Finite Group Databasebeta

← #373695

Subgroups — #373695

Order 512.

2462 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34173C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3C2×#1857C212yesmaximal, complemented
4#20844C212yesmaximal
5C2×#1784C211yesmaximal, characteristic
6#11136C214yesmaximal, complemented
7#10750C212yesmaximal, complemented
8#10719C212yesmaximal
9C24×Q8C2211yescharacteristic
10C2×#519C2211yescharacteristic
11C2×#519C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#428C2214yes—
15C22×#101C2212yes—
16C22×#101C2212yes—
17C2×#407C2212yes—
18Q8×C22⋊C4—24no—
19Q8×C22⋊C4—24no—
20#2310—22no—
21#2310C2212yes—
22C2×#391—28no—
23C2×#391—22no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26#2280—22no—
27#2280C2212yes—
28C22×#95—22no—
29#1857—28no—
30#1853—24no—
31#1852—24no—
32#1784C418yescomplemented
33#1736C2214yes—
34C2×#339C2214yes—
35C2×#339C2214yes—
36C23×Q8—24no—
37C23×Q8—24no—
38C23×Q8—24no—
39C23×Q8—24no—
40C23×Q8—42no—
41C23×Q8—42no—
42C23×Q8C2×C414yes—
43C23×Q8D412yes—
44C23×Q8Q814yes—
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C24×C4—21no—
48C24×C4—21no—
49C24×C4D412yes—
50C24×C4C2311yescharacteristic
51C22×M16C2314yes—
52#519—24no—
53#519—24no—
54#519—42no—
55#519—42no—
56C2×C4×Q8—24no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117C2312yes—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42—24no—
70C22×C42—24no—
71C22×C42—42no—
72C22×C42C2312yes—
73#428—28no—
74#428—44no—
75C2×#107—24no—
76C2×#106—24no—
77C2×#101—24no—
78C2×#101—24no—
79C2×#101—42no—
80C2×#101—42no—
81C2×#101C2×C418yescomplemented
82C2×#101C2×C418yescomplemented
83#407—24no—
84#407C2×C418yes—
85C2×C22⋊C8—24no—
86#391—24no—
87#391—28no—
88#391—24no—
89#391—216no—
90#391—24no—
91#391—24no—
92#391—22no—
93#391—22no—
94#391—24no—
95#391—24no—
96#391D418yes—
97#381—22no—
98#381—24no—
99#380—24no—
100#380—24no—
101#380D414yes—
102#380D414yes—
103#379—22no—
104C4×C22⋊C4—28no—
105C4×C22⋊C4—24no—
106C2×#95—216no—
107C2×#95—24no—
108C2×#95—24no—
109#340—28no—
110#339—28no—
111#339—44no—
112C25C2×D411yescharacteristic
113C22×Q8—216no—
114C22×Q8—216no—
115C22×Q8—216no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—216no—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—48no—
134C22×Q8—48no—
135C22×Q8—416no—
136C22×Q8—416no—
137C22×Q8—416no—
138C22×Q8—416no—
139C22×Q8C4⋊C418yes—
140C22×Q8C22×C412yes—
141C22×Q8C22×C412yes—
142C22×Q8C2×D414yes—
143C22×Q8C2×Q812yes—
144C22×Q8C2×Q812yes—
145C22×Q8#4114yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—28no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—48no—
166C23×C4—42no—
167C23×C4C4⋊C418yes—
168C23×C4C22×C412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D414yes—
173C23×C4C2×Q812yes—
174C23×C4#4114yes—
175C23×C4C2411yescharacteristic
176C2×M16—24no—
177C2×M16—24no—
178C2×M16—24no—
179C2×M16—24no—
180C2×M16—24no—
181C2×M16C22×C418yes—
182C22×C8—24no—
183C4×Q8—48no—
184C4×Q8—48no—
185C4×Q8—416no—
186#117—24no—
187#117—24no—
188#117—24no—
189#117—24no—
190#117—44no—
191#117—44no—
192#117—44no—
193#117—44no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—416no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—416no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42C2×D414yes—
224#107—48no—
225#106—48no—
226#101—216no—
227#101—48no—
228C22⋊C8—216no—
229#95—216no—
230#95—28no—
231#95—28no—
232#95—216no—
233#95—24no—
234#95—24no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—24no—
240C24—24no—
241C24C2×C4⋊C412yes—
242C24#12012yes—
243C24#12312yes—
