Finite Group Databasebeta

← #6690877

Subgroups — #6690877

Order 512.

1453 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39506C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic, complemented
3#33068C211yesmaximal, characteristic, complemented
4#32237C214yesmaximal, complemented
5#32141C214yesmaximal, complemented
6#30230C211yesmaximal, characteristic, complemented
7#29868C211yesmaximal, characteristic, complemented
8#27557C214yesmaximal
9#27299C214yesmaximal, complemented
10#26277C212yesmaximal, complemented
11#25470C212yesmaximal, complemented
12#24327C212yesmaximal
13#22519C212yesmaximal, complemented
14#22238C211yesmaximal, characteristic, complemented
15#21113C211yesmaximal, characteristic
16C2×#554C2212yes—
17C2×#476C2212yes—
18C2×C8×Q8C2212yes—
19C2×C4×D8C2211yescharacteristic
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857—21no—
23#2783—24no—
24#2621—21no—
25#2567C2212yes—
26#2552C2214yes—
27#2538C2212yes—
28#2513C2214yes—
29#2508C2214yes—
30#2502—22no—
31#2489—22no—
32#2440—24no—
33#2355C2212yes—
34D4×C4⋊C4C2214yes—
35#2314—21no—
36C4×C4⋊Q8C2211yescharacteristic
37#2293C2211yescharacteristic
38C2×#384—21no—
39C42×D4C2211yescharacteristic
40#2101C2214yes—
41#2100C2214yes—
42#2100C2214yes—
43#2088C2214yes—
44#2085C2214yes—
45#2084C2214yes—
46#2071C2214yes—
47#2065C2214yes—
48#2059C2214yes—
49#2025C2214yes—
50#1991C2214yes—
51#1990C2212yes—
52#1990C2212yes—
53#1989C2214yes—
54#1987C2212yes—
55#1984C2212yes—
56#1948C2212yes—
57#1941C2214yes—
58#1934C2214yes—
59#1925C2212yes—
60#1920C2214yes—
61#1919C2214yes—
62#1918C2212yes—
63#1918C2214yes—
64#1916C2214yes—
65#1916C2214yes—
66#1879C2214yes—
67#1879C2212yes—
68#1864C2214yes—
69#1864C2214yes—
70#1862C2212yes—
71#1831C2211yescharacteristic
72#1804C2214yes—
73#1802C2212yes—
74C4×#107C2211yescharacteristic
75C4×#102C2211yescharacteristic
76C4×#102C2211yescharacteristic
77C4×#102C2214yes—
78#1744C2211yescharacteristic
79C22×D8C2311yescharacteristic
80#554—28no—
81#554—44no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8D411yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×#125—21no—
86C2×#125C2314yes—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—24no—
90C2×C4×Q8C2312yes—
91C2×C4×D4C2312yes—
92C2×C4×D4C2311yescharacteristic
93C2×C4×D4C2314yes—
94C2×C4×D4C2311yescharacteristic
95C2×#117—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—24no—
98C22×C42—21no—
99#476—44no—
100C8×Q8—44no—
101C4×D8—24no—
102C2×#107C2311yescharacteristic
103C2×#107C2312yes—
104C2×#106C2314yes—
105C2×#106C2314yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2314yes—
108C2×#102C2314yes—
109C2×#102C2312yes—
110C2×#102C2312yes—
111C2×#102C2311yescharacteristic
112C2×#102C2311yescharacteristic
113C2×#102C2314yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2314yes—
117#400—24no—
118#398—24no—
119#398—24no—
120#398—24no—
121#397—24no—
122#397—24no—
123#396—24no—
124#395—24no—
125#391—22no—
126#391—22no—
127#391D412yes—
128#390C2314yes—
129#390C2312yes—
130#389—24no—
131#389—22no—
132#389D414yes—
133#389C2314yes—
134#389C2314yes—
135#389C2314yes—
136#389C2312yes—
137#389C2314yes—
138#388—24no—
139#388—22no—
140#388—24no—
141#387—24no—
142#387—24no—
143#385—24no—
144#385—24no—
145#384—22no—
146#384—24no—
147#384—21no—
148#384—21no—
149#384D411yescharacteristic
150#384C2312yes—
151#384C2314yes—
152#382—21no—
153#382—24no—
154#382C2314yes—
155#380—22no—
156#380—24no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4C2311yescharacteristic
159C4×C4⋊C4C2311yescharacteristic
160C4×C4⋊C4C2314yes—
161C4×C4⋊C4C2312yes—
162C4×C4⋊C4C2314yes—
163C4×C22⋊C4—22no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166C43C2311yescharacteristic
167#340C2314yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2312yes—
175#340C2314yes—
176#340C2312yes—
177#340C2312yes—
178#336C2314yes—
179#336C2314yes—
180#336C2314yes—
181#336C2312yes—
182#336C2312yes—
183C22×Q8—22no—
184C22×Q8C2×D412yes—
185C22×D4C2411yescharacteristic
186C22×D4C2411yescharacteristic
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—24no—
