Finite Group Databasebeta

← #161835

Subgroups — #161835

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3#20772C211yesmaximal, characteristic, complemented
4#7722C212yesmaximal, complemented
5#7719C212yesmaximal, complemented
6#7710C214yesmaximal, complemented
7#7584C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C2×#532C411yescharacteristic
10C2×#431—21no—
11#2862C411yescharacteristic
12#2862C2211yescharacteristic
13#2861—21no—
14#2861—21no—
15#2858C411yescharacteristic
16#2858C2211yescharacteristic
17#2850—21no—
18#2850—21no—
19#2850C412yes—
20#2850C2212yes—
21C4×#124—21no—
22C4×#124—21no—
23C4×#124C412yes—
24C4×#124C2212yes—
25#2297—21no—
26#2297—21no—
27C2×C43C2211yescharacteristic
28#1842—21no—
29#1839—21no—
30#1839—21no—
31#1839C2212yes—
32#1839C2212yes—
33#1831C2212yes—
34#1797—21no—
35#1797C2212yes—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2212yes—
38#1728—21no—
39#1728—21no—
40#1728C2212yes—
41#1272C2218yes—
42#1270C2218yes—
43C22×#41—21no—
44C22×M16—22no—
45#533—44no—
46#532—44no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#125—21no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—21no—
55C2×#124C2×C412yes—
56C2×#124C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×#121—21no—
60C2×#121C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×#117—21no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73#431—44no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C4×C8—22no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391C2×C412yes—
86#391C2312yes—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390C2×C412yes—
94#390C2312yes—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#382—21no—
108#382—21no—
109#382C2×C412yes—
110#382C2×C412yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C43C2×C411yescharacteristic
116C43C2×C411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336C2314yes—
130#336C2314yes—
131C2×#41—44no—
132C2×#41—44no—
133C2×#41—44no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8C22×C412yes—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—44no—
150C2×M16—44no—
151C2×M16—44no—
152C22×C8—24no—
153C22×C8—24no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157#127—42no—
158#125—44no—
159#124—28no—
160#124—44no—
161#124—44no—
162#124—44no—
163#124—42no—
164#124—42no—
165C22⋊Q8—44no—
166C22⋊Q8—44no—
167#121—44no—
168#121—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×D4—44no—
172C4×D4—44no—
173#117—44no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
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—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
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×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
229C2×C42C2411yescharacteristic
230C4⋊C8—44no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C22⋊C8—44no—
234C4×C8—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24—22no—
244C24C2×C22⋊C411yescharacteristic
245#41—816no—
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×D4—24no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—42no—
266C2×D4—42no—
267C22×C4—22no—
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—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—88no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C42—22no—
337C42—21no—
338C42—21no—
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—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—44no—
363C42#10014yes—
364C42#13114yes—
365C42AGL(1,ℤ/8)12yes—
366C42#13712yes—
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#42112yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—84no—
383Q8—84no—
384Q8—84no—
385D4—84no—
386D4—84no—
387D4—84no—
388D4—88no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
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—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
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—88no—
437C2×C4#40812yes—
438C2×C4#40812yes—
439C2×C4#41012yes—
440C2×C4C2×#10012yes—
441C2×C4C2×#10012yes—
442C2×C4C2×#10012yes—
443C2×C4#41712yes—
444C2×C4#41712yes—
445C2×C4#42112yes—
446C2×C4C2×#13112yes—
447C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C2×C4#57512yes—
450C8—84no—
451C8—84no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22C2×#40811yescharacteristic
467C22C2×#41011yescharacteristic
468C22C22×#10011yescharacteristic
469C22C2×#41711yescharacteristic
470C22#289611yescharacteristic
471C22C2×#42111yescharacteristic
472C22C2×#42111yescharacteristic
473C4—22no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—41no—
487C4—41no—
488C4—82no—
489C4—82no—
490C4—82no—
491C2—42no—
492C2—162no—
493C2#1738411yescharacteristic
494C2#1746611yescharacteristic
495C2#1747711yescharacteristic
496C2#1758511yescharacteristic
497C2#1759311yescharacteristic
498C2#1766611yescharacteristic
499C2#1775011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. #20772C2semidirect
4. #7722C2semidirect
5. #7719C2semidirect
6. #7710C2semidirect
7. #7584C2semidirect

Maximal subgroups

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

subgroupindex
#366392
#208082
#207722
#77222
#77192
#77102
#75842