Finite Group Databasebeta

← #215496

Subgroups — #215496

Order 512.

1002 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#9710C212yesmaximal, complemented
5#9709C212yesmaximal, complemented
6#9700C212yesmaximal, complemented
7#9699C212yesmaximal, complemented
8#8343C212yesmaximal, complemented
9#8341C212yesmaximal
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#2862C2211yescharacteristic
13#2862C2211yescharacteristic
14#2861—21no—
15#2861—21no—
16#2858C2211yescharacteristic
17#2858C2211yescharacteristic
18#2850—21no—
19#2850—21no—
20#2850C412yes—
21#2850C412yes—
22C4×#124—21no—
23C4×#124—21no—
24C4×#124C412yes—
25C4×#124C412yes—
26#2297—21no—
27#2297—21no—
28C2×C43C2211yescharacteristic
29#1839—21no—
30#1839—21no—
31#1831C2212yes—
32#1831C2212yes—
33#1817—21no—
34#1817—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39#1749—21no—
40C4×C22⋊C8—21no—
41#1745—21no—
42#1744—21no—
43C2×#333C2214yes—
44C2×#332C2214yes—
45C2×#332C2214yes—
46C2×#331C2214yes—
47C22×#41—21no—
48#533—44no—
49#532—44no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#127C2311yescharacteristic
54C2×#125—21no—
55C2×#124—21no—
56C2×#124—21no—
57C2×#124—21no—
58C2×#124C2×C412yes—
59C2×#124C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C412yes—
62C2×#121—21no—
63C2×#121C2×C412yes—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C412yes—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×#117—21no—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8C2314yes—
77C2×C4⋊C8C2314yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391C2×C412yes—
89#391C2×C412yes—
90#390—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2×C412yes—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#382—21no—
111#382—21no—
112#382C2×C412yes—
113#382C2×C412yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#333—44no—
135#332—44no—
136#332—44no—
137#331—44no—
138C2×#41—44no—
139C2×#41—44no—
140C2×#41—44no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8C22×C412yes—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C412yes—
148C23×C4—22no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—24no—
157C22×C8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161#127—24no—
162#125—44no—
163#124—44no—
164#124—44no—
165#124—44no—
166#124—44no—
167#124—44no—
168C22⋊Q8—44no—
169C22⋊Q8—44no—
170#121—44no—
171#121—44no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×D4—44no—
175C4×D4—44no—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2411yescharacteristic
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C22⋊C8—44no—
237C22⋊C8—44no—
238#97—44no—
239C4×C8—44no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248C24—22no—
249C24C2×C22⋊C411yescharacteristic
250#41—816no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4#13514yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—44no—
366C42H(5,𝔽2)14yes—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23#41612yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—84no—
383Q8—84no—
384Q8—84no—
385D4—84no—
386D4—88no—
387D4—84no—
388D4—84no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—28no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—88no—
438C2×C4#41312yes—
439C2×C4#41312yes—
440C2×C4#41612yes—
441C2×C4#51812yes—
442C2×C4#53512yes—
443C2×C4#53512yes—
444C2×C4C2×#13512yes—
445C2×C4#57512yes—
446C2×C4#57512yes—
447C2×C4#57512yes—
448C8—84no—
449C8—84no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22#230911yescharacteristic
465C22C2×#41311yescharacteristic
466C22C2×#41311yescharacteristic
467C22C2×#41611yescharacteristic
468C22#289611yescharacteristic
469C22#289611yescharacteristic
470C22#289611yescharacteristic
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4#321114yes—
487C4#321914yes—
488C2—42no—
489C2—162no—
490C2#1771311yescharacteristic
491C2#1771611yescharacteristic
492C2#1774011yescharacteristic
493C2#1775611yescharacteristic
494C2#2180011yescharacteristic
495C2#2180611yescharacteristic
496C2#2181211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #20777C2semidirect
4. #9710C2semidirect
5. #9709C2semidirect
6. #9700C2semidirect
7. #9699C2semidirect
8. #8343C2semidirect

Maximal subgroups

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

subgroupindex
#366392
#207782
#207772
#97102
#97092
#97002
#96992
#83432
#83412