Finite Group Databasebeta

← #163967

Subgroups — #163967

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#22591C212yesmaximal, complemented
3#7653C211yesmaximal, characteristic, complemented
4#7649C211yesmaximal, characteristic, complemented
5#7639C211yesmaximal, characteristic, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7624C212yesmaximal, complemented
8#7618C212yesmaximal, complemented
9#7595C211yesmaximal, characteristic, complemented
10#7578C212yesmaximal
11C2×#1272C211yesmaximal, characteristic, complemented
12C2×#536C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×#519C411yescharacteristic
15#2362—22no—
16#2360—22no—
17#2359—22no—
18#2359—22no—
19#2327C411yescharacteristic
20#2327C411yescharacteristic
21#2326C411yescharacteristic
22#2326C411yescharacteristic
23#2317C411yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#124C2211yescharacteristic
27C4×#124C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1917—22no—
32#1917—22no—
33#1850—22no—
34#1839C2212yes—
35#1831C2212yes—
36#1817C2212yes—
37#1767C2212yes—
38#1752—22no—
39#1745C2212yes—
40C2×#336C2212yes—
41#1272C2212yes—
42#1272C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51C22×#41C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C23×C8—22no—
54#536—24no—
55#536—24no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#123—22no—
61C2×C22⋊Q8—22no—
62#520—44no—
63#519—44no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×#97—22no—
89C2×#97—22no—
90#391—22no—
91#391C2×C411yescharacteristic
92#391C2×C411yescharacteristic
93#391C2×C411yescharacteristic
94#391C2311yescharacteristic
95#391C2311yescharacteristic
96#391C2311yescharacteristic
97#390C2×C411yescharacteristic
98#390C2311yescharacteristic
99#389—22no—
100#389—22no—
101#389—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#386—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#382C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D412yes—
127#376C2×C411yescharacteristic
128#376C2×C411yescharacteristic
129C2×#95—22no—
130C43C2311yescharacteristic
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336C2312yes—
139#336C2312yes—
140#336C2312yes—
141#336C2312yes—
142C2×#41—22no—
143C2×#41—42no—
144C2×#41—42no—
145C2×#41M1614yes—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8M1614yes—
151C22×Q8C22×C411yescharacteristic
152C22×Q8C22×C411yescharacteristic
153C22×Q8C22×C411yescharacteristic
154C22×D4—21no—
155C22×D4C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—44no—
168C22×C8—44no—
169C22×C8—82no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174#124—22no—
175#124—22no—
176#124—22no—
177#124—22no—
178#123—216no—
179C22⋊Q8—216no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—44no—
191C4×D4—44no—
192#117—22no—
193#117—22no—
194#117—24no—
195#117—44no—
196#117—44no—
197#117—44no—
198#117—42no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4C22⋊C411yescharacteristic
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C411yescharacteristic
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—42no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2411yescharacteristic
248C4⋊C8—44no—
249C4⋊C8—82no—
250C22⋊C8—44no—
251#97—44no—
252#97—82no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265C24—21no—
266C24C2×C22⋊C411yescharacteristic
267#41—48no—
268#41—48no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—48no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—48no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C2×M1612yes—
284C2×Q8C2×M1612yes—
285C2×Q8C2×M1612yes—
286C2×D4—24no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4C2×M1612yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×M1612yes—
327C22×C4C2×M1612yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D412yes—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—88no—
343C4⋊C4—28no—
344C4⋊C4—28no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C22⋊C4—28no—
362C22⋊C4—28no—
363C22⋊C4—24no—
364C22⋊C4—22no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—82no—
371C22⋊C4#10014yes—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42—42no—
384C42—41no—
385C42—42no—
386C42—41no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23#40712yes—
402C23#41012yes—
403C23C2×#10012yes—
404C23C22×C22⋊C411yescharacteristic
405Q8—42no—
406Q8—42no—
407Q8—42no—
408Q8—48no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412D4—42no—
413D4—42no—
414D4—48no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—48no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—162no—
460C2×C4—162no—
461C2×C4—162no—
462C2×C4#40712yes—
463C2×C4C2×#10012yes—
464C2×C4C2×#10012yes—
465C2×C4C22×M1611yescharacteristic
466C2×C4C22×M1611yescharacteristic
467C2×C4#57712yes—
468C2×C4#57812yes—
469C8—44no—
470C8—82no—
471C8—82no—
472C8—82no—
473C22—22no—
474C22—24no—
475C22—22no—
476C22—24no—
477C22—44no—
478C22—42no—
479C22—81no—
480C22—81no—
481C22—82no—
482C22#146614yes—
483C22C2×#40711yescharacteristic
484C22C2×#41011yescharacteristic
485C22C22×#10011yescharacteristic
486C22#289812yes—
487C22#290112yes—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—21no—
494C4—21no—
495C4—41no—
496C4—42no—
497C4—42no—
498C4—41no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C2—24no—
508C2—82no—
509C2C2×#146611yescharacteristic
510C2#1745412yes—
511C2#1760912yes—
512C2#1768511yescharacteristic
513C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22591C2semidirect
3. #7653C2semidirect
4. #7649C2semidirect
5. #7639C2semidirect
6. #7635C2semidirect
7. #7624C2semidirect
8. #7618C2semidirect
9. #7595C2semidirect
11. C2×#1272C2semidirect

Maximal subgroups

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

subgroupindex
#342762
#225912
#76532
#76492
#76392
#76352
#76242
#76182
#75952
#75782
C2×#12722