Finite Group Databasebeta

← #6667459

Subgroups — #6667459

Order 512.

1209 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35337C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#32854C214yesmaximal, complemented
4#29947C214yesmaximal
5#29945C214yesmaximal
6#29884C212yesmaximal
7#29883C212yesmaximal
8#29293C214yesmaximal, complemented
9#29226C211yesmaximal, characteristic, complemented
10#29225C211yesmaximal, characteristic, complemented
11#29224C212yesmaximal, complemented
12#28695C211yesmaximal, characteristic, complemented
13#28695C211yesmaximal, characteristic, complemented
14#28692C212yesmaximal, complemented
15#20811C211yesmaximal, characteristic
16C2×#501C2212yes—
17C2×#501C2212yes—
18C2×#432C2212yes—
19#2868C2211yescharacteristic
20#2862C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2212yes—
24#2851C2211yescharacteristic
25#2851C2211yescharacteristic
26#2647C2212yes—
27#2647C2212yes—
28#2628C2212yes—
29#2628C2212yes—
30#2589—22no—
31#2589—22no—
32#2588—22no—
33#2588—22no—
34#2511—22no—
35#2492—22no—
36#2451—22no—
37#2423—22no—
38#2331C2212yes—
39#2331C2212yes—
40#2314C2212yes—
41#2314C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×#125C2211yescharacteristic
44C4×#125C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2101C2214yes—
48#2100C2214yes—
49#2099C2214yes—
50#2088C2214yes—
51#2088C2214yes—
52#2087C2214yes—
53#2086C2214yes—
54#2085C2214yes—
55#2085C2214yes—
56#2084C2214yes—
57#2083C2214yes—
58#2077C2214yes—
59#2077C2214yes—
60#2076C2214yes—
61#2075C2214yes—
62#1993C2212yes—
63#1993C2212yes—
64#1992C2212yes—
65#1992C2212yes—
66#1988C2214yes—
67#1988C2214yes—
68#1987C2214yes—
69#1987C2214yes—
70#1949C2214yes—
71#1949C2214yes—
72#1949C2214yes—
73#1949C2214yes—
74#1843C2212yes—
75#1836C2212yes—
76#1836C2212yes—
77#1832C2212yes—
78#1798C2212yes—
79#1798C2212yes—
80#1767C2212yes—
81#1767C2211yescharacteristic
82#1767C2211yescharacteristic
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#125C2311yescharacteristic
88C2×#125C2311yescharacteristic
89C2×C22⋊Q8—22no—
90C2×C4×Q8—22no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#501—44no—
95#501—44no—
96#432—44no—
97C2×#107C2314yes—
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2312yes—
107C2×#97C2312yes—
108#400—22no—
109#400—22no—
110#400—22no—
111#400—22no—
112#400—24no—
113#399—22no—
114#399—22no—
115#398—22no—
116#398—22no—
117#398—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—22no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391—24no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389C2312yes—
136#389C2312yes—
137#389C2314yes—
138#389C2312yes—
139#389C2312yes—
140#389C2314yes—
141#388—22no—
142#388—22no—
143#388—24no—
144#387—22no—
145#387—22no—
146#387—24no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—24no—
154#385—22no—
155#385—22no—
156#384—22no—
157#384C2312yes—
158#384C2312yes—
159#384C2312yes—
160#382C2311yescharacteristic
161#382C2311yescharacteristic
162#382C2312yes—
163#382C2312yes—
164#382C2312yes—
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#336C2312yes—
198#336C2312yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C23×C4C2×D412yes—
203C23×C4C2411yescharacteristic
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—24no—
207C4⋊Q8—28no—
208C4⋊Q8—28no—
209C4⋊Q8—24no—
210C4⋊Q8—84no—
211#125—22no—
212#125—22no—
213#125—22no—
214#125—22no—
215C22⋊Q8—84no—
216C4×Q8—84no—
217#117—24no—
218#117—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
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—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266#107—44no—
267#107—44no—
268#106—44no—
269#106—44no—
270C4⋊C8—24no—
271C4⋊C8—28no—
272C4⋊C8—24no—
273C4⋊C8—44no—
274C22⋊C8—44no—
275#97—24no—
276#97—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#95—22no—
297#95—22no—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D412yes—
301#95C2×D412yes—
302C24C22×D411yescharacteristic
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—84no—
309C2×Q8—82no—
310C2×Q8—82no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C2511yescharacteristic
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—84no—
374C42—24no—
375C42—21no—
376C42—21no—
377C42—22no—
378C42—22no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—82no—
384C42—82no—
385C42—84no—
386C42H(5,𝔽2)12yes—
387C422−1+414yes—
388C422−1+414yes—
389C422−1+412yes—
390C23—22no—
391C23—22no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23C23×D411yescharacteristic
396Q8—88no—
397Q8—164no—
398Q8—164no—
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—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—84no—
439C2×C4#53612yes—
440C2×C4#53614yes—
441C2×C4#53612yes—
442C2×C4#53714yes—
443C2×C4C2×H(5,𝔽2)11yescharacteristic
444C2×C4C2×2−1+412yes—
445C2×C4C2×2−1+412yes—
446C2×C4C2×2−1+411yescharacteristic
447C8—44no—
448C8—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—42no—
453C22—42no—
454C22C2×#53611yescharacteristic
455C22C2×#53611yescharacteristic
456C22C2×#53711yescharacteristic
457C22#358512yes—
458C22#358612yes—
459C4—22no—
460C4—22no—
461C4—24no—
462C4—24no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—42no—
467C4—42no—
468C4—84no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4—84no—
474C4#355012yes—
475C4#355612yes—
476C2—81no—
477C2#3764111yescharacteristic
478C2#6088311yescharacteristic
479C2#6088411yescharacteristic
480C2#6088512yes—
481C2#6091912yes—
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. #32854C2semidirect
8. #29293C2semidirect
9. #29226C2semidirect
10. #29225C2semidirect
11. #29224C2semidirect
12. #28695C2semidirect
13. #28695C2semidirect
14. #28692C2semidirect

Maximal subgroups

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

subgroupindex
#353372
#352832
#328542
#299472
#299452
#298842
#298832
#292932
#292262
#292252
#292242
#286952
#286922
#208112