Finite Group Databasebeta

← #356491

Subgroups — #356491

Order 512.

2248 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2C2×#2344C212yesmaximal
3C2×#1740C212yesmaximal
4#10854C212yesmaximal
5#10846C211yesmaximal, characteristic
6#10736C214yesmaximal
7#10726C212yesmaximal
8C4×#342C211yesmaximal, characteristic
9C22×#117C2214yes—
10C22×#117C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12#2344—28no—
13#2344—22no—
14#2344—24no—
15#2344—22no—
16#2344C4116yescomplemented
17C2×#389—22no—
18C2×#389C412yescomplemented
19C2×#387—24no—
20C2×#387C414yescomplemented
21C2×#381C2214yes—
22C2×#381C2212yes—
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4—22no—
26C2×C4×C4⋊C4C412yescomplemented
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1740—28no—
30C2×#342C2212yes—
31C2×#342C2214yes—
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2214yes—
36C2×#342C2211yescharacteristic
37C24×C4C2312yes—
38C24×C4C2311yescharacteristic
39C2×#117—24no—
40C2×#117—24no—
41C2×#117—28no—
42C2×#117—24no—
43C2×#117—24no—
44C2×#117—28no—
45C2×#117—28no—
46C2×#117—44no—
47C2×#117—44no—
48C2×#117—44no—
49C2×#117C2×C4116yes—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C418yes—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4D414yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D414yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4Q814yes—
63C22×C22⋊C4C2312yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2311yescharacteristic
68C22×C22⋊C4C2312yes—
69C22×C42—24no—
70C22×C42—22no—
71C22×C42C2×C414yes—
72C22×C42C2×C412yes—
73#389—24no—
74#389—24no—
75#389—216no—
76#387—28no—
77#387—28no—
78#387—232no—
79#381—28no—
80#381—24no—
81#381—24no—
82#381C2×C4116yes—
83#380—24no—
84#380—22no—
85#380—22no—
86#380C2×C418yes—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—216no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2×C418yes—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95—22no—
98C2×#95—24no—
99C2×#95C2×C412yes—
100C2×#95C2×C414yes—
101C2×#95C2×C412yes—
102C2×#95C2×C414yes—
103#342—24no—
104#342—28no—
105#342—22no—
106#342—24no—
107#342—42no—
108#342—44no—
109#342D414yes—
110C25C2411yescharacteristic
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—44no—
121C23×C4C22×C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C418yes—
124C23×C4C22×C412yes—
125C23×C4C22×C414yes—
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C414yes—
130C23×C4C2×D414yes—
131C23×C4C2×D412yes—
132C23×C4C2×D414yes—
133C23×C4C2×D414yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×Q814yes—
137C23×C4C2×Q812yes—
138#117—232no—
139#117—416no—
140#117—48no—
141#117—48no—
142#117—48no—
143#117—48no—
144#117—416no—
145#117—416no—
146C2×C4⋊C4—216no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4C4214yes—
158C2×C4⋊C4C4214yes—
159C2×C4⋊C4C4218yes—
160C2×C4⋊C4C22⋊C418yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C4116yes—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C22⋊C418yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C4⋊C418yes—
171C2×C4⋊C4C4⋊C418yes—
172C2×C4⋊C4C4⋊C414yes—
173C2×C4⋊C4C4⋊C414yes—
174C2×C4⋊C4C4⋊C4116yes—
175C2×C4⋊C4C4⋊C418yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4C22×C418yes—
200C2×C22⋊C4C22×C414yes—
201C2×C22⋊C4C22×C414yes—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D414yes—
204C2×C22⋊C4#4112yes—
205C2×C22⋊C4#4114yes—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—216no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—28no—
212C2×C42—28no—
213C2×C42—44no—
214C2×C42—48no—
215C2×C42—48no—
216C2×C42—48no—
217C2×C42—44no—
218#95—28no—
219#95—216no—
220#95—28no—
221#95—216no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—28no—
227C24—42no—
228C24—42no—
229C24C23×C411yescharacteristic
230C24C23×C412yes—
231C24C22×D412yes—
