Finite Group Databasebeta

← #365652

Subgroups — #365652

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2370C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#10908C212yesmaximal
9#10908C212yesmaximal
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2370—24no—
17#2370—24no—
18C2×#389—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—21no—
22C2×#387—21no—
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1992—24no—
32#1989C418yescomplemented
33#1948—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1442—21no—
44#1442—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125C2×C414yes—
54C2×#125D414yes—
55C2×#125Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8C2×C414yes—
83C2×#97—22no—
84C2×#97—22no—
85C2×#97—22no—
86#389—28no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#387—28no—
92#387—28no—
93#384—22no—
94#384—22no—
95#382—44no—
96C4×C4⋊C4—28no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C414yes—
106C4×C4⋊C4D414yes—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340C2×C418yes—
125#340C2×C418yes—
126#336—24no—
127#336—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C22×C414yes—
154C22×C8#4114yes—
155#125—28no—
156#125—28no—
157#125—28no—
158#125C4⋊C4116yes—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C4⋊C4C2×Q812yes—
212C2×C4⋊C4#4114yes—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—82no—
230C2×C42—82no—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×Q812yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237#106—48no—
238C4⋊C8—28no—
239#97—44no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—44no—
249#95—44no—
250#95—82no—
251#95—82no—
252C24C22×D411yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4#12012yes—
316C22×C4#12012yes—
317C22×C4C22⋊Q812yes—
318C22×C4C22⋊Q812yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4#13712yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C42−1+412yes—
330C22×C42−1+412yes—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—216no—
354C4⋊C4—216no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4C2×C4⋊C414yes—
373C4⋊C4C2×C4⋊C414yes—
374C4⋊C4#12414yes—
375C42—48no—
376C42—48no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—42no—
382C42—42no—
383C42—84no—
384C42—84no—
385C42C2×C4⋊C414yes—
386C42C4×D414yes—
387C42C22⋊Q814yes—
388C42#12314yes—
389C23C2×C4×D411yescharacteristic
390C23C2×#12011yescharacteristic
391C23C2×C22⋊Q811yescharacteristic
392C23#54712yes—
393C23#54812yes—
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×#13711yescharacteristic
396C23#57511yescharacteristic
397C23#57511yescharacteristic
398C23#57712yes—
399C23#57712yes—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#38514yes—
467C2×C4#38514yes—
468C2×C4C22×C4⋊C411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C4×#4112yes—
471C2×C4#51912yes—
472C2×C4C2×C22⋊Q811yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4C2×#12411yescharacteristic
475C2×C4#52912yes—
476C2×C4#53211yescharacteristic
477C2×C4#53611yescharacteristic
478C2×C4#53611yescharacteristic
479C2×C4#54112yes—
480C2×C4#54812yes—
481C2×C4#54812yes—
482C2×C4#55212yes—
483C2×C4#55612yes—
484C2×C4#55712yes—
485C2×C4#55712yes—
486C8—44no—
487C8—44no—
488C22C2×#38511yescharacteristic
489C22#236212yes—
490C22#243211yescharacteristic
491C22#243611yescharacteristic
492C22#245312yes—
493C22C4×AGL(1,ℤ/8)11yescharacteristic
494C22C4×#13711yescharacteristic
495C22#295212yes—
496C22#300611yescharacteristic
497C22#300711yescharacteristic
498C22#300911yescharacteristic
499C22#301011yescharacteristic
500C22#307911yescharacteristic
501C22#308011yescharacteristic
502C22#308412yes—
503C22#334412yes—
504C22#334512yes—
505C22#334812yes—
506C22#335012yes—
507C22#336812yes—
508C22#337112yes—
509C22#337212yes—
510C22#337612yes—
511C4—24no—
512C4—24no—
513C4—44no—
514C4—42no—
515C4—42no—
516C4—44no—
517C4—44no—
518C4—84no—
519C4—81no—
520C4—82no—
521C4—81no—
522C4—162no—
523C4C4×#12411yescharacteristic
524C4#232711yescharacteristic
525C4#235112yes—
526C4#242112yes—
527C4#256511yescharacteristic
528C4#256811yescharacteristic
529C2#1397811yescharacteristic
530C2#2087511yescharacteristic
531C2#2088211yescharacteristic
532C2#2307212yes—
533C2#2307812yes—
534C2#2571411yescharacteristic
535C2#2571511yescharacteristic
536C2#2594411yescharacteristic
537C2#2594511yescharacteristic
538C2#2652212yes—
539C2#2652512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #1989C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23702
C2×#19922
C2×#19892
C2×#19482
#139112
#112932
#112922
#109082
#108352
#107032