Finite Group Databasebeta

← #6498080

Subgroups — #6498080

Order 512.

1253 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35337C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic
3#33317C211yesmaximal, characteristic, complemented
4#33295C211yesmaximal, characteristic, complemented
5#33294C211yesmaximal, characteristic, complemented
6#33293C211yesmaximal, characteristic, complemented
7#29885C214yesmaximal
8#29187C214yesmaximal, complemented
9#29184C212yesmaximal, complemented
10#29183C212yesmaximal, complemented
11#27886C214yesmaximal
12#27881C212yesmaximal
13#27880C212yesmaximal
14#22255C214yesmaximal, complemented
15#20465C211yesmaximal, characteristic
16C2×#501C2214yes—
17C2×#433C2212yes—
18#2868C2211yescharacteristic
19#2862C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2851C2211yescharacteristic
24#2851C2211yescharacteristic
25#2647C2212yes—
26#2629C2212yes—
27#2628C2212yes—
28#2628C2212yes—
29#2589—22no—
30#2589—22no—
31#2588—22no—
32#2588—22no—
33#2570—22no—
34#2550—22no—
35#2441—22no—
36#2439—22no—
37#2331C2212yes—
38#2331C2212yes—
39#2330C2212yes—
40#2314C2212yes—
41C4×#125C2211yescharacteristic
42C4×#125C2211yescharacteristic
43#2285C2212yes—
44C4×#117C2211yescharacteristic
45#2088C2214yes—
46#2088C2214yes—
47#2087C2214yes—
48#2086C2214yes—
49#2077C2214yes—
50#2077C2214yes—
51#2076C2214yes—
52#2075C2214yes—
53#1993C2214yes—
54#1992C2212yes—
55#1992C2212yes—
56#1992C2212yes—
57#1992C2212yes—
58#1991C2214yes—
59#1990C2214yes—
60#1990C2214yes—
61#1921C2214yes—
62#1921C2214yes—
63#1920C2214yes—
64#1920C2214yes—
65#1919C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1918C2214yes—
69#1847C2212yes—
70#1847C2212yes—
71#1846C2214yes—
72#1836C2212yes—
73#1836C2214yes—
74#1836C2212yes—
75#1836C2214yes—
76#1768C2212yes—
77#1767C2212yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125—22no—
87C2×#125C2311yescharacteristic
88C2×#125C2311yescharacteristic
89C2×#125C2312yes—
90C2×#117—22no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#501—216no—
95#433—28no—
96C2×#107C2314yes—
97C2×#107C2314yes—
98C2×#106C2314yes—
99C2×#106C2314yes—
100C2×C4⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×#97C2312yes—
103C2×#97C2312yes—
104C2×#97C2312yes—
105C2×#97C2312yes—
106C2×C4×C8C2312yes—
107C2×C4×C8C2312yes—
108#400—22no—
109#400—22no—
110#400—22no—
111#400—22no—
112#399—22no—
113#399—22no—
114#398—24no—
115#398—22no—
116#398—22no—
117#396—24no—
118#395—22no—
119#395—22no—
120#393—22no—
121#393—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—24no—
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#389C2312yes—
138#389C2312yes—
139#389C2314yes—
140#388—22no—
141#388—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—24no—
149#387—24no—
150#387—22no—
151#387—22no—
152#385—22no—
153#385—22no—
154#384—22no—
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#384C2312yes—
159#382—22no—
160#382—22no—
161#382C2311yescharacteristic
162#382C2311yescharacteristic
163#382C2312yes—
164#382C2312yes—
165#382C2312yes—
166#380—22no—
167#380C2312yes—
168#380C2312yes—
169#380C2314yes—
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×D412yes—
201C23×C4C2411yescharacteristic
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—28no—
205C4⋊Q8—28no—
206C4⋊Q8—84no—
207#125—28no—
208#125—24no—
209#125—24no—
210#125—84no—
211#117—24no—
212#117—24no—
213#117—84no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
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—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2414yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C22⋊C4C2×D411yescharacteristic
246C2×C22⋊C4C2×D412yes—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2411yescharacteristic
257C2×C42C2411yescharacteristic
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262#107—28no—
263#107—28no—
264#106—28no—
265#106—28no—
266C4⋊C8—28no—
267C22⋊C8—28no—
268#97—24no—
269#97—24no—
270#97—24no—
271#97#4118yes—
272C4×C8—24no—
273C4×C8—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—22no—
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—84no—
302C2×Q8—82no—
303C2×Q8—82no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D412yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C2511yescharacteristic
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8C2×#4118yes—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—28no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C22⋊C4—24no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—84no—
366C42—24no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—82no—
375C42—82no—
376C42—84no—
377C42C2×#4114yes—
378C422−1+414yes—
379C422−1+414yes—
380C23—22no—
381C23—22no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23C23×D411yescharacteristic
386Q8—164no—
387Q8—164no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
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—21no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4#53214yes—
428C2×C4#53614yes—
429C2×C4#53612yes—
430C2×C4#53612yes—
431C2×C4C22×#4112yes—
432C2×C4C2×2−1+412yes—
433C2×C4C2×2−1+412yes—
434C8—28no—
435C8—44no—
436C22—42no—
437C22—42no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22C2×#53211yescharacteristic
442C22C2×#53611yescharacteristic
443C22C2×#53611yescharacteristic
444C22#358512yes—
445C22#358612yes—
446C4—24no—
447C4—24no—
448C4—22no—
449C4—22no—
450C4—42no—
451C4—42no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—84no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—84no—
461C4#350914yes—
462C2—81no—
463C2#3532211yescharacteristic
464C2#6072712yes—
465C2#6086712yes—
466C2#6088311yescharacteristic
467C2#6088411yescharacteristic
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. #33295C2semidirect
5. #33294C2semidirect
6. #33293C2semidirect
8. #29187C2semidirect
9. #29184C2semidirect
10. #29183C2semidirect
14. #22255C2semidirect

Maximal subgroups

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

subgroupindex
#353372
#352942
#333172
#332952
#332942
#332932
#298852
#291872
#291842
#291832
#278862
#278812
#278802
#222552
#204652