Finite Group Databasebeta

← #430985

Subgroups — #430985

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#1864C212yesmaximal
5C2×#1847C211yesmaximal, characteristic, complemented
6#15196C211yesmaximal, characteristic, complemented
7#11372C211yesmaximal, characteristic, complemented
8#11325C212yesmaximal, complemented
9#11123C212yesmaximal
10#10927C211yesmaximal, characteristic
11#10875C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20#2563—24no—
21#2563—24no—
22C2×#393—22no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#380—22no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2093—24no—
32#2057—28no—
33#1864—28no—
34#1847—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1444—22no—
42#1441—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—42no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×#107—42no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102Q818yescomplemented
88C2×#97—24no—
89C2×#97—22no—
90#393—216no—
91#390—22no—
92#390—22no—
93#390—22no—
94#390D414yes—
95#389—24no—
96#389—24no—
97#384—22no—
98#384—22no—
99#382—22no—
100#382—22no—
101#382—22no—
102#382D414yes—
103#380—216no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—24no—
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—24no—
115#336—24no—
116#336—22no—
117#336D418yes—
118C25—21no—
119C22×Q8—24no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—84no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4C2×D412yes—
130C22×D4C2×Q814yes—
131C23×C4—24no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—42no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—44no—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C2×D414yes—
157C22×C8C2×D414yes—
158C22×C8C2×Q814yes—
159C22×C8#4114yes—
160C4⋊Q8—48no—
161C4⋊Q8—48no—
162C4×D4—48no—
163C4×D4—48no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—84no—
204C2×C22⋊C4—84no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D414yes—
217#107—48no—
218#107—48no—
219#106—48no—
220#102—216no—
221#102—48no—
222#97—28no—
223#95—28no—
224#95—28no—
225#95—24no—
226#95—44no—
227#95—44no—
228#95—84no—
229C24—22no—
230C24—24no—
231C24—41no—
232C24—41no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—44no—
237C24C22×D411yescharacteristic
238C2×Q8—48no—
239C2×Q8—88no—
240C2×Q8—88no—
241C2×Q8—88no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4C4⋊Q814yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—48no—
278C22×C4—416no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—84no—
299C22×C4—88no—
300C22×C4—84no—
301C22×C4#12014yes—
302C22×C4#12712yes—
303C22×C4#12712yes—
304C22×C4#13714yes—
305C22×C4C22×D412yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×Q812yes—
309C22×C4C2×#4112yes—
310C22×C4H(5,𝔽2)12yes—
311C22×C42−1+412yes—
312C22×C42−1+412yes—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8C22⋊Q818yes—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—88no—
344C4⋊C4—88no—
345C22⋊C4—48no—
346C22⋊C4—88no—
347C22⋊C4—88no—
348C42—22no—
349C42—22no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—41no—
354C42—41no—
355C42—42no—
356C23—24no—
357C23—48no—
358C23—48no—
359C23—42no—
360C23—42no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—84no—
366C23—84no—
367C23—81no—
368C23—81no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23C2×#12011yescharacteristic
373C23C2×#12711yescharacteristic
374C23#53511yescharacteristic
375C23#54712yes—
376C23D4×Q812yes—
377C23C2×#13712yes—
378C23#57512yes—
379C23#57614yes—
380Q8—816no—
381D4—28no—
382D4—44no—
383D4—44no—
384D4—44no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—816no—
412C2×C4—84no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4#49714yes—
428C2×C4C2×C22⋊Q812yes—
429C2×C4#52511yescharacteristic
430C2×C4C2×C4⋊Q811yescharacteristic
431C2×C4#53611yescharacteristic
432C2×C4#53612yes—
433C2×C4#53611yescharacteristic
434C2×C4#53912yes—
435C2×C4#54412yes—
436C2×C4D4212yes—
437C2×C4D4212yes—
438C2×C4#54712yes—
439C2×C4#54712yes—
440C2×C4D4×Q812yes—
441C2×C4D4×Q812yes—
442C2×C4#55412yes—
443C2×C4#56312yes—
444C8—28no—
445C8—44no—
446C22—44no—
447C22—48no—
448C22—84no—
449C22—88no—
450C22—84no—
451C22—84no—
452C22—82no—
453C22—82no—
454C22#240611yescharacteristic
455C22#243311yescharacteristic
456C22#243411yescharacteristic
457C22#254212yes—
458C22#272012yes—
459C22#300712yes—
460C22#300812yes—
461C22C2×#49711yescharacteristic
462C22#315311yescharacteristic
463C22#315512yes—
464C22#321512yes—
465C22#322012yes—
466C22#334714yes—
467C22#335714yes—
468C22#338514yes—
469C22#338914yes—
470C4—24no—
471C4—44no—
472C4—42no—
473C4—82no—
474C4—84no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—162no—
479C4#244012yes—
480C4#244912yes—
481C4#267712yes—
482C4#268012yes—
483C2—84no—
484C2#1543011yescharacteristic
485C2#2410512yes—
486C2#2573011yescharacteristic
487C2#2573411yescharacteristic
488C2#2585012yes—
489C2#2853912yes—
490C2#2854412yes—
491C2#3237314yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
3. C2×#2057C2semidirect
5. C2×#1847C2semidirect
6. #15196C2semidirect
7. #11372C2semidirect
8. #11325C2semidirect
11. #10875C2semidirect
87. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#20932
C2×#20572
C2×#18642
C2×#18472
#151962
#113722
#113252
#111232
#109272
#108752