Finite Group Databasebeta

← #224434

Subgroups — #224434

Order 512.

1023 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic, complemented
3#13933C211yesmaximal, characteristic, complemented
4#9912C212yesmaximal, complemented
5#9909C211yesmaximal, characteristic, complemented
6#9885C212yesmaximal, complemented
7#9885C211yesmaximal, characteristic, complemented
8#8571C211yesmaximal, characteristic
9#8571C212yesmaximal
10#8220C212yesmaximal
11C4×#327C211yesmaximal, characteristic
12C2×#554—21no—
13#2857C2211yescharacteristic
14#2857C2212yes—
15#2564—22no—
16#2556—22no—
17#2551C411yescharacteristic, complemented
18#2542C411yescharacteristic, complemented
19#2506C412yescomplemented
20#2492—21no—
21#2452C412yescomplemented
22#2417—21no—
23#2365C411yescharacteristic, complemented
24#2360—21no—
25D4×C4⋊C4C411yescharacteristic, complemented
26C4×#121C2211yescharacteristic
27C4×#121C2212yes—
28C2×#389—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1939—21no—
31#1916—22no—
32#1797C2211yescharacteristic
33#1797C2212yes—
34C8×C4⋊C4—21no—
35C4×C22⋊C8C2211yescharacteristic
36C4×C22⋊C8C2212yes—
37C2×#327C2211yescharacteristic
38C2×#327C2212yes—
39C2×#327C2211yescharacteristic
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—
53#554—44no—
54#554—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C412yes—
58C2×#125—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—21no—
64C2×#121C2311yescharacteristic
65C2×C4×Q8—21no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C412yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2312yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2311yescharacteristic
82C2×C22⋊C8C2311yescharacteristic
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85#400—22no—
86#399—22no—
87#398—22no—
88#396—22no—
89#395—22no—
90#393—22no—
91#391—21no—
92#391—21no—
93#391—22no—
94#390C2×C411yescharacteristic
95#390C2×C412yes—
96#389—21no—
97#389—22no—
98#389—21no—
99#389—44no—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C411yescharacteristic
104#389C2×C412yes—
105#388—22no—
106#388—21no—
107#387—22no—
108#387—22no—
109#386—21no—
110#386—22no—
111#385—22no—
112#385—22no—
113#384C2×C411yescharacteristic
114#384C2×C412yes—
115#380—21no—
116#380—22no—
117#380—21no—
118#380C2312yes—
119#380C2311yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95D412yes—
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139#336—21no—
140#336—22no—
141#336—21no—
142#336—22no—
143#327—22no—
144#327—24no—
145#327Q814yescomplemented
146C22×Q8—21no—
147C22×Q8—22no—
148C22×D4—21no—
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C4⋊Q8—42no—
164C4⋊Q8—44no—
165C4⋊Q8—82no—
166#125—48no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169C22⋊Q8—42no—
170C22⋊Q8—44no—
171C22⋊Q8—42no—
172C22⋊Q8—48no—
173C22⋊Q8—44no—
174#121—21no—
175#121—21no—
176#121C2×Q812yes—
177#121#4112yes—
178C4×Q8—44no—
179C4×D4—42no—
180C4×D4—44no—
181C4×D4—42no—
182C4×D4—44no—
183C4×D4—42no—
184C4×D4—82no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4D814yes—
211C2×C4⋊C4SD1614yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C412yes—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—82no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C4⋊C8—42no—
242C4⋊C8—82no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—21no—
248C22⋊C8C2×Q812yes—
249C4×C8—42no—
250C4×C8—82no—
251#95—21no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95—42no—
266#95—44no—
267#95—82no—
268#95—81no—
269#95—82no—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C2×Q812yes—
275C24—21no—
276C24C22×D411yescharacteristic
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4C4×Q812yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×D812yes—
348C22×C4C2×D814yes—
349C22×C4C2×SD1612yes—
350C22×C4C2×SD1614yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—81no—
384C4⋊C4—82no—
385C4⋊C4—84no—
386C4⋊C4—88no—
387C4⋊C4#10214yes—
388C4⋊C4#10218yes—
389C4⋊C4#10214yes—
390C4⋊C4C4×Q812yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—42no—
394C22⋊C4—44no—
395C22⋊C4—42no—
396C22⋊C4—41no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—81no—
402C22⋊C4—82no—
403C42—22no—
404C42—22no—
405C42—42no—
406C42—42no—
407C42—41no—
408C42—41no—
409C42—42no—
410C42—42no—
411C42—44no—
412C42—42no—
413C42—84no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23C22×C22⋊C411yescharacteristic
426C23#54811yescharacteristic
427C23D4×Q811yescharacteristic
428C23C22×D811yescharacteristic
429C23C22×SD1611yescharacteristic
430C23#57711yescharacteristic
431C23#57811yescharacteristic
432Q8—84no—
433Q8—164no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—84no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—164no—
512C2×C4#41012yes—
513C2×C4C2×#10214yes—
514C2×C4C2×#10212yes—
515C2×C4C2×#10214yes—
516C2×C4C2×#10212yes—
517C2×C4#42112yes—
518C2×C4C2×C4×Q811yescharacteristic
519C2×C4#51911yescharacteristic
520C2×C4#54811yescharacteristic
521C2×C4D4×Q811yescharacteristic
522C8—42no—
523C8—81no—
524C8—82no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22Q8×C22⋊C411yescharacteristic
545C22C2×#41011yescharacteristic
546C22#288311yescharacteristic
547C22C22×#10211yescharacteristic
548C22#289811yescharacteristic
549C22C2×#42111yescharacteristic
550C22#290111yescharacteristic
551C22#335811yescharacteristic
552C22#335911yescharacteristic
553C22Q8×D811yescharacteristic
554C22Q8×SD1611yescharacteristic
555C4—22no—
556C4—24no—
557C4—24no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4—82no—
571C4—82no—
572C4—81no—
573C4—162no—
574C4#151714yes—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—82no—
580C2C2×#151711yescharacteristic
581C2#1760011yescharacteristic
582C2#1761211yescharacteristic
583C2#1761311yescharacteristic
584C2#2251311yescharacteristic
585C2Q8×#10211yescharacteristic
586C2#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. #13933C2semidirect
4. #9912C2semidirect
5. #9909C2semidirect
6. #9885C2semidirect
7. #9885C2semidirect
17. #2551C4semidirect
18. #2542C4semidirect
19. #2506C4semidirect
21. #2452C4semidirect
23. #2365C4semidirect
25. D4×C4⋊C4C4semidirect
145. #327Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
#226252
#139332
#99122
#99092
#98852
#85712
#82202
C4×#3272