232C24C22×D411yescharacteristic
233C24C22×D411yescharacteristic
234C24C22×Q811yescharacteristic
235C24C2×#4111yescharacteristic
236C24C2×#4112yes—
237C24H(5,𝔽2)12yes—
238C24H(5,𝔽2)11yescharacteristic
239C242−1+411yescharacteristic
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—28no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—28no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—48no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4C2×C4214yes—
303C22×C4C2×C4214yes—
304C22×C4C2×C4212yes—
305C22×C4C2×C4212yes—
306C22×C4C2×C22⋊C418yes—
307C22×C4C2×C22⋊C418yes—
308C22×C4C2×C22⋊C414yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C414yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C4⋊C414yes—
317C22×C4C2×C4⋊C418yes—
318C22×C4C2×C4⋊C418yes—
319C22×C4C2×C4⋊C414yes—
320C22×C4C2×C4⋊C412yes—
321C22×C4C2×C4⋊C414yes—
322C22×C4C2×C4⋊C412yes—
323C22×C4C2×C4⋊C414yes—
324C22×C4C4×D414yes—
325C4⋊C4—416no—
326C4⋊C4—432no—
327C4⋊C4—44no—
328C4⋊C4—416no—
329C4⋊C4—44no—
330C4⋊C4#9518yes—
331C4⋊C4#95116yes—
332C4⋊C4#9518yes—
333C4⋊C4#95116yes—
334C4⋊C4#9514yes—
335C4⋊C4#9518yes—
336C4⋊C4#9514yes—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—28no—
343C22⋊C4—24no—
344C22⋊C4—28no—
345C22⋊C4—24no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4C4×D418yes—
355C42—48no—
356C42—48no—
357C42—432no—
358C42—44no—
359C42—416no—
360C42—44no—
361C23—22no—
362C23—24no—
363C23—22no—
364C23—24no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—42no—
373C23—44no—
374C23—44no—
375C23—42no—
376C23—42no—
377C23C22×C4211yescharacteristic
378C23C22×C22⋊C412yes—
379C23C22×C22⋊C411yescharacteristic
380C23C22×C4⋊C411yescharacteristic
381C23C22×C4⋊C412yes—
382C23C2×C4×D411yescharacteristic
383C23C2×C4×D412yes—
384C23C4×#4111yescharacteristic
385C23C4×#4112yes—
386C23#51811yescharacteristic
387C23#51812yes—
388C23#52011yescharacteristic
389C23#52012yes—
390C23D4212yes—
391C23D4211yescharacteristic
392C23#54612yes—
393C23#54612yes—
394C23#54612yes—
395C23#54712yes—
396C23#54711yescharacteristic
397C23D4×Q811yescharacteristic
398C23#55411yescharacteristic
399C23#55412yes—
400C2×C4—28no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—88no—
455C2×C4—84no—
456C2×C4—88no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4C2×#9512yes—
463C2×C4C2×#9514yes—
464C2×C4C2×#9514yes—
465C2×C4C2×#9518yes—
466C2×C4C2×#9514yes—
467C2×C4C2×#9512yes—
468C2×C4#41014yes—
469C22—22no—
470C22—21no—
471C22—22no—
472C22—21no—
473C22—22no—
474C22—21no—
475C22—22no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—24no—
482C22—22no—
483C22—22no—
484C22—24no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—42no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—42no—
495C22C22×#9511yescharacteristic
496C22C42×D411yescharacteristic
497C22#227511yescharacteristic
498C22D4×C22⋊C412yes—
499C22D4×C22⋊C411yescharacteristic
500C22#233912yes—
501C22#233911yescharacteristic
502C22D4×C4⋊C411yescharacteristic
503C22D4×C4⋊C412yes—
504C22#234911yescharacteristic
505C22#234912yes—
506C22C2×#41012yes—
507C22#288412yes—
508C4—22no—
509C4—24no—
510C4—24no—
511C4—28no—
512C4—24no—
513C4—48no—
514C4—44no—
515C4—88no—
516C4—84no—
517C4—84no—
518C4—82no—
519C4#174014yescomplemented
520C2—22no—
521C2—22no—
522C2—21no—
523C2—22no—
524C2—21no—
525C2—22no—
526C2—21no—
527C2—21no—
528C2D4×#9511yescharacteristic
529C2C2×#174011yescharacteristic
530C2#2035511yescharacteristic
531C2#2240612yes—
532C2#2247812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, prime radical, cosocleC25C24
LCSG ▷ C23 ▷ 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
16. #2344C4semidirect
18. C2×#389C4semidirect
20. C2×#387C4semidirect
26. C2×C4×C4⋊C4C4semidirect
519. C4#1740semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C2×#17402
#108542
#108462
#107362
#107262
C4×#3422