191C2×D8—22no—
192C2×D8—24no—
193C22×C8C2411yescharacteristic
194C22×C8C2411yescharacteristic
195C22×C8C2414yes—
196C22×C8C2412yes—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—84no—
201#125—28no—
202#125—44no—
203#125—82no—
204C22⋊Q8—416no—
205C22⋊Q8—48no—
206C22⋊Q8—84no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—28no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—44no—
216C4×D4#4114yes—
217#117—44no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—88no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2411yescharacteristic
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2411yescharacteristic
241C2×C4⋊C4C2412yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2414yes—
259C2×C42C2412yes—
260C2×C42C2411yescharacteristic
261#107—22no—
262#107—24no—
263#106—28no—
264#106—44no—
265C4⋊C8—24no—
266C4⋊C8—44no—
267#102—22no—
268#102—28no—
269#102—24no—
270#102—22no—
271#102—24no—
272#102—44no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—44no—
276#95—21no—
277#95—22no—
278#95—22no—
279#95—24no—
280#95—21no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95C2×D414yes—
288#95C2×D414yes—
289C24—21no—
290C24—21no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—44no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4C2×#4112yes—
304C2×D42−1+414yes—
305C22×C4—28no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—44no—
314C22×C4—48no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4C22×D414yes—
324C22×C4C22×D412yes—
325C22×C4C22×D414yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D412yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C2511yescharacteristic
330D8—44no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—28no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—44no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—82no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4C2×#4112yes—
368C4⋊C42−1+414yes—
369C22⋊C4—28no—
370C22⋊C4—24no—
371C22⋊C4—48no—
372C22⋊C4—42no—
373C22⋊C4—44no—
374C22⋊C4—82no—
375C22⋊C4—84no—
376C22⋊C4—84no—
377C42—22no—
378C42—24no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42—44no—
384C42—42no—
385C42—42no—
386C42—84no—
387C42#13514yes—
388C42C2×#4112yes—
389C42H(5,𝔽2)12yes—
390C23—22no—
391C23—24no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23C23×D411yescharacteristic
401Q8—48no—
402Q8—164no—
403D4—22no—
404D4—24no—
405D4—41no—
406D4—41no—
407D4—42no—
408D4#54114yes—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—416no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—84no—
442C2×C4—168no—
443C2×C4#54612yes—
444C2×C4#54612yes—
445C2×C4C2×#13514yes—
446C2×C4C2×#13512yes—
447C2×C4#57712yes—
448C2×C4C22×#4111yescharacteristic
449C2×C4C2×H(5,𝔽2)11yescharacteristic
450C2×C4C2×2−1+412yes—
451C2×C4C2×2−1+412yes—
452C8—41no—
453C8—41no—
454C8—42no—
455C8—44no—
456C22—24no—
457C22—42no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—81no—
462C22—81no—
463C22—82no—
464C22—82no—
465C22—81no—
466C22C2×#54611yescharacteristic
467C22#348612yes—
468C22C22×#13511yescharacteristic
469C22C2×#57711yescharacteristic
470C22#358512yes—
471C4—22no—
472C4—24no—
473C4—44no—
474C4—44no—
475C4—42no—
476C4—44no—
477C4—42no—
478C4—84no—
479C4—82no—
480C4—84no—
481C4—81no—
482C4#330314yes—
483C4C2×#54111yescharacteristic
484C4#350911yescharacteristic
485C4#355212yes—
486C2—44no—
487C2—161no—
488C2#3769511yescharacteristic
489C2C2×#330311yescharacteristic
490C2#6122311yescharacteristic
491C2#6181912yes—
492C2#6186012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #39506C2semidirect
2. #36180C2semidirect
3. #33068C2semidirect
4. #32237C2semidirect
5. #32141C2semidirect
6. #30230C2semidirect
7. #29868C2semidirect
9. #27299C2semidirect
10. #26277C2semidirect
11. #25470C2semidirect
13. #22519C2semidirect
14. #22238C2semidirect

Maximal subgroups

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

subgroupindex
#395062
#361802
#330682
#322372
#321412
#302302
#298682
#275572
#272992
#262772
#254702
#243272
#225192
#222382
#211132