Finite Group Databasebeta

← #223081

Subgroups — #223081

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3#13921C211yesmaximal, characteristic, complemented
4#9908C211yesmaximal, characteristic, complemented
5#9884C211yesmaximal, characteristic, complemented
6#9157C212yesmaximal
7#9114C212yesmaximal
8#8714C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10C4×#327C211yesmaximal, characteristic
11#8025C212yesmaximal, complemented
12C2×#431C2212yes—
13#2857C2211yescharacteristic
14#2854C2212yes—
15#2571C411yescharacteristic, complemented
16#2563C411yescharacteristic, complemented
17#2532—22no—
18#2513C412yescomplemented
19#2508C412yescomplemented
20#2495—21no—
21#2488—22no—
22#2486—21no—
23#2369C411yescharacteristic, complemented
24#2354C411yescharacteristic, complemented
25#2352—21no—
26#2346—21no—
27C4×#121C2211yescharacteristic
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×#327C2211yescharacteristic
38C2×#327C2211yescharacteristic
39C2×#327C2212yes—
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×#123—22no—
55C2×C22⋊Q8—21no—
56C2×#121C2311yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×#117—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#431—24no—
67#431—42no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×C22⋊C8C2311yescharacteristic
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75#400—22no—
76#400—22no—
77#398—22no—
78#397—22no—
79#397—22no—
80#393—22no—
81#391—21no—
82#391—22no—
83#390C2×C412yes—
84#390C2×C412yes—
85#390C2×C411yescharacteristic
86#389—21no—
87#389—21no—
88#389—21no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C411yescharacteristic
93#388—22no—
94#388—22no—
95#388—22no—
96#388—21no—
97#388—22no—
98#388—22no—
99#387—22no—
100#386—22no—
101#386—21no—
102#384C2×C411yescharacteristic
103#383—22no—
104#383—24no—
105#383—42no—
106#382—21no—
107#382—22no—
108#382C2×C412yes—
109#382C2×C412yes—
110#381—22no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380C2312yes—
116#380C2311yescharacteristic
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#327—22no—
145#327—22no—
146#327—42no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C4⋊Q8—42no—
163#123—82no—
164#123—82no—
165C22⋊Q8—81no—
166C22⋊Q8—81no—
167#121—21no—
168#121—21no—
169#121—21no—
170#121—21no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174#117—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C412yes—
222C2×C42C22×C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C4⋊C8—44no—
229C4⋊C8—42no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—21no—
235C22⋊C8#4112yes—
236C4×C8—44no—
237C4×C8—42no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—44no—
260#95—82no—
261#95—81no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266#95#4112yes—
267C24—21no—
268C24C22×D411yescharacteristic
269C2×Q8—42no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4#13512yes—
338C22×C4#13512yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C42−1+411yescharacteristic
345C2×C8—22no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—41no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—82no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C22⋊C4—44no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—42no—
383C22⋊C4—81no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—41no—
398C42—41no—
399C42—44no—
400C42—42no—
401C42—82no—
402C42—84no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23C22×C22⋊C411yescharacteristic
415C23#54611yescharacteristic
416C23#54711yescharacteristic
417C23C2×#13511yescharacteristic
418C23C2×#13511yescharacteristic
419C23#57611yescharacteristic
420C23#57711yescharacteristic
421Q8—162no—
422D4—81no—
423D4—81no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
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—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—84no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#41012yes—
504C2×C4#41612yes—
505C2×C4#41612yes—
506C2×C4#42112yes—
507C2×C4C4×#4111yescharacteristic
508C2×C4#52011yescharacteristic
509C2×C4#54611yescharacteristic
510C2×C4#54711yescharacteristic
511C8—42no—
512C8—82no—
513C8—81no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22#233911yescharacteristic
534C22C2×#41011yescharacteristic
535C22#288411yescharacteristic
536C22C2×#41611yescharacteristic
537C22#289711yescharacteristic
538C22C2×#42111yescharacteristic
539C22#290111yescharacteristic
540C22#330311yescharacteristic
541C22#330511yescharacteristic
542C22#334011yescharacteristic
543C22#334111yescharacteristic
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4—81no—
558C4—82no—
559C4—82no—
560C4—81no—
561C4—81no—
562C4—162no—
563C4#152114yes—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—161no—
569C2C2×#152111yescharacteristic
570C2#1760211yescharacteristic
571C2#1760711yescharacteristic
572C2#1761611yescharacteristic
573C2#2247811yescharacteristic
574C2#2248911yescharacteristic
575C2#2250711yescharacteristic
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. #22599C2semidirect
3. #13921C2semidirect
4. #9908C2semidirect
5. #9884C2semidirect
8. #8714C2semidirect
11. #8025C2semidirect
15. #2571C4semidirect
16. #2563C4semidirect
18. #2513C4semidirect
19. #2508C4semidirect
23. #2369C4semidirect
24. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#361762
#225992
#139212
#99082
#98842
#91572
#91142
#87142
#85712
C4×#3272
#80252