Finite Group Databasebeta

← #374161

Subgroups — #374161

Order 512.

1773 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4#13795C211yesmaximal, characteristic, complemented
5#11203C212yesmaximal, complemented
6#11043C212yesmaximal
7#10960C212yesmaximal
8#10912C212yesmaximal, complemented
9#10736C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×#117C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2214yes—
14#2623—24no—
15#2623—24no—
16C2×#400C412yescomplemented
17C2×#399C412yescomplemented
18C2×#398C412yescomplemented
19C2×#397C412yescomplemented
20#2313—28no—
21C2×#387—24no—
22C2×#386C2211yescharacteristic
23C2×#386C2211yescharacteristic
24C2×#385C2211yescharacteristic
25C2×#385C2211yescharacteristic
26C2×#381C2212yes—
27C2×#381C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2212yes—
30#1899—28no—
31C2×#342C2214yes—
32C2×#342C2214yes—
33C2×#342C2214yes—
34C4⋊#95—24no—
35C22⋊#95C2212yes—
36C22⋊#95C2212yes—
37C23×D4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C2×#123—22no—
42C2×#123—22no—
43C2×#123—24no—
44C2×#123—24no—
45C2×#117—28no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—44no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4D414yes—
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C22⋊C4C2311yescharacteristic
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2314yes—
70C22×C42—22no—
71C22×C42—22no—
72C2×#99—28no—
73C2×#99—28no—
74C2×#99—44no—
75#400—28no—
76#399—28no—
77#398—28no—
78#397—28no—
79#387—232no—
80#386—22no—
81#386—22no—
82#386—22no—
83#386—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#385—22no—
88#381—24no—
89#381—22no—
90#381—22no—
91#381D418yes—
92#380—28no—
93C4×C22⋊C4—28no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—24no—
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—28no—
106#342—28no—
107#342—44no—
108C25C22×C411yescharacteristic
109C25C2411yescharacteristic
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—24no—
115C22×D4—22no—
116C22×D4—22no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D414yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4#4114yes—
145#123—44no—
146#123—44no—
147#123—44no—
148#123—44no—
149#117—416no—
150#117—416no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
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—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—84no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22⋊C418yes—
171C2×C4⋊C4C22⋊C418yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4C2×D418yes—
202C2×C22⋊C4#4118yes—
203C2×C42—28no—
204C2×C42—28no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42—84no—
208#99—416no—
209#95—28no—
210#95—28no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—216no—
218#95—84no—
219#95—84no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C23×C411yescharacteristic
229C24C22×D411yescharacteristic
230C24C22×D411yescharacteristic
231C24C22×D412yes—
232C24C2×#4112yes—
233C24C2×#4114yes—
234C24H(5,𝔽2)12yes—
235C242−1+412yes—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C414yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C4×D414yes—
315C22×C4C4×D414yes—
316C22×C4#12014yes—
317C22×C4#12114yes—
318C22×C4#12114yes—
319C22×C4#12314yes—
320C4⋊C4—216no—
321C4⋊C4—216no—
322C4⋊C4—416no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—88no—
328C22⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C42—416no—
349C42—88no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—48no—
360C23—42no—
361C23—44no—
362C23—44no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—44no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23—82no—
373C23C22×C22⋊C411yescharacteristic
374C23C2×C4×D412yes—
375C23C4×#4112yes—
376C23#52012yes—
377C23C2×#12011yescharacteristic
378C23C2×#12112yes—
379C23C2×#12311yescharacteristic
380C23#52512yes—
381C23#53212yes—
382C23#53511yescharacteristic
383C23#53611yescharacteristic
384C23#53712yes—
385C23#54614yes—
386C23#54714yes—
387C23#55114yes—
388C23#55214yes—
389D4—82no—
390D4—82no—
391D4—82no—
392D4—82no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—416no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—88no—
439C2×C4—88no—
440C2×C4—88no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#38614yes—
448C2×C4#38614yes—
449C22—22no—
450C22—24no—
451C22—24no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—21no—
464C22—44no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—81no—
469C22—82no—
470C22—82no—
471C22—81no—
472C22—81no—
473C22—164no—
474C22C2×#38611yescharacteristic
475C22#229711yescharacteristic
476C22#231111yescharacteristic
477C22#235914yes—
478C22#241112yes—
479C22#241312yes—
480C22#243012yes—
481C22#249812yes—
482C22#288312yes—
483C22#288412yes—
484C4—24no—
485C4—24no—
486C4—44no—
487C4—44no—
488C4—88no—
489C4—88no—
490C4—81no—
491C4—82no—
492C4—82no—
493C4—81no—
494C4—81no—
495C2—22no—
496C2—22no—
497C2—24no—
498C2—22no—
499C2—22no—
500C2—161no—
501C2#1413511yescharacteristic
502C2#2098511yescharacteristic
503C2#2098611yescharacteristic
504C2#2142312yes—
505C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
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
2. C2×#2313C2semidirect
4. #13795C2semidirect
5. #11203C2semidirect
8. #10912C2semidirect
9. #10736C2semidirect
16. C2×#400C4semidirect
17. C2×#399C4semidirect
18. C2×#398C4semidirect
19. C2×#397C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C2×#23132
C2×#18992
#137952
#112032
#110432
#109602
#109122
#107362