Finite Group Databasebeta

← #224384

Subgroups — #224384

Order 512.

861 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic
3#13933C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9909C211yesmaximal, characteristic
6#9887C212yesmaximal
7#9885C211yesmaximal, characteristic
8#8578C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8217C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2851C2212yes—
14#2590—22no—
15#2568C411yescharacteristic, complemented
16#2552C411yescharacteristic, complemented
17#2511—21no—
18#2508C412yescomplemented
19#2505—22no—
20#2502—21no—
21#2453C412yescomplemented
22#2365C411yescharacteristic, complemented
23#2360—21no—
24#2355—21no—
25Q8×C4⋊C4C411yescharacteristic, complemented
26C4×C22⋊Q8C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C2×#389—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1939—21no—
31#1917—22no—
32#1797C2211yescharacteristic
33#1797C2212yes—
34#1770—21no—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2211yescharacteristic
38C2×#328C2212yes—
39C2×#328C2211yescharacteristic
40#1442—22no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2311yescharacteristic
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8C2311yescharacteristic
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×#97—22no—
73C2×C4×C8—21no—
74#400—22no—
75#400—22no—
76#400—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391C2×C412yes—
85#391C2×C411yescharacteristic
86#389—22no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—44no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C411yescharacteristic
95#389C2×C412yes—
96#387—22no—
97#387—21no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—21no—
107#384—21no—
108#384—21no—
109#384C2×C411yescharacteristic
110#382—22no—
111#382C2×C412yes—
112#380—21no—
113#380C2311yescharacteristic
114#380C2312yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95D412yes—
134C2×#95C2311yescharacteristic
135C2×#95C2312yes—
136C2×#95C2311yescharacteristic
137#336—21no—
138#336—22no—
139#336—21no—
140#336—22no—
141#328—22no—
142#328—24no—
143#328Q814yescomplemented
144C22×Q8—21no—
145C22×Q8C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C4⋊Q8—42no—
158#125—82no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8C2×Q812yes—
164C22⋊Q8#4112yes—
165C4×Q8—42no—
166C4×Q8—44no—
167C4×Q8—82no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—82no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C4⋊C8—42no—
222C4⋊C8—82no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—22no—
226C22⋊C8—21no—
227C22⋊C8—22no—
228C22⋊C8C2×Q812yes—
229#97—82no—
230C4×C8—42no—
231#95—21no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—21no—
250#95—22no—
251#95—42no—
252#95—44no—
253#95—82no—
254#95—81no—
255#95—82no—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C2×Q812yes—
261C24C22×D411yescharacteristic
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8C4×Q812yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4#13712yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C42−1+411yescharacteristic
327C22×C42−1+411yescharacteristic
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—81no—
356C4⋊C4—82no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—84no—
363C4⋊C4—88no—
364C4⋊C4C4×Q812yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C42—22no—
372C42—22no—
373C42—42no—
374C42—42no—
375C42—41no—
376C42—41no—
377C42—42no—
378C42—42no—
379C42—44no—
380C42—42no—
381C42—42no—
382C42—82no—
383C42—84no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23C22×C22⋊C411yescharacteristic
389C23#54811yescharacteristic
390C23D4×Q811yescharacteristic
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23C2×#13711yescharacteristic
393C23#57711yescharacteristic
394C23#57811yescharacteristic
395Q8—42no—
396Q8—42no—
397Q8—84no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#41012yes—
474C2×C4#41712yes—
475C2×C4#41712yes—
476C2×C4#42112yes—
477C2×C4C2×C4×Q811yescharacteristic
478C2×C4#51911yescharacteristic
479C2×C4#54811yescharacteristic
480C2×C4D4×Q811yescharacteristic
481C8—42no—
482C8—81no—
483C8—82no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22Q8×C22⋊C411yescharacteristic
497C22C2×#41011yescharacteristic
498C22#288311yescharacteristic
499C22C2×#41711yescharacteristic
500C22#289811yescharacteristic
501C22C2×#42111yescharacteristic
502C22#290111yescharacteristic
503C22#336811yescharacteristic
504C22#337411yescharacteristic
505C22#338611yescharacteristic
506C22#338711yescharacteristic
507C4—22no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—81no—
524C4—81no—
525C4—82no—
526C4—162no—
527C4#151214yes—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2C2×#151211yescharacteristic
533C2#1760011yescharacteristic
534C2#1761211yescharacteristic
535C2#1763011yescharacteristic
536C2#2251311yescharacteristic
537C2#2252411yescharacteristic
538C2#2252911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2568C4semidirect
16. #2552C4semidirect
18. #2508C4semidirect
21. #2453C4semidirect
22. #2365C4semidirect
25. Q8×C4⋊C4C4semidirect
143. #328Q8semidirect

Maximal subgroups

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

subgroupindex
#365522
#226292
#139332
#99132
#99092
#98872
#98852
#85782
#85722
#82172
C4×#3282