Finite Group Databasebeta

← #194143

Subgroups — #194143

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3038C211yesmaximal, characteristic
2C4⋊C4×#41C211yesmaximal, characteristic, complemented
3#21803C211yesmaximal, characteristic, complemented
4C2×#1982C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic, complemented
6#8979C214yesmaximal, complemented
7#8977C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×AGL(1,ℤ/8)—21no—
11C22×#127C2211yescharacteristic
12C2×C4×#41—21no—
13C2×C4×#41C2211yescharacteristic
14#3038—24no—
15#3038C418yescomplemented
16C2×#460—21no—
17C2×#460C2212yes—
18C2×#459—21no—
19C2×#459C2212yes—
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#418—21no—
24C22×#102—22no—
25#2351—21no—
26Q8×C4⋊C4C2212yescomplemented
27#2349—22no—
28#2349—21no—
29D4×C4⋊C4—22no—
30D4×C4⋊C4C2212yescomplemented
31#2296—22no—
32#2296—21no—
33#2285—21no—
34#2285C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1982—21no—
38#1982—21no—
39#1982—22no—
40#1982C2214yes—
41#1957C2214yes—
42#1786—22no—
43#1477—24no—
44C2×Q8⋊C8C2212yescomplemented
45C2×D4⋊C8C2212yescomplemented
46#1296C2214yescomplemented
47#1279C2214yescomplemented
48C22×#41D411yescharacteristic
49C23×D4—21no—
50C23×D4—21no—
51C23×D4—21no—
52C2×AGL(1,ℤ/8)—28no—
53C2×AGL(1,ℤ/8)—44no—
54C2×AGL(1,ℤ/8)—44no—
55C22×SD16—22no—
56C22×D8—22no—
57C22×M16—21no—
58C23×C8—21no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—21no—
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—22no—
65C2×#127C2×C414yescomplemented
66C2×#127C2312yes—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41—44no—
70C4×#41—44no—
71C4×#41C2×C414yes—
72C4×#41D414yes—
73C2×C4×Q8—22no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—24no—
79C2×C4×D4—22no—
80C2×C4×D4D411yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×#117—22no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C22×C4⋊C4—22no—
87C22×C4⋊C4D412yes—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#460—28no—
93#460—24no—
94#460C2×C418yescomplemented
95#459—28no—
96#459—24no—
97#459C2×C418yescomplemented
98#431—22no—
99#431—22no—
100#423—22no—
101#423C2×C414yescomplemented
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2312yes—
109#418—28no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—24no—
113C2×#102—24no—
114C2×#102—28no—
115C2×#102—48no—
116C2×C22⋊C8—21no—
117C2×C4×C8C2312yes—
118#389—24no—
119#389—22no—
120#389—22no—
121#389—24no—
122#385—24no—
123#385—24no—
124#385—24no—
125#384—22no—
126#384C2312yes—
127#382—22no—
128#382—21no—
129#380—22no—
130#380—21no—
131#380—22no—
132#380D412yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2312yes—
144C2×#95—22no—
145C2×#95—21no—
146Q8⋊C8—24no—
147D4⋊C8—24no—
148C25—41no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41C22⋊C414yes—
153C2×#41D814yes—
154C2×#41SD1614yes—
155C22×Q8C2×D411yescharacteristic
156C22×D4—21no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—84no—
176C22×D4C2×D411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185AGL(1,ℤ/8)—432no—
186C2×SD16—28no—
187C2×SD16—44no—
188C2×SD16—44no—
189C2×D8—28no—
190C2×D8—44no—
191C2×D8—44no—
192C2×M16—24no—
193C2×M16—42no—
194C2×M16—42no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—44no—
199C22×C8—42no—
200#127—24no—
201#127—24no—
202#127—24no—
203#127—22no—
204#127—22no—
205#127—22no—
206#127—42no—
207#127—42no—
208#127—42no—
209#127C22×C414yes—
210C4×Q8—22no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—44no—
214C4×Q8—44no—
215C4×Q8C22⋊C414yes—
216C4×Q8C22×C412yes—
217C4×Q8C2×D412yes—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—416no—
226C4×D4—44no—
227C4×D4—44no—
228C4×D4C22⋊C414yes—
229C4×D4C22×C412yes—
230C4×D4C2×D412yes—
231#117—22no—
232#117—24no—
233#117—24no—
234#117—48no—
235#117—42no—
236#117—42no—
237#117C22⋊C414yes—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—48no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C22×C412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4112yes—
279C2×C42#4112yes—
280C2×C42C2411yescharacteristic
281C4⋊C8—24no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C4⋊C8C22×C414yes—
288#102—28no—
289#102—28no—
290#102—48no—
291#102—48no—
292#102—88no—
293C22⋊C8—42no—
294C4×C8—22no—
295C4×C8—22no—
296#95—24no—
297#95—24no—
298#95—21no—
299#95—22no—
300#95—21no—
301#95—22no—
302#95—22no—
303C24—21no—
304C24—42no—
305C24—44no—
306C24—84no—
307C24—84no—
308C24—82no—
309C24—82no—
310C24#12011yescharacteristic
311#41—28no—
312#41—28no—
313#41—28no—
314#41#102116yescomplemented
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8C2×C22⋊C412yes—
319C2×Q8C2×D812yes—
320C2×Q8C2×SD1612yes—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—48no—
331C2×D4—416no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—82no—
353C2×D4—82no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4C2×C22⋊C412yes—
357C2×D4C2×D812yes—
358C2×D4C2×SD1612yes—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—48no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—84no—
383C22×C4C2×C22⋊C412yes—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4#12011yescharacteristic
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12112yes—
390C22×C4#12112yes—
391C22×C4#12312yes—
392C22×C4C2×D812yes—
393C22×C4C2×SD1612yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4#13712yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399SD16—416no—
400D8—416no—
401M16—48no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—44no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—44no—
409C2×C8—84no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—84no—
427C4⋊C4C2×C22⋊C412yes—
428C4⋊C4C2×C22⋊C412yes—
429C4⋊C4AGL(1,ℤ/8)14yescomplemented
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C42—22no—
437C42—22no—
438C42—24no—
439C42—24no—
440C42—48no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42—42no—
446C42C2×C22⋊C412yes—
447C42C4×D412yes—
448C42C4×D412yes—
449C42#12112yes—
450C42#12112yes—
451C42#12312yes—
452C42C23×C411yescharacteristic
453C42C22×D411yescharacteristