Finite Group Databasebeta

← #223080

Subgroups — #223080

Order 512.

849 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic
3#13921C211yesmaximal, characteristic
4#9909C211yesmaximal, characteristic
5#9885C211yesmaximal, characteristic
6#9153C212yesmaximal
7#9112C212yesmaximal
8#8714C212yesmaximal
9#8572C211yesmaximal, characteristic
10C4×#328C211yesmaximal, characteristic
11#8025C212yesmaximal
12C2×#431C2212yes—
13#2857C2211yescharacteristic
14#2855C2212yes—
15#2571C411yescharacteristic, complemented
16#2568C411yescharacteristic, complemented
17#2533—22no—
18#2514—21no—
19#2514C412yescomplemented
20#2509C412yescomplemented
21#2503—21no—
22#2493—22no—
23#2365C411yescharacteristic, complemented
24#2360—21no—
25#2356—21no—
26#2355C411yescharacteristic, complemented
27C4×C22⋊Q8C2211yescharacteristic
28#2283C2212yes—
29C2×#383C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1916—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35C8×C4⋊C4—21no—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2211yescharacteristic
38C2×#328C2211yescharacteristic
39C2×#328C2212yes—
40#1443—22no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95C2211yescharacteristic
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2211yescharacteristic
51#1294C2212yes—
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2311yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64#431—24no—
65#431—42no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2312yes—
69C2×C22⋊C8C2311yescharacteristic
70C2×C22⋊C8C2311yescharacteristic
71C2×C4×C8—22no—
72C2×C4×C8—21no—
73#401—22no—
74#400—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#400—22no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C411yescharacteristic
86#389—21no—
87#389—22no—
88#389—21no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C411yescharacteristic
93#387—22no—
94#387—22no—
95#387—22no—
96#387—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#385—21no—
102#384C2×C411yescharacteristic
103#383—22no—
104#383—22no—
105#383—24no—
106#383—42no—
107#382—21no—
108#382—21no—
109#382C2×C412yes—
110#382C2×C412yes—
111#381—22no—
112#381—22no—
113#380—21no—
114#380C2312yes—
115#380C2311yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129#376—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95D412yes—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140#336—21no—
141#336—22no—
142#336—22no—
143#336—21no—
144#328—22no—
145#328—22no—
146#328—42no—
147C22×Q8—21no—
148C22×Q8C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C4⋊Q8—42no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C4×Q8—42no—
168C4×Q8—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C4⋊C8—44no—
219C4⋊C8—42no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—21no—
225C22⋊C8#4112yes—
226C4×C8—44no—
227C4×C8—42no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—44no—
253#95—44no—
254#95—42no—
255#95—44no—
256#95—82no—
257#95—81no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262#95#4112yes—
263C24C22×D411yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
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—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#13512yes—
325C22×C4#13512yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+411yescharacteristic
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—82no—
363C4⋊C4—84no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—41no—
381C42—41no—
382C42—44no—
383C42—42no—
384C42—42no—
385C42—82no—
386C42—84no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23C22×C22⋊C411yescharacteristic
392C23#54611yescharacteristic
393C23#54711yescharacteristic
394C23C2×#13511yescharacteristic
395C23C2×#13511yescharacteristic
396C23#57711yescharacteristic
397C23#57811yescharacteristic
398Q8—81no—
399Q8—81no—
400Q8—84no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
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—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#41012yes—
479C2×C4#41612yes—
480C2×C4#41612yes—
481C2×C4#42112yes—
482C2×C4C4×#4111yescharacteristic
483C2×C4#52011yescharacteristic
484C2×C4#54611yescharacteristic
485C2×C4#54711yescharacteristic
486C8—42no—
487C8—82no—
488C8—81no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22#233911yescharacteristic
502C22C2×#41011yescharacteristic
503C22#288411yescharacteristic
504C22C2×#41611yescharacteristic
505C22#289811yescharacteristic
506C22C2×#42111yescharacteristic
507C22#290111yescharacteristic
508C22#330211yescharacteristic
509C22#330411yescharacteristic
510C22#334011yescharacteristic
511C22#334211yescharacteristic
512C4—22no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—41no—
518C4—41no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—82no—
530C4—81no—
531C4—162no—
532C4#152114yes—
533C2—21no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2C2×#152111yescharacteristic
538C2#1760111yescharacteristic
539C2#1761211yescharacteristic
540C2#1761611yescharacteristic
541C2#2247811yescharacteristic
542C2#2249011yescharacteristic
543C2#2250711yescharacteristic
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. #2571C4semidirect
16. #2568C4semidirect
19. #2514C4semidirect
20. #2509C4semidirect
23. #2365C4semidirect
26. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#361772
#225992
#139212
#99092
#98852
#91532
#91122
#87142
#85722
C4×#3282
#80252