Finite Group Databasebeta

← #224140

Subgroups — #224140

Order 512.

1241 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#121C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#13817C211yesmaximal, characteristic, complemented
4#9909C211yesmaximal, characteristic, complemented
5#9908C212yesmaximal, complemented
6#9888C212yesmaximal, complemented
7#9885C211yesmaximal, characteristic, complemented
8#8573C212yesmaximal
9#8571C211yesmaximal, characteristic
10#8218C212yesmaximal
11C4×#327C211yesmaximal, characteristic
12C2×D4×Q8—21no—
13C2×D4×Q8C411yescharacteristic, complemented
14C22×C4×Q8C2211yescharacteristic
15C2×#446—21no—
16#2857C2211yescharacteristic
17#2857C2212yes—
18#2546C411yescharacteristic, complemented
19#2546C412yescomplemented
20#2538C411yescharacteristic, complemented
21#2538C412yescomplemented
22#2429—21no—
23#2429—22no—
24#2417—21no—
25#2417—22no—
26Q8×C4⋊C4C411yescharacteristic, complemented
27Q8×C22⋊C4—21no—
28C4×#121C2211yescharacteristic
29C4×#121C2212yes—
30C2×#391—21no—
31C2×C4×C4⋊C4—21no—
32#1985—22no—
33#1984—21no—
34#1797C2211yescharacteristic
35#1797C2212yes—
36#1749C2212yes—
37C4×C22⋊C8C2211yescharacteristic
38C2×#327C2211yescharacteristic
39C2×#327C2212yes—
40C2×#327C2211yescharacteristic
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43#1434C2212yes—
44C4×#95C2211yescharacteristic
45C4×#95C2212yes—
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8—21no—
54C23×Q8D411yescharacteristic
55D4×Q8—44no—
56D4×Q8—44no—
57D4×Q8—44no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—21no—
68C2×#121C2311yescharacteristic
69C2×C4×Q8—21no—
70C2×C4×Q8—44no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D412yes—
77C2×C4×D4—22no—
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C412yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D412yes—
84C22×C42—21no—
85C22×C42—22no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#446—44no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94C2×#97—22no—
95C2×C4×C8—21no—
96#398—22no—
97#398—22no—
98#398—22no—
99#396—22no—
100#396—22no—
101#396—22no—
102#391—28no—
103#391—21no—
104#391—22no—
105#391—21no—
106#391—22no—
107#390C2×C411yescharacteristic
108#390C2×C412yes—
109#389—22no—
110#389—21no—
111#389—22no—
112#389—21no—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C412yes—
116#389C2×C411yescharacteristic
117#389C2×C412yes—
118#388—22no—
119#388—21no—
120#386—22no—
121#386—21no—
122#384C2×C411yescharacteristic
123#384C2×C412yes—
124#380—22no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380C2311yescharacteristic
129#380C2312yes—
130C4×C4⋊C4—44no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C412yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143#336—22no—
144#336—21no—
145#336—22no—
146#336—21no—
147#327—22no—
148#327—24no—
149#327Q814yescomplemented
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×Q8—44no—
159C22×Q8C22⋊C411yescharacteristic
160C22×Q8C22⋊C412yes—
161C22×Q8C22⋊C411yescharacteristic
162C22×D4—21no—
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C4⋊Q8—42no—
178C4⋊Q8—44no—
179C4⋊Q8—48no—
180C4⋊Q8—44no—
181C22⋊Q8—48no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—48no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—44no—
189#121—21no—
190#121—21no—
191#121C2×Q812yes—
192#121#4112yes—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—44no—
200C4×D4—42no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—42no—
205C4×D4—42no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—82no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C412yes—
255C2×C42C22×C412yes—
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C4⋊C8—42no—
262C4⋊C8—44no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—22no—
266C22⋊C8—21no—
267C22⋊C8—22no—
268C22⋊C8C2×Q812yescomplemented
269#97—44no—
270C4×C8—42no—
271#95—28no—
272#95—28no—
273#95—21no—
274#95—22no—
275#95—21no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—21no—
281#95—22no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—21no—
286#95—22no—
287#95—42no—
288#95—44no—
289#95—81no—
290#95C2×Q812yescomplemented
291C24—21no—
292C24C22×D411yescharacteristic
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—42no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8—816no—
310C2×Q8#10414yes—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4C4×Q812yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4AGL(1,ℤ/8)12yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×Q811yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C42−1+411yescharacteristic
384C22×C42−1+411yescharacteristic
385C2×C8—22no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—42no—
394C4⋊C4—21no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—82no—
416C4⋊C4—81no—
417C4⋊C4#9914yes—
418C4⋊C4C4×Q812yes—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—42no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—82no—
430C22⋊C4—81no—
431C42—22no—
432C42—24no—
433C42—24no—
434C42—22no—
435C42—41no—
436C42—42no—
437C42—42no—
438C42—41no—
439C42—48no—
440C42—44no—
441C42—42no—
442C42—88no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23C22×C22⋊C411yescharacteristic
455C23#54811yescharacteristic
456C23D4×Q811yescharacteristic
457C23C2×AGL(1,ℤ/8)11yescharacteristic
458C23C2×AGL(1,ℤ/8)11yescharacteristic
459C23#57711yescharacteristic
460C23#57811yescharacteristic
461Q8—28no—
462Q8—48no—
463Q8—44no—
464Q8—88no—
465Q8—84no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—88no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4C2×#9914yes—
545C2×C4C2×#9912yes—
546C2×C4#41812yes—
547C2×C4#41812yes—
548C2×C4C2×#10414yes—
549C2×C4C2×#10412yes—
550C2×C4C2×C4×Q811yescharacteristic
551C2×C4#51911yescharacteristic
552C2×C4#54811yescharacteristic
553C2×C4D4×Q811yescharacteristic
554C8—42no—
555C8—81no—
556C8—82no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22Q8×C22⋊C411yescharacteristic
577C22C22×#9911yescharacteristic
578C22#288411yescharacteristic
579C22C2×#41811yescharacteristic
580C22#289811yescharacteristic
581C22C22×#10411yescharacteristic
582C22#290111yescharacteristic
583C22#336711yescharacteristic
584C22#337311yescharacteristic
585C22#338411yescharacteristic
586C22#338611yescharacteristic
587C4—22no—
588C4—24no—
589C4—24no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—84no—
601C4—82no—
602C4—82no—
603C4—81no—
604C4—82no—
605C4—81no—
606C4#150714yes—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—82no—
612C2C2×#150711yescharacteristic
613C2#1754911yescharacteristic
614C2#1760811yescharacteristic
615C2#1763111yescharacteristic
616C2Q8×#9911yescharacteristic
617C2#2252311yescharacteristic
618C2Q8×#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. #13817C2semidirect
4. #9909C2semidirect
5. #9908C2semidirect
6. #9888C2semidirect
7. #9885C2semidirect
13. C2×D4×Q8C4semidirect
18. #2546C4semidirect
19. #2546C4semidirect
20. #2538C4semidirect
21. #2538C4semidirect
26. Q8×C4⋊C4C4semidirect
149. #327Q8semidirect
268. C22⋊C8C2×Q8semidirect
290. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1212
#225102
#138172
#99092
#99082
#98882
#98852
#85732
#85712
#82182
C4×#3272