Finite Group Databasebeta

← #6498053

Subgroups — #6498053

Order 512.

1253 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35336C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#33317C211yesmaximal, characteristic, complemented
4#33297C211yesmaximal, characteristic, complemented
5#33296C211yesmaximal, characteristic, complemented
6#33294C211yesmaximal, characteristic, complemented
7#29884C212yesmaximal
8#29883C212yesmaximal
9#29188C214yesmaximal, complemented
10#29187C214yesmaximal, complemented
11#27888C214yesmaximal
12#27886C214yesmaximal
13#22251C212yesmaximal, complemented
14#22250C212yesmaximal, complemented
15#20467C211yesmaximal, characteristic
16C2×#501C2212yes—
17C2×#501C2212yes—
18C2×#432C2212yes—
19#2868C2211yescharacteristic
20#2862C2211yescharacteristic
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2212yes—
24#2851C2211yescharacteristic
25#2851C2211yescharacteristic
26#2647C2212yes—
27#2647C2212yes—
28#2629C2212yes—
29#2628C2212yes—
30#2590—22no—
31#2590—22no—
32#2587—22no—
33#2587—22no—
34#2511—22no—
35#2492—22no—
36#2451—22no—
37#2423—22no—
38#2331C2212yes—
39#2330C2212yes—
40#2314C2212yes—
41#2314C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×#125C2211yescharacteristic
44C4×#125C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2088C2214yes—
47#2088C2214yes—
48#2087C2214yes—
49#2086C2214yes—
50#2077C2214yes—
51#2077C2214yes—
52#2076C2214yes—
53#2075C2214yes—
54#1993C2214yes—
55#1993C2212yes—
56#1993C2212yes—
57#1992C2214yes—
58#1992C2212yes—
59#1992C2212yes—
60#1991C2214yes—
61#1990C2214yes—
62#1921C2214yes—
63#1921C2214yes—
64#1920C2214yes—
65#1920C2214yes—
66#1919C2214yes—
67#1919C2214yes—
68#1918C2214yes—
69#1918C2214yes—
70#1847C2214yes—
71#1846C2214yes—
72#1836C2214yes—
73#1836C2212yes—
74#1836C2212yes—
75#1834C2214yes—
76#1768C2212yes—
77#1768C2212yes—
78#1750C2212yes—
79#1750C2212yes—
80#1745C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125C2311yescharacteristic
87C2×#125C2311yescharacteristic
88C2×#125C2312yes—
89C2×C22⋊Q8—22no—
90C2×C4×Q8—22no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#501—28no—
95#501—28no—
96#432—28no—
97C2×#107C2314yes—
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×C4⋊C8C2314yes—
102C2×C22⋊C8C2314yes—
103C2×#97C2312yes—
104C2×#97C2312yes—
105C2×#97C2312yes—
106C2×#97C2312yes—
107C2×C4×C8C2312yes—
108C2×C4×C8C2312yes—
109#400—22no—
110#400—22no—
111#400—22no—
112#400—24no—
113#399—22no—
114#398—22no—
115#398—22no—
116#398—22no—
117#398—22no—
118#398—24no—
119#396—22no—
120#396—22no—
121#395—22no—
122#393—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#391—24no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389C2312yes—
137#389C2312yes—
138#389C2314yes—
139#389C2312yes—
140#389C2312yes—
141#389C2314yes—
142#388—22no—
143#388—22no—
144#388—24no—
145#387—22no—
146#387—22no—
147#387—24no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—24no—
155#385—22no—
156#385—22no—
157#384—22no—
158#384C2312yes—
159#384C2312yes—
160#384C2312yes—
161#384C2312yes—
162#382C2311yescharacteristic
163#382C2311yescharacteristic
164#382C2312yes—
165#382C2312yes—
166#382C2312yes—
167#380—22no—
168#380C2312yes—
169#380C2314yes—
170#380C2312yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4D412yes—
180C4×C22⋊C4D412yes—
181#376C2311yescharacteristic
182#376C2311yescharacteristic
183C43C2311yescharacteristic
184C43C2311yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C23×C4C2×D412yes—
202C23×C4C2411yescharacteristic
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—24no—
206C4⋊Q8—24no—
207C4⋊Q8—28no—
208C4⋊Q8—84no—
209#125—24no—
210#125—24no—
211#125—28no—
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—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
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⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2414yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C2×D411yescharacteristic
248C2×C22⋊C4C2×D412yes—
249C2×C22⋊C4C2×D411yescharacteristic
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2411yescharacteristic
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265#107—28no—
266#107—28no—
267#106—28no—
268#106—28no—
269C4⋊C8—28no—
270C22⋊C8—28no—
271#97—24no—
272#97—24no—
273#97—24no—
274#97#4118yes—
275C4×C8—24no—
276C4×C8—24no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—24no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95C2×D412yes—
297#95C2×D412yes—
298#95C2×D412yes—
299#95C2×D412yes—
300C24C22×D411yescharacteristic
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—84no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D412yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C2511yescharacteristic
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8C2×#4118yes—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C22⋊C4—24no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—84no—
371C42—24no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—82no—
380C42—82no—
381C42—84no—
382C42C2×#4114yes—
383C422−1+414yes—
384C422−1+414yes—
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—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—84no—
432C2×C4#53214yes—
433C2×C4#53612yes—
434C2×C4#53614yes—
435C2×C4#53612yes—
436C2×C4C22×#4112yes—
437C2×C4C2×2−1+412yes—
438C2×C4C2×2−1+412yes—
439C8—28no—
440C8—44no—
441C22—42no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22C2×#53211yescharacteristic
447C22C2×#53611yescharacteristic
448C22C2×#53611yescharacteristic
449C22#358512yes—
450C22#358612yes—
451C4—24no—
452C4—22no—
453C4—22no—
454C4—24no—
455C4—42no—
456C4—41no—
457C4—41no—
458C4—42no—
459C4—41no—
460C4—82no—
461C4—82no—
462C4—84no—
463C4—84no—
464C4—82no—
465C4—82no—
466C4#350914yes—
467C2—81no—
468C2#3532211yescharacteristic
469C2#6070312yes—
470C2#6088311yescharacteristic
471C2#6088411yescharacteristic
472C2#6088712yes—
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. #33317C2semidirect
4. #33297C2semidirect
5. #33296C2semidirect
6. #33294C2semidirect
9. #29188C2semidirect
10. #29187C2semidirect
13. #22251C2semidirect
14. #22250C2semidirect

Maximal subgroups

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

subgroupindex
#353362
#352832
#333172
#332972
#332962
#332942
#298842
#298832
#291882
#291872
#278882
#278862
#222512
#222502
#204672