Finite Group Databasebeta

← #368990

Subgroups — #368990

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic
3C2×#1998C211yesmaximal, characteristic
4C2×#1998C211yesmaximal, characteristic
5#21303C211yesmaximal, characteristic
6#11204C212yesmaximal
7#11204C212yesmaximal
8#10913C212yesmaximal
9#10913C212yesmaximal
10#10847C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14Q8×M16—24no—
15Q8×M16—24no—
16C2×#446—22no—
17C2×#446—21no—
18C2×C8×Q8—21no—
19C2×#423C2212yes—
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×M16C2212yes—
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2584—21no—
26#2584—22no—
27#2584—21no—
28#2584—22no—
29#2584—21no—
30#2584—21no—
31C2×#398C412yes—
32C2×#395C412yes—
33C2×#389C414yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1998—24no—
38#1998—24no—
39#1985—21no—
40#1985—21no—
41#1984—22no—
42C2×#343C2211yescharacteristic
43C2×#343C2211yescharacteristic
44C2×#343C2212yes—
45C2×#343C2212yes—
46C2×#343C2211yescharacteristic
47C2×#343C2211yescharacteristic
48C2×#343C2212yes—
49C2×#343C2212yes—
50C2×#343C2212yes—
51C2×#343C2212yes—
52#1728C2212yes—
53#1728C2212yes—
54C23×Q8C2×C411yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2312yes—
60C22×M16C2311yescharacteristic
61C2×#125—21no—
62C2×#125—21no—
63C2×#125—21no—
64C2×#125—21no—
65C2×#125—22no—
66C2×#125—22no—
67C2×#125—22no—
68C2×#125—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D412yes—
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D412yes—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C22×C4⋊C4C2×C412yes—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C412yes—
83C22×C4⋊C4C2×C414yes—
84C22×C4⋊C4C2×C414yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C42C2×C412yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#446—216no—
92#446—28no—
93C8×Q8—28no—
94#423—24no—
95#423—24no—
96#423—22no—
97#423—22no—
98#423—22no—
99#423D414yes—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C4×M16—24no—
107C4×M16—24no—
108C4×M16—22no—
109C4×M16—22no—
110C4×M16—22no—
111C4×M16D414yes—
112C2×#97—22no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—22no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118#398—22no—
119#398—24no—
120#398—22no—
121#395—22no—
122#395—24no—
123#395—22no—
124#389—24no—
125#389—24no—
126#389—24no—
127#389—24no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C2×#95C2×C412yes—
141C2×#95C2×C414yes—
142C2×#95C2×C414yes—
143#343—24no—
144#343—24no—
145#343—22no—
146#343—22no—
147#343—22no—
148#343—22no—
149#343—24no—
150#343—24no—
151#343—24no—
152#343Q818yescomplemented
153#336—22no—
154#336—22no—
155#336—22no—
156#336—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—44no—
160C22×Q8C22⋊C411yescharacteristic
161C22×Q8C22⋊C412yes—
162C22×Q8C22⋊C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C414yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2411yescharacteristic
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—21no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16C2×D414yes—
195C2×M16C2×D412yes—
196C2×M16C2×D412yes—
197C2×M16C2×Q814yes—
198C2×M16C2×Q814yes—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205#125—42no—
206#125—42no—
207#125—42no—
208#125—42no—
209#125—42no—
210#125—42no—
211#125—42no—
212#125—42no—
213C4×Q8—22no—
214C4×Q8—24no—
215C4×Q8—22no—
216C4×Q8—24no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—24no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—24no—
224C4×Q8—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—84no—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22⋊C412yes—
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C412yes—
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C2×Q814yes—
267C2×C4⋊C4#4114yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—82no—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4112yes—
286C2×C42#4112yes—
287C4⋊C8—28no—
288C4⋊C8—28no—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293#97—28no—
294#97—24no—
295#97—42no—
296C4×C8—28no—
297C4×C8—24no—
298C4×C8—42no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305C24C23×C411yescharacteristic
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—48no—
314C2×Q8—48no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4C4×Q818yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C42−1+412yes—
361C22×C42−1+412yes—
362M16—22no—
363M16—24no—
364M16—22no—
365M16—24no—
366M16—22no—
367M16—24no—
368M16—24no—
369M16—24no—
370M16—42no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—42no—
380M16—42no—
381M16C4⋊Q814yes—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—48no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—82no—
432C42—82no—
433C42#12012yes—
434C42#12012yes—
435C42#12112yes—
436C42#12112yes—
437C42#12112yes—
438C42#12112yes—
439C42#12312yes—
440C42#12312yes—
441C23—22no—
442C23—21no—
443C23—21no—
444C23C22×C22⋊C411yescharacteristic
445C23C2×C4×D411yescharacteristic
446C23C2×C4×D411yescharacteristic
447C23C2×C4×Q812yes—
448C23#51912yes—
449Q8—28no—
450Q8—48no—
451Q8—44no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#38612yes—
492C2×C4#38612yes—
493C2×C4#38612yes—
494C2×C4#38612yes—
495C2×C4#41314yes—
496C2×C4C2×#12011yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4C2×C4⋊Q811yescharacteristic
501C2×C4#53211yescharacteristic
502C2×C4#53312yes—
503C2×C4#53611yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#54112yes—
506C2×C4#54712yes—
507C2×C4#54812yes—
508C2×C4#54812yes—
509C2×C4D4×Q812yes—
510C2×C4D4×Q812yes—
511C2×C4D4×Q812yes—
512C2×C4#55612yes—
513C2×C4#55712yes—
514C8—24no—
515C8—44no—
516C8—42no—
517C8—44no—
518C8—81no—
519C8—81no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22C2×#38611yescharacteristic
529C22C4×C4⋊Q811yescharacteristic
530C22#232711yescharacteristic
531C22Q8×C22⋊C412yes—
532C22#236512yes—
533C22C2×#41312yes—
534C22#288811yescharacteristic
535C22#288911yescharacteristic
536C4—22no—
537C4—22no—
538C4—22no—
539C4—22no—
540C4—24no—
541C4—42no—
542C4—42no—
543C4—48no—
544C4—42no—
545C4—82no—
546C4—82no—
547C4#242912yes—
548C4#243211yescharacteristic
549C4#243411yescharacteristic
550C4#244612yes—
551C4#254511yescharacteristic
552C4#254611yescharacteristic
553C2—21no—
554C2—21no—
555C2—22no—
556C2#1402411yescharacteristic
557C2#2098911yescharacteristic
558C2#2099211yescharacteristic
559C2#2251512yes—
560C2#2251712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
152. #343Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
C2×#25842
C2×#19982
#213032
#112042
#109132
#108472
C4×#3432