescharacteristic
445C42#12111yescharacteristic
446C42#12311yescharacteristic
447C42#12311yescharacteristic
448C42#12311yescharacteristic
449C42#12311yescharacteristic
450C23—41no—
451C23—41no—
452C23C22×C22⋊C411yescharacteristic
453C23C2×C4×D411yescharacteristic
454C23C2×C4×D411yescharacteristic
455C23C2×C4×Q812yes—
456C23#51912yes—
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—82no—
464Q8—82no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4#38611yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4#38611yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×#12311yescharacteristic
507C2×C4C2×C4⋊Q811yescharacteristic
508C2×C4#53211yescharacteristic
509C2×C4#53312yes—
510C2×C4#53611yescharacteristic
511C2×C4#53611yescharacteristic
512C2×C4#54112yes—
513C2×C4#54712yes—
514C2×C4#54812yes—
515C2×C4#54812yes—
516C2×C4D4×Q812yes—
517C2×C4D4×Q812yes—
518C2×C4D4×Q812yes—
519C2×C4#55612yes—
520C2×C4#55712yes—
521C8—24no—
522C8—44no—
523C8—42no—
524C8—44no—
525C8—82no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—42no—
534C22C2×#38611yescharacteristic
535C22C4×C4⋊Q811yescharacteristic
536C22#232711yescharacteristic
537C22Q8×C22⋊C412yes—
538C22#236512yes—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—48no—
547C4—44no—
548C4—44no—
549C4—44no—
550C4—82no—
551C4#189112yes—
552C4#242912yes—
553C4#243211yescharacteristic
554C4#243411yescharacteristic
555C4#244612yes—
556C4#254511yescharacteristic
557C4#254611yescharacteristic
558C2—22no—
559C2—21no—
560C2—21no—
561C2—22no—
562C2#1402411yescharacteristic
563C2C2×#189111yescharacteristic
564C2#2096411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
145. #344Q8semidirect
395. C2×C8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#601922
#349532
#213372
#213032
#112232
#109192
#108562
C4×#3442