Finite Group Databasebeta

← #6667458

Subgroups — #6667458

Order 512.

1205 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35337C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic
3#32857C212yesmaximal, complemented
4#32856C212yesmaximal, complemented
5#29949C212yesmaximal
6#29948C212yesmaximal
7#29945C214yesmaximal
8#29884C212yesmaximal
9#29883C212yesmaximal
10#29292C214yesmaximal, complemented
11#29227C212yesmaximal, complemented
12#29224C212yesmaximal, complemented
13#28694C212yesmaximal, complemented
14#28692C212yesmaximal, complemented
15#20811C211yesmaximal, characteristic
16C2×#501C2212yes—
17C2×#501C2212yes—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2647C2211yescharacteristic
27#2647C2211yescharacteristic
28#2647C2212yes—
29#2629C2212yes—
30#2628C2212yes—
31#2589—22no—
32#2589—22no—
33#2588—22no—
34#2588—22no—
35#2507—22no—
36#2497—22no—
37#2446—22no—
38#2429—22no—
39#2331C2212yes—
40#2330C2212yes—
41#2314C2211yescharacteristic
42#2314C2211yescharacteristic
43#2314C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#125C2212yes—
46C4×#117C2211yescharacteristic
47#2101C2214yes—
48#2101C2214yes—
49#2101C2214yes—
50#2101C2214yes—
51#2088C2214yes—
52#2088C2214yes—
53#2087C2214yes—
54#2086C2214yes—
55#2084C2214yes—
56#2084C2214yes—
57#2083C2214yes—
58#2083C2214yes—
59#2077C2214yes—
60#2077C2214yes—
61#2076C2214yes—
62#2075C2214yes—
63#1993C2212yes—
64#1993C2212yes—
65#1992C2212yes—
66#1992C2212yes—
67#1989C2214yes—
68#1989C2214yes—
69#1988C2214yes—
70#1987C2214yes—
71#1949C2214yes—
72#1949C2214yes—
73#1948C2214yes—
74#1947C2214yes—
75#1843C2212yes—
76#1836C2212yes—
77#1836C2212yes—
78#1832C2212yes—
79#1798C2212yes—
80#1798C2212yes—
81#1767C2212yes—
82#1767C2212yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125C2312yes—
87C2×#125C2312yes—
88C2×C22⋊Q8—22no—
89C2×C4×Q8—22no—
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#501—44no—
94#501—44no—
95#432—42no—
96#432—42no—
97C2×#107C2314yes—
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2312yes—
107C2×#97C2312yes—
108C2×#97C2312yes—
109#400—22no—
110#400—22no—
111#400—22no—
112#400—24no—
113#399—22no—
114#398—24no—
115#398—22no—
116#398—22no—
117#398—24no—
118#396—24no—
119#395—22no—
120#393—22no—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—24no—
134#389C2312yes—
135#389C2312yes—
136#389C2314yes—
137#389C2314yes—
138#389C2312yes—
139#389C2312yes—
140#388—22no—
141#388—22no—
142#388—24no—
143#387—22no—
144#387—22no—
145#387—24no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—24no—
151#387—22no—
152#387—22no—
153#385—22no—
154#385—22no—
155#384—22no—
156#384C2312yes—
157#384C2311yescharacteristic
158#384C2312yes—
159#384C2311yescharacteristic
160#384C2312yes—
161#382C2312yes—
162#382C2312yes—
163#382C2311yescharacteristic
164#382C2311yescharacteristic
165#382C2312yes—
166#380—22no—
167#380C2312yes—
168#380C2314yes—
169#380C2312yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2314yes—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4D412yes—
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C43C2311yescharacteristic
183C43C2311yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198C22×Q8—22no—
199C22×Q8—22no—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2411yescharacteristic
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—24no—
206C4⋊Q8—28no—
207C4⋊Q8—24no—
208C4⋊Q8—84no—
209#125—24no—
210#125—28no—
211#125—24no—
212C22⋊Q8—84no—
213C4×Q8—84no—
214#117—24no—
215#117—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2414yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2411yescharacteristic
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263#107—44no—
264#107—44no—
265#106—44no—
266#106—44no—
267C4⋊C8—28no—
268C4⋊C8—28no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273#97—24no—
274#97—24no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—22no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95C2×D412yes—
295#95C2×D412yes—
296#95C2×D412yes—
297#95C2×D412yes—
298C24C22×D411yescharacteristic
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—82no—
304C2×Q8—82no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2511yescharacteristic
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—28no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—84no—
368C42—22no—
369C42—22no—
370C42—21no—
371C42—21no—
372C42—22no—
373C42—22no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—82no—
379C42—82no—
380C42—84no—
381C42H(5,𝔽2)12yes—
382C422−1+414yes—
383C422−1+414yes—
384C422−1+412yes—
385C23—22no—
386C23—22no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23C23×D411yescharacteristic
391Q8—88no—
392Q8—164no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#53614yes—
435C2×C4#53614yes—
436C2×C4#53712yes—
437C2×C4#53712yes—
438C2×C4C2×H(5,𝔽2)11yescharacteristic
439C2×C4C2×2−1+412yes—
440C2×C4C2×2−1+412yes—
441C2×C4C2×2−1+411yescharacteristic
442C8—44no—
443C8—44no—
444C22—41no—
445C22—42no—
446C22—42no—
447C22—41no—
448C22—41no—
449C22C2×#53611yescharacteristic
450C22C2×#53611yescharacteristic
451C22C2×#53711yescharacteristic
452C22#358512yes—
453C22#358612yes—
454C4—24no—
455C4—24no—
456C4—22no—
457C4—22no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—41no—
462C4—41no—
463C4—82no—
464C4—82no—
465C4—84no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4#355012yes—
470C4#355612yes—
471C2—81no—
472C2#3764111yescharacteristic
473C2#6088512yes—
474C2#6088612yes—
475C2#6092011yescharacteristic
476C2#6092111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32857C2semidirect
4. #32856C2semidirect
10. #29292C2semidirect
11. #29227C2semidirect
12. #29224C2semidirect
13. #28694C2semidirect
14. #28692C2semidirect

Maximal subgroups

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

subgroupindex
#353372
#352842
#328572
#328562
#299492
#299482
#299452
#298842
#298832
#292922
#292272
#292242
#286942
#286922
#208112