244C24C22×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—28no—
256C2×Q8—28no—
257C2×Q8—28no—
258C2×Q8—28no—
259C2×Q8—216no—
260C2×Q8—216no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—464no—
267C2×Q8—464no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8C4×D418yes—
273C2×Q8C22⋊Q818yes—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—216no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—216no—
300C22×C4—216no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—416no—
313C22×C4—416no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—48no—
318C22×C4—416no—
319C22×C4—48no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—88no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4C2×C4⋊C414yes—
326C22×C4C2×C4⋊C414yes—
327C22×C4C2×C4⋊C412yes—
328C22×C4C4×D418yes—
329C22×C4#12014yes—
330C22×C4#12012yes—
331C22×C4C22⋊Q818yes—
332C22×C4#12312yes—
333C22×C4#12314yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342M16—28no—
343M16—28no—
344M16—44no—
345M16—44no—
346C2×C8—28no—
347C2×C8—44no—
348C2×C8—44no—
349C4⋊C4—24no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—48no—
357C4⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C42—28no—
365C42—48no—
366C42—48no—
367C42—48no—
368C42—44no—
369C42—44no—
370C42—44no—
371C23—24no—
372C23—24no—
373C23—28no—
374C23—24no—
375C23—24no—
376C23—28no—
377C23—22no—
378C23—28no—
379C23—28no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23#38514yes—
390C23#45414yes—
391C23#45614yes—
392C23C22×C4⋊C411yescharacteristic
393C23C2×#12011yescharacteristic
394C23C2×#12011yescharacteristic
395C23C2×#12311yescharacteristic
396C23C2×#12311yescharacteristic
397C23#53411yescharacteristic
398C23#53411yescharacteristic
399Q8—216no—
400Q8—216no—
401Q8—432no—
402Q8—48no—
403Q8—48no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—216no—
417C2×C4—216no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—432no—
423C2×C4—432no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—48no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—84no—
442C2×C4—816no—
443C2×C4—84no—
444C2×C4#38518yes—
445C2×C4#38514yes—
446C2×C4C2×C4×D412yes—
447C2×C4C2×C4×D412yes—
448C2×C4#51812yes—
449C2×C4C2×C22⋊Q812yes—
450C2×C4C2×C22⋊Q812yes—
451C2×C4#54312yes—
452C2×C4D4214yes—
453C2×C4#54614yes—
454C2×C4#54614yes—
455C2×C4#55114yes—
456C8—48no—
457C22—28no—
458C22—28no—
459C22—28no—
460C22—28no—
461C22—24no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—22no—
469C22—22no—
470C22—24no—
471C22—22no—
472C22—22no—
473C22—24no—
474C22—24no—
475C22#185718yescomplemented
476C22C2×#38511yescharacteristic
477C22C2×#38511yescharacteristic
478C22#231511yescharacteristic
479C22#240311yescharacteristic
480C22#240511yescharacteristic
481C22#240511yescharacteristic
482C22#248311yescharacteristic
483C22C2×#45412yes—
484C22C2×#45612yes—
485C4—24no—
486C4—28no—
487C4—48no—
488C4—44no—
489C4—82no—
490C4—84no—
491C4—82no—
492C4—84no—
493C4#235814yes—
494C4#243714yes—
495C2—21no—
496C2—21no—
497C2—28no—
498C2—21no—
499C2—21no—
500C2#1412711yescharacteristic
501C2C2×#185712yes—
502C2#2391011yescharacteristic
503C2#2391111yescharacteristic
504C2#2401912yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1857C2semidirect
6. #11136C2semidirect
7. #10750C2semidirect
32. #1784C4semidirect
81. C2×#101C2×C4semidirect
82. C2×#101C2×C4semidirect
475. C22#1857semidirect

Maximal subgroups

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

subgroupindex
#341732
C22×#3912
C2×#18572
#208442
C2×#17842
#111362
#107502
#107192