Finite Group Databasebeta

← #224134

Subgroups — #224134

Order 512.

1229 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36544C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3Q8×#95C211yesmaximal, characteristic, complemented
4#9910C212yesmaximal, complemented
5#9908C211yesmaximal, characteristic, complemented
6#9887C212yesmaximal, complemented
7#9884C211yesmaximal, characteristic, complemented
8#8577C212yesmaximal
9#8571C211yesmaximal, characteristic
10#8218C212yesmaximal
11C4×#327C211yesmaximal, characteristic
12C2×#554C411yescharacteristic, complemented
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#446—21no—
16#2857C2211yescharacteristic
17#2851C2212yes—
18#2584C412yescomplemented
19#2580C412yescomplemented
20#2567C411yescharacteristic, complemented
21#2538C411yescharacteristic, complemented
22#2465—22no—
23#2449—22no—
24#2423—21no—
25#2409—21no—
26#2360—21no—
27#2355C411yescharacteristic, complemented
28D4×C4⋊C4—21no—
29C4×#121C2211yescharacteristic
30C4×#121C2212yes—
31#1985—22no—
32#1984—21no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#327C2211yescharacteristic
38C2×#327C2212yes—
39C2×#327C2211yescharacteristic
40C4⋊#95—21no—
41C4⋊#95—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4×#95C2211yescharacteristic
45C4×#95C2212yes—
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#554—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×#121C2311yescharacteristic
62C2×C4×Q8—28no—
63C2×C4×Q8—44no—
64C2×C4×Q8—44no—
65C2×C4×Q8—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D412yes—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C412yes—
75C2×C4×D4C2×C411yescharacteristic
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C42—22no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#446—44no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2311yescharacteristic
91C2×#97—22no—
92C2×C4×C8—21no—
93#399—22no—
94#398—22no—
95#396—22no—
96#395—22no—
97#395—22no—
98#393—22no—
99#391—21no—
100#391—21no—
101#391—21no—
102#390C2×C411yescharacteristic
103#390C2×C412yes—
104#389—22no—
105#389—22no—
106#389—21no—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2×C411yescharacteristic
111#389C2×C412yes—
112#388—22no—
113#388—21no—
114#387—22no—
115#387—22no—
116#387—22no—
117#386—22no—
118#386—21no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#384—21no—
124#384C2×C411yescharacteristic
125#382C2×C412yes—
126#380—21no—
127#380—21no—
128#380C2311yescharacteristic
129#380C2312yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C412yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146#327—22no—
147#327—24no—
148#327Q814yescomplemented
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C411yescharacteristic
156C22×Q8C22⋊C411yescharacteristic
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C4⋊Q8—42no—
173#125—44no—
174C22⋊Q8—48no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177C22⋊Q8—81no—
178C22⋊Q8—81no—
179#121—21no—
180#121—21no—
181#121C2×Q812yes—
182#121#4112yes—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—44no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—44no—
189C4×Q8—432no—
190C4×D4—42no—
191C4×D4—44no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—82no—
195C4×D4—82no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C4⋊C8—42no—
255C4⋊C8—44no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—22no—
259C22⋊C8—21no—
260C22⋊C8—22no—
261C22⋊C8C2×Q812yes—
262#97—44no—
263C4×C8—42no—
264#95—21no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—21no—
279#95—22no—
280#95—42no—
281#95—44no—
282#95—82no—
283#95—81no—
284#95C2×Q812yes—
285C24—21no—
286C24C22×D411yescharacteristic
287C2×Q8—216no—
288C2×Q8—216no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—42no—
297C2×Q8—81no—
298C2×Q8—81no—
299C2×Q8#9914yes—
300C2×Q8#10414yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4C4×Q812yescomplemented
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4#13712yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C42−1+411yescharacteristic
374C22×C42−1+411yescharacteristic
375C2×C8—22no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—42no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—416no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—81no—
408C4⋊C4C4×Q812yescomplemented
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—42no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—81no—
420C22⋊C4—82no—
421C42—22no—
422C42—24no—
423C42—24no—
424C42—22no—
425C42—41no—
426C42—42no—
427C42—41no—
428C42—42no—
429C42—416no—
430C42—48no—
431C42—42no—
432C42—82no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23C22×C22⋊C411yescharacteristic
445C23#54811yescharacteristic
446C23D4×Q811yescharacteristic
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×#13711yescharacteristic
449C23#57611yescharacteristic
450C23#57711yescharacteristic
451Q8—216no—
452Q8—28no—
453Q8—28no—
454Q8—162no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—84no—
532C2×C4—164no—
533C2×C4C2×#9914yes—
534C2×C4C2×#9912yes—
535C2×C4#41712yes—
536C2×C4#41712yes—
537C2×C4C2×#10414yes—
538C2×C4C2×#10412yes—
539C2×C4C2×C4×Q811yescharacteristic
540C2×C4#51911yescharacteristic
541C2×C4#54811yescharacteristic
542C2×C4D4×Q811yescharacteristic
543C8—42no—
544C8—81no—
545C8—82no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22Q8×C22⋊C411yescharacteristic
566C22C22×#9911yescharacteristic
567C22#288411yescharacteristic
568C22C2×#41711yescharacteristic
569C22#289711yescharacteristic
570C22C22×#10411yescharacteristic
571C22#290111yescharacteristic
572C22#336611yescharacteristic
573C22#337211yescharacteristic
574C22#338411yescharacteristic
575C22#338511yescharacteristic
576C4—24no—
577C4—22no—
578C4—24no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—44no—
584C4—42no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—42no—
589C4—84no—
590C4—82no—
591C4—81no—
592C4—82no—
593C4—82no—
594C4—81no—
595C4#150414yes—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—82no—
601C2C2×#150411yescharacteristic
602C2#1755011yescharacteristic
603C2#1761111yescharacteristic
604C2#1762811yescharacteristic
605C2Q8×#9911yescharacteristic
606C2#2252211yescharacteristic
607C2Q8×#10411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. Q8×#95C2semidirect
4. #9910C2semidirect
5. #9908C2semidirect
6. #9887C2semidirect
7. #9884C2semidirect
12. C2×#554C4semidirect
18. #2584C4semidirect
19. #2580C4semidirect
20. #2567C4semidirect
21. #2538C4semidirect
27. #2355C4semidirect
148. #327Q8semidirect
310. C2×D4C4×Q8semidirect
408. C4⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#365442
#225102
Q8×#952
#99102
#99082
#98872
#98842
#85772
#85712
#82182
C4×#3272