Finite Group Databasebeta

← #164721

Subgroups — #164721

Order 512.

1031 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36493C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic, complemented
3#20684C211yesmaximal, characteristic, complemented
4#7730C212yesmaximal, complemented
5#7684C211yesmaximal, characteristic, complemented
6#7670C212yesmaximal, complemented
7#7664C211yesmaximal, characteristic, complemented
8#7647C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7582C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×Q8—21no—
15C2×#408—21no—
16#2856C2211yescharacteristic
17#2850C2212yes—
18#2541C411yescharacteristic
19#2540C412yes—
20#2538C411yescharacteristic
21#2535C412yes—
22#2471—22no—
23#2456—22no—
24#2448—21no—
25#2431—21no—
26#2366—21no—
27#2363—21no—
28#2355C411yescharacteristic
29C4×C22⋊Q8C2211yescharacteristic
30C4×#121C2212yes—
31#1954—21no—
32#1954—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1797C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39C4×C22⋊C8C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56#559—44no—
57C2×#127—21no—
58C2×#127C2×C411yescharacteristic
59C2×#124—21no—
60C2×#124C2×C412yes—
61C2×#123—22no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C412yes—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C2×#121C2×C412yes—
69C4×#41—44no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D412yes—
80C2×#117D412yes—
81C2×#117D411yescharacteristic
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84C8×Q8—44no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87#408—44no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96#399—22no—
97#397—22no—
98#396—22no—
99#392—22no—
100#392—22no—
101#392—22no—
102#391—21no—
103#391C2311yescharacteristic
104#390—21no—
105#390—21no—
106#390—22no—
107#390—22no—
108#390—21no—
109#390—21no—
110#390C2312yes—
111#389—21no—
112#389C2311yescharacteristic
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—21no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#385—21no—
124#384C2×C411yescharacteristic
125#382C2×C412yes—
126#380C2×C411yescharacteristic
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C411yescharacteristic
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22⋊C412yes—
157C22×D4C22×C412yes—
158C23×C4—22no—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C2×M16—42no—
166C2×M16—42no—
167C2×M16—42no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177#127—42no—
178#127—44no—
179#124—44no—
180#124—44no—
181#123—48no—
182C22⋊Q8—22no—
183C22⋊Q8—41no—
184C22⋊Q8—82no—
185#121—24no—
186#121—42no—
187#121—42no—
188#121—42no—
189#121—81no—
190#121—81no—
191#121—82no—
192#121—82no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—42no—
196C4×D4—44no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—82no—
203C4×D4—82no—
204#117—42no—
205#117—44no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C412yes—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—44no—
239C2×C42—42no—
240C2×C42C22×C412yes—
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2411yescharacteristic
247C4⋊C8—44no—
248C4⋊C8—42no—
249C22⋊C8—44no—
250C22⋊C8—42no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—82no—
254C22⋊C8—81no—
255C4×C8—44no—
256C4×C8—42no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—21no—
266C24—22no—
267C24C2×C22⋊C412yes—
268C24C2×C22⋊C411yescharacteristic
269#41—48no—
270#41—48no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8#9914yes—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—81no—
297C2×D4—81no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—82no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4#13512yes—
340C22×C4#13512yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344M16—84no—
345C2×C8—22no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—84no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—81no—
378C4⋊C4—82no—
379C4⋊C4#13112yes—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—81no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C42—24no—
398C42—22no—
399C42—24no—
400C42—22no—
401C42—41no—
402C42—42no—
403C42—41no—
404C42—42no—
405C42—42no—
406C42—82no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23#41612yes—
428C23C22×C22⋊C411yescharacteristic
429Q8—28no—
430Q8—81no—
431Q8—81no—
432D4—48no—
433D4—44no—
434D4—82no—
435D4—82no—
436D4—88no—
437D4—84no—
438D4—162no—
439D4—164no—
440D4—164no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—164no—
489C2×C4C2×#9914yes—
490C2×C4C2×#9912yes—
491C2×C4#41612yes—
492C2×C4#42112yes—
493C2×C4C2×#13111yescharacteristic
494C2×C4#56811yescharacteristic
495C2×C4C2×#13511yescharacteristic
496C2×C4C2×#13511yescharacteristic
497C2×C4#57611yescharacteristic
498C2×C4#57711yescharacteristic
499C8—42no—
500C8—81no—
501C8—82no—
502C8—81no—
503C8—81no—
504C8—82no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22#288011yescharacteristic
527C22C22×#9911yescharacteristic
528C22#288811yescharacteristic
529C22C2×#41611yescharacteristic
530C22#289711yescharacteristic
531C22C2×#42111yescharacteristic
532C22#290111yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—24no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—41no—
542C4—42no—
543C4—42no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—82no—
548C4—81no—
549C4#151614yes—
550C2—42no—
551C2—82no—
552C2—162no—
553C2#1741911yescharacteristic
554C2#1751011yescharacteristic
555C2#1752211yescharacteristic
556C2C2×#151611yescharacteristic
557C2#1761111yescharacteristic
558C2#1768111yescharacteristic
559C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36493C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22530C2semidirect
3. #20684C2semidirect
4. #7730C2semidirect
5. #7684C2semidirect
6. #7670C2semidirect
7. #7664C2semidirect
8. #7647C2semidirect
9. #7617C2semidirect
10. #7582C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364932
#225302
#206842
#77302
#76842
#76702
#76642
#76472
#76172
#75822
#75662