Finite Group Databasebeta

← #363049

Subgroups — #363049

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2#13923C212yesmaximal
3#11353C212yesmaximal
4#11353C212yesmaximal
5#10854C212yesmaximal
6#10846C211yesmaximal, characteristic
7#10801C211yesmaximal, characteristic
8#10801C211yesmaximal, characteristic
9#10801C211yesmaximal, characteristic
10#10801C211yesmaximal, characteristic
11C4×#342C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13#2345—24no—
14#2345—24no—
15C2×#389C412yescomplemented
16C2×#387C412yescomplemented
17C2×#387C412yescomplemented
18C2×#380C2212yes—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C412yescomplemented
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×#342C2211yescharacteristic
28C2×#342C2211yescharacteristic
29C2×#342C2211yescharacteristic
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2212yes—
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2212yes—
36C2×#342C2211yescharacteristic
37#1441—24no—
38#1436—24no—
39C22⋊#95C2212yes—
40C22⋊#95C2212yes—
41C22⋊#95C2212yes—
42C22⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4D414yes—
58C22×C4⋊C4D414yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2312yes—
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2311yescharacteristic
63C22×C22⋊C4C2311yescharacteristic
64C22×C22⋊C4C2312yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2311yescharacteristic
67C22×C42—22no—
68C22×C42C2×C412yes—
69C22×C42C2×C412yes—
70C22×C42C2×C412yes—
71#389—24no—
72#389—24no—
73#387—24no—
74#387—24no—
75#387—24no—
76#387—24no—
77#380—24no—
78#380—22no—
79#380—22no—
80#380—24no—
81#380—24no—
82#380—24no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—24no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105#342—22no—
106#342—24no—
107#342—24no—
108#342—24no—
109#342—22no—
110#342—22no—
111#342—42no—
112#342—42no—
113#342—42no—
114#342Q814yes—
115C25C2411yescharacteristic
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—44no—
127C23×C4—82no—
128C23×C4—82no—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D414yes—
140C23×C4C2×D414yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4#4112yes—
144C23×C4#4114yes—
145C23×C4#4112yes—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—84no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C414yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C414yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C2×Q814yes—
210C2×C22⋊C4C2×Q812yes—
211C2×C22⋊C4#4114yes—
212C2×C22⋊C4#4112yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—84no—
222C2×C42—82no—
223C2×C42—82no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—24no—
232#95—48no—
233#95—48no—
234#95—44no—
235#95—44no—
236#95—84no—
237#95—84no—
238C24—42no—
239C24—42no—
240C24C23×C411yescharacteristic
241C24C22×D412yes—
242C24C22×Q811yescharacteristic
243C24C2×#4112yes—
244C24C2×#4111yescharacteristic
245C24C2×#4111yescharacteristic
246C24C2×#4111yescharacteristic
247C24C2×#4112yes—
248C24H(5,𝔽2)11yescharacteristic
249C242−1+412yes—
250C242−1+411yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—82no—
318C22×C4—84no—
319C22×C4—82no—
320C22×C4—84no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4#11712yes—
331C22×C4#11712yes—
332C22×C4#11712yes—
333C22×C4#11712yes—
334C22×C4C4×Q814yes—
335C22×C4#12314yes—
336C22×C4#12314yes—
337C22×C4#12314yes—
338C22×C4#12314yes—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—28no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4—88no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C42—44no—
375C42—44no—
376C42—82no—
377C42—82no—
378C42—82no—
379C42—82no—
380C42—88no—
381C23—22no—
382C23—24no—
383C23—22no—
384C23—24no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23—42no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23C22×C22⋊C411yescharacteristic
398C23C2×#11711yescharacteristic
399C23C2×#11711yescharacteristic
400C23C2×C4×Q811yescharacteristic
401C23C4×#4111yescharacteristic
402C23#51911yescharacteristic
403C23#52011yescharacteristic
404C23C2×#12312yes—
405C23C2×#12312yes—
406C23#52912yes—
407C23#53312yes—
408C23#54112yes—
409C23#54212yes—
410C23#54812yes—
411C23#54812yes—
412C23D4×Q812yes—
413C23#55012yes—
414C23#55212yes—
415C23#55411yescharacteristic
416C23#55611yescharacteristic
417C23#55612yes—
418C23#55711yescharacteristic
419C23#55911yescharacteristic
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#38112yes—
493C2×C4#38112yes—
494C2×C4#38114yes—
495C2×C4#41014yes—
496C22—24no—
497C22—22no—
498C22—22no—
499C22—24no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—41no—
521C22—41no—
522C22C2×#38111yescharacteristic
523C22#229511yescharacteristic
524C22#232011yescharacteristic
525C22Q8×C22⋊C411yescharacteristic
526C22#234111yescharacteristic
527C22#235511yescharacteristic
528C22#235611yescharacteristic
529C22#242112yes—
530C22#242312yes—
531C22#244612yes—
532C22#244812yes—
533C22C2×#41012yes—
534C22#288312yes—
535C4—24no—
536C4—22no—
537C4—22no—
538C4—24no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—44no—
543C4—84no—
544C4—84no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—162no—
549C4—162no—
550C2—22no—
551C2—22no—
552C2—22no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—22no—
557C2—21no—
558C2#1393111yescharacteristic
559C2#2068811yescharacteristic
560C2#2068911yescharacteristic
561C2#2251312yes—
562C2#2253512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#389C4semidirect
16. C2×#387C4semidirect
17. C2×#387C4semidirect
22. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
#139232
#113532
#108542
#108462
#108012
C4×#3422