Finite Group Databasebeta

← #215468

Subgroups — #215468

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3C4×#432C211yesmaximal, characteristic, complemented
4#9688C214yesmaximal, complemented
5#9687C214yesmaximal, complemented
6#8335C212yesmaximal, complemented
7#8334C212yesmaximal
8C22×#125C2211yescharacteristic
9C22×#124C2211yescharacteristic
10C2×#432—21no—
11#2861—21no—
12#2861—21no—
13#2861—21no—
14#2861—21no—
15#2861C412yes—
16#2861C412yes—
17#2861C2212yes—
18#2861C2212yes—
19C2×#391C411yescharacteristic
20C2×#391C411yescharacteristic
21C2×#390C411yescharacteristic
22C2×#390C411yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×#388—21no—
26C2×#388—21no—
27C2×C43C2211yescharacteristic
28#1839—21no—
29#1831—21no—
30#1817—21no—
31#1797—21no—
32#1767C2212yescomplemented
33#1767C2212yescomplemented
34C4×C4⋊C8—21no—
35#1749C2212yes—
36#1749C2212yes—
37C4×C22⋊C8—21no—
38#1745—21no—
39#1745C2212yescomplemented
40C4×#97C2212yescomplemented
41C2×#330C2218yescomplemented
42C2×#329C2218yes—
43C23×Q8C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C2×#125—21no—
46C2×#125—21no—
47C2×#125—44no—
48C2×#125C2×C412yes—
49C2×#125C2312yes—
50C2×#124—21no—
51C2×#124—21no—
52C2×#124—44no—
53C2×#124C2×C412yes—
54C2×#124C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#432—28no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—22no—
78C2×#97C2314yes—
79C2×#97C2314yes—
80C2×C4×C8—22no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390C2×C412yes—
94#390C2×C412yes—
95#390C2×C412yes—
96#390C2×C412yes—
97#388—22no—
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#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#330—48no—
132#329—48no—
133C25—21no—
134C22×Q8—21no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8C22×C412yes—
139C22×D4—21no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4C22×C412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C22×C8—24no—
155C22×C8—24no—
156#125—28no—
157#125—44no—
158#125—44no—
159#125—44no—
160#124—28no—
161#124—44no—
162#124—44no—
163#124—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2411yescharacteristic
229C4⋊C8—28no—
230C4⋊C8—44no—
231C22⋊C8—28no—
232C22⋊C8—44no—
233#97—28no—
234#97—28no—
235#97—28no—
236C4×C8—44no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245C24—22no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24C2×C22⋊C411yescharacteristic
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—88no—
261C2×D4—24no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—88no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
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—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—28no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊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—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
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—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42H(5,𝔽2)14yes—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23—84no—
386C23—84no—
387C23#41312yes—
388C23C2×#10412yes—
389C23#42112yes—
390C23C22×C22⋊C411yescharacteristic
391Q8—84no—
392Q8—84no—
393D4—84no—
394D4—84no—
395C2×C4—28no—
396C2×C4—21no—
397C2×C4—21no—
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—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4#41312yes—
449C2×C4C2×#10412yes—
450C2×C4C2×#10412yes—
451C2×C4#42112yes—
452C2×C4#42112yes—
453C2×C4#51812yes—
454C2×C4#53512yes—
455C2×C4#53512yes—
456C8—48no—
457C8—84no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—168no—
472C22#152914yes—
473C22#230911yescharacteristic
474C22C2×#41311yescharacteristic
475C22C2×#41311yescharacteristic
476C22C22×#10411yescharacteristic
477C22C2×#42111yescharacteristic
478C22C2×#42111yescharacteristic
479C22C2×#42111yescharacteristic
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—82no—
495C4—82no—
496C4—82no—
497C2—24no—
498C2—162no—
499C2C2×#152911yescharacteristic
500C2#1765011yescharacteristic
501C2#1765911yescharacteristic
502C2#1766811yescharacteristic
503C2#2179811yescharacteristic
504C2#2181511yescharacteristic
505C2#2181611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
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. C4×#432C2semidirect
4. #9688C2semidirect
5. #9687C2semidirect
6. #8335C2semidirect
32. #1767C22semidirect
33. #1767C22semidirect
39. #1745C22semidirect
40. C4×#97C22semidirect
41. C2×#330C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#207782
C4×#4322
#96882
#96872
#83352
#83342