Finite Group Databasebeta

← #220109

Subgroups — #220109

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36122C211yesmaximal, characteristic
2#22751C212yesmaximal, complemented
3C2×#1560C211yesmaximal, characteristic, complemented
4C2×#1558C211yesmaximal, characteristic, complemented
5#9699C212yesmaximal, complemented
6#9142C212yesmaximal, complemented
7#9135C212yesmaximal, complemented
8#8594C211yesmaximal, characteristic, complemented
9#8344C211yesmaximal, characteristic, complemented
10#8342C211yesmaximal, characteristic, complemented
11C4×#333C211yesmaximal, characteristic, complemented
12C2×#548C412yes—
13C2×#532C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#432C2212yes—
16C2×#423C2212yes—
17#2858C2211yescharacteristic
18#2857C2211yescharacteristic
19#2856C2211yescharacteristic
20#2545C412yes—
21#2491—22no—
22#2452—22no—
23#2436C412yes—
24#2432C412yes—
25#2416—22no—
26#2361—22no—
27C4×C4⋊Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1923—22no—
30#1917—22no—
31#1870—22no—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35C4×C4⋊C8C2212yes—
36#1753—22no—
37#1560—24no—
38#1558—24no—
39C2×#333C2211yescharacteristic
40C2×#333C2211yescharacteristic
41C2×#333C2212yes—
42C2×#333C2211yescharacteristic
43C2×#333C2212yes—
44C2×#333C2211yescharacteristic
45C2×#331C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×#41C2×C412yes—
50C23×Q8C2×C412yes—
51C22×M16—22no—
52#548—48no—
53#532—22no—
54#532—22no—
55#532—42no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#124—22no—
65C2×#124C2×C412yes—
66C2×#124C2×C412yes—
67C2×#124C2×C412yes—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8C2×C412yes—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121C2×C412yes—
74C2×C4×Q8C2×C412yes—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D412yes—
79C22×C4⋊C4—22no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#432—24no—
83#432—42no—
84#423—24no—
85#423—42no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×#97—22no—
95C2×C4×C8—22no—
96#396—24no—
97#394—24no—
98#393—24no—
99#391C2×C412yes—
100#391C2×C412yes—
101#390—22no—
102#390—22no—
103#390—22no—
104#390C2×C412yes—
105#389—22no—
106#389—22no—
107#388—24no—
108#386—24no—
109#385—24no—
110#384C2311yescharacteristic
111#380—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4D412yes—
115C4×C22⋊C4C2×C411yescharacteristic
116C4×C22⋊C4C2×C411yescharacteristic
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D412yes—
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#333—22no—
126#333—22no—
127#333—22no—
128#333—24no—
129#333—42no—
130#333D414yescomplemented
131#331—24no—
132#331—42no—
133C2×#41—24no—
134C2×#41—42no—
135C2×#41—44no—
136C2×#41—82no—
137C22×Q8—22no—
138C22×Q8—24no—
139C22×Q8—42no—
140C22×Q8—44no—
141C22×Q8—82no—
142C22×Q8C22×C412yes—
143C22×Q8C22×C412yes—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C22⋊C411yescharacteristic
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C2×M16—44no—
156C2×M16—44no—
157C2×M16—82no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C4⋊Q8—24no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—42no—
170C4⋊Q8C2×D412yes—
171C4⋊Q8#4112yes—
172#127—21no—
173#127—21no—
174#127—22no—
175#124—24no—
176#124—42no—
177#124—44no—
178#124—48no—
179#124—82no—
180C22⋊Q8—48no—
181C22⋊Q8—44no—
182#121—44no—
183#121—44no—
184#121—44no—
185#121—44no—
186#121—44no—
187#121—48no—
188C4×Q8—44no—
189C4×D4—24no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—48no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4C22⋊C411yescharacteristic
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C411yescharacteristic
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—44no—
241C4⋊C8C2×D414yes—
242C22⋊C8—44no—
243C22⋊C8—44no—
244C22⋊C8—82no—
245#97—44no—
246C4×C8—82no—
247#95—24no—
248#95—24no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24—22no—
253C24C2×C22⋊C411yescharacteristic
254C24C2×C22⋊C411yescharacteristic
255#41—88no—
256#41—88no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—82no—
267C2×Q8—88no—
268C2×Q8—88no—
269C2×Q8C4×D414yes—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—44no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—48no—
279C2×D4—44no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—82no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D412yes—
325M16—88no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C22⋊C4—22no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—48no—
354C22⋊C4—48no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—84no—
360C22⋊C4#10014yes—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—41no—
375C42—44no—
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C42H(5,𝔽2)12yes—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—22no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—22no—
387C23—22no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23C2×#10012yes—
394C23C22×C22⋊C411yescharacteristic
395Q8—44no—
396Q8—44no—
397Q8—82no—
398Q8—82no—
399Q8—88no—
400D4—44no—
401D4—82no—
402D4—82no—
403D4—88no—
404D4—88no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—82no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
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—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—88no—
449C2×C4—88no—
450C2×C4C2×#10012yes—
451C2×C4C2×#10012yes—
452C2×C4C2×C4×D411yescharacteristic
453C2×C4#51811yescharacteristic
454C2×C4D4212yes—
455C2×C4#54612yes—
456C2×C4#57512yes—
457C2×C4#57512yes—
458C2×C4#57712yes—
459C2×C4#57812yes—
460C8—44no—
461C8—82no—
462C8—82no—
463C8—82no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—41no—
470C22—42no—
471C22—41no—
472C22—41no—
473C22—42no—
474C22—82no—
475C22—84no—
476C22—82no—
477C22—82no—
478C22—84no—
479C22D4×C22⋊C411yescharacteristic
480C22C22×#10011yescharacteristic
481C22#288811yescharacteristic
482C22#289612yes—
483C22#289812yes—
484C4—22no—
485C4—21no—
486C4—21no—
487C4—22no—
488C4—41no—
489C4—42no—
490C4—41no—
491C4—44no—
492C4—42no—
493C4—42no—
494C4—82no—
495C4—82no—
496C4#327512yes—
497C4#327712yes—
498C4#332612yes—
499C4#332712yes—
500C2—42no—
501C2—81no—
502C2—162no—
503C2#1773811yescharacteristic
504C2#1776812yes—
505C2#1777911yescharacteristic
506C2D4×#10011yescharacteristic
507C2#2242912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36122C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22751C2semidirect
3. C2×#1560C2semidirect
4. C2×#1558C2semidirect
5. #9699C2semidirect
6. #9142C2semidirect
7. #9135C2semidirect
8. #8594C2semidirect
9. #8344C2semidirect
10. #8342C2semidirect
11. C4×#333C2semidirect
130. #333D4semidirect

Maximal subgroups

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

subgroupindex
#361222
#227512
C2×#15602
C2×#15582
#96992
#91422
#91352
#85942
#83442
#83422
C4×#3332