Finite Group Databasebeta

← #190958

Subgroups — #190958

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34141C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#8706C212yesmaximal, complemented
5#8567C212yesmaximal, complemented
6#8563C212yesmaximal, complemented
7#8313C212yesmaximal
8#8183C211yesmaximal, characteristic, complemented
9#8177C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×C4×#41—21no—
14C2×#433—21no—
15#2367—22no—
16#2366C412yes—
17#2365C412yes—
18#2364—22no—
19#2361C412yes—
20#2361C412yes—
21#2297—21no—
22#2293C2211yescharacteristic
23#2293C2211yescharacteristic
24#2290—21no—
25#2290—21no—
26C42×D4C2211yescharacteristic
27C42×D4C2211yescharacteristic
28C4×#117C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1923—22no—
31#1916—22no—
32#1842C2212yes—
33#1832—21no—
34#1798—21no—
35#1797C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×#41—21no—
52#532—24no—
53#532—24no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#121—21no—
58C2×#121C2×C411yescharacteristic
59C2×#121C2×C411yescharacteristic
60C4×#41—44no—
61C4×#41—44no—
62C2×C4×Q8—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4C2×C412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#433—44no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C2×C4×C8C2312yes—
92C2×C4×C8C2312yes—
93#391—22no—
94#390—22no—
95#390C2×C412yes—
96#390C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128#376D411yescharacteristic
129#376D411yescharacteristic
130C2×#95—21no—
131C43C2311yescharacteristic
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139D4⋊C8—24no—
140D4⋊C8—24no—
141D4⋊C8—24no—
142D4⋊C8—24no—
143C2×#41—28no—
144C2×#41—44no—
145C2×#41—42no—
146C2×#41—42no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4C22×C411yescharacteristic
153C22×D4C22×C411yescharacteristic
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168#127—22no—
169#127—22no—
170#124—28no—
171#121—28no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176C4×Q8—48no—
177C4×D4—28no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4—48no—
187C4×D4#4114yes—
188C4×D4#4114yes—
189#117—42no—
190#117—42no—
191#117—42no—
192#117—42no—
193#117—42no—
194#117D814yes—
195#117SD1614yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42C22×C412yes—
234C2×C42C22×C412yes—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—44no—
250C4⋊C8—44no—
251C4⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254#97—42no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—44no—
260C4×C8—42no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272C24—21no—
273C24—21no—
274C24C2×C22⋊C411yescharacteristic
275#41—416no—
276#41—416no—
277C2×Q8—24no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—82no—
297C2×D4#11712yes—
298C2×D4#11712yes—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×D812yes—
336C22×C4C2×SD1612yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—24no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4#10218yes—
363C4⋊C4#11712yes—
364C4⋊C4#11712yes—
365C4⋊C4#13114yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4#10218yes—
378C42—21no—
379C42—21no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—42no—
389C42—42no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—44no—
395C42—44no—
396C42—44no—
397C42C2×D812yes—
398C42C2×D812yes—
399C42C2×SD1612yes—
400C42C2×SD1612yes—
401C42C2×#4112yes—
402C42C2×#4112yes—
403C23—22no—
404C23—21no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23C2×#10212yes—
418C23C22×C22⋊C411yescharacteristic
419Q8—48no—
420D4—22no—
421D4—22no—
422D4—24no—
423D4—48no—
424D4—48no—
425D4—48no—
426D4—41no—
427D4—41no—
428D4—42no—
429D4—84no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—82no—
469C2×C4—88no—
470C2×C4—88no—
471C2×C4—88no—
472C2×C4C2×#10212yes—
473C2×C4C2×#10214yes—
474C2×C4C2×#10214yes—
475C2×C4C2×#11711yescharacteristic
476C2×C4C2×#11711yescharacteristic
477C2×C4#52512yes—
478C2×C4#52512yes—
479C2×C4C2×#13112yes—
480C2×C4#56812yes—
481C2×C4C22×D811yescharacteristic
482C2×C4C22×SD1611yescharacteristic
483C2×C4#57512yes—
484C8—44no—
485C8—82no—
486C8—82no—
487C8—82no—
488C22—24no—
489C22—22no—
490C22—41no—
491C22—42no—
492C22—41no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22#228011yescharacteristic
504C22#288012yes—
505C22C22×#10211yescharacteristic
506C22#289611yescharacteristic
507C22#290112yes—
508C4—22no—
509C4—24no—
510C4—24no—
511C4—22no—
512C4—41no—
513C4—44no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—82no—
520C4#292211yescharacteristic
521C4#292311yescharacteristic
522C4#292512yes—
523C4#310012yes—
524C4#310112yes—
525C2—42no—
526C2—82no—
527C2—82no—
528C2#1750712yes—
529C2#1751612yes—
530C2#2072011yescharacteristic
531C2#2073211yescharacteristic
532C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21430C2semidirect
3. #20812C2semidirect
4. #8706C2semidirect
5. #8567C2semidirect
6. #8563C2semidirect
8. #8183C2semidirect
9. #8177C2semidirect
10. #8174C2semidirect
11. C4×D4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#341412
#214302
#208122
#87062
#85672
#85632
#83132
#81832
#81772
#81742
C4×D4⋊C82