Finite Group Databasebeta

← #224432

Subgroups — #224432

Order 512.

887 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic, complemented
3#13936C211yesmaximal, characteristic, complemented
4#9913C212yesmaximal, complemented
5#9908C211yesmaximal, characteristic, complemented
6#9886C212yesmaximal, complemented
7#9884C211yesmaximal, characteristic, complemented
8#8575C212yesmaximal
9#8571C211yesmaximal, characteristic
10#8220C212yesmaximal
11C4×#327C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2851C2212yes—
14#2589—22no—
15#2563C411yescharacteristic, complemented
16#2552C411yescharacteristic, complemented
17#2509C412yescomplemented
18#2497—21no—
19#2487—21no—
20#2454C412yescomplemented
21#2450—22no—
22#2369C411yescharacteristic, complemented
23D4×C4⋊C4—21no—
24D4×C4⋊C4C411yescharacteristic, complemented
25#2345—21no—
26C4×#121C2211yescharacteristic
27C4×#121C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—21no—
30#1916—22no—
31#1797C2211yescharacteristic
32#1797C2212yes—
33C8×C4⋊C4—21no—
34C4×C22⋊C8C2211yescharacteristic
35C4×C22⋊C8C2212yes—
36C2×#327C2211yescharacteristic
37C2×#327C2212yes—
38C2×#327C2211yescharacteristic
39#1443—22no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43#1434C2212yes—
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×#121C2311yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C412yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×C22⋊C8C2311yescharacteristic
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75#400—22no—
76#400—22no—
77#399—22no—
78#398—22no—
79#396—22no—
80#393—22no—
81#391—21no—
82#391—22no—
83#390C2×C412yes—
84#390C2×C411yescharacteristic
85#389—22no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C411yescharacteristic
93#389C2×C412yes—
94#388—21no—
95#388—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#386—21no—
100#386—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#384—21no—
106#384C2×C411yescharacteristic
107#382C2×C412yes—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—22no—
113#380C2311yescharacteristic
114#380C2312yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—22no—
125C4×C22⋊C4—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95D412yes—
133C2×#95C2311yescharacteristic
134C2×#95C2312yes—
135C2×#95C2311yescharacteristic
136#336—21no—
137#336—22no—
138#336—21no—
139#336—22no—
140#327—22no—
141#327—24no—
142#327Q814yescomplemented
143C22×Q8—21no—
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C4⋊Q8—42no—
160#125—82no—
161C22⋊Q8—81no—
162C22⋊Q8—81no—
163#121—21no—
164#121—21no—
165#121C2×Q812yes—
166#121#4112yes—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—82no—
171C4×D4—82no—
172C4×D4—82no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
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×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—82no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C412yes—
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C4⋊C8—42no—
226C4⋊C8—82no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—21no—
232C22⋊C8C2×Q812yes—
233C4×C8—42no—
234C4×C8—82no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—21no—
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#95—82no—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C2×Q812yes—
262C24—21no—
263C24C22×D411yescharacteristic
264C2×Q8—42no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4C4×Q812yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—44no—
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—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4#13512yes—
334C22×C4#13512yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C42−1+411yescharacteristic
340C22×C42−1+411yescharacteristic
341C2×C8—22no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C4⋊C4—21no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4C4×Q812yes—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—81no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C42—22no—
386C42—22no—
387C42—42no—
388C42—42no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—82no—
395C42—84no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C22×C22⋊C411yescharacteristic
408C23#54811yescharacteristic
409C23D4×Q811yescharacteristic
410C23C2×#13511yescharacteristic
411C23C2×#13511yescharacteristic
412C23#57611yescharacteristic
413C23#57711yescharacteristic
414Q8—162no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—84no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—164no—
495C2×C4#41012yes—
496C2×C4#41612yes—
497C2×C4#41612yes—
498C2×C4#42112yes—
499C2×C4C2×C4×Q811yescharacteristic
500C2×C4#51911yescharacteristic
501C2×C4#54811yescharacteristic
502C2×C4D4×Q811yescharacteristic
503C8—42no—
504C8—81no—
505C8—82no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22Q8×C22⋊C411yescharacteristic
526C22C2×#41011yescharacteristic
527C22#288311yescharacteristic
528C22C2×#41611yescharacteristic
529C22#289711yescharacteristic
530C22C2×#42111yescharacteristic
531C22#290111yescharacteristic
532C22#336011yescharacteristic
533C22#336111yescharacteristic
534C22#338011yescharacteristic
535C22#338111yescharacteristic
536C4—22no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4—81no—
551C4—82no—
552C4—82no—
553C4—81no—
554C4—162no—
555C4#152114yes—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2—82no—
561C2C2×#152111yescharacteristic
562C2#1759911yescharacteristic
563C2#1761011yescharacteristic
564C2#1761711yescharacteristic
565C2#2251311yescharacteristic
566C2#2251911yescharacteristic
567C2#2252911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
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
2. #22625C2semidirect
3. #13936C2semidirect
4. #9913C2semidirect
5. #9908C2semidirect
6. #9886C2semidirect
7. #9884C2semidirect
15. #2563C4semidirect
16. #2552C4semidirect
17. #2509C4semidirect
20. #2454C4semidirect
22. #2369C4semidirect
24. D4×C4⋊C4C4semidirect
142. #327Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
#226252
#139362
#99132
#99082
#98862
#98842
#85752
#85712
#82202
C4×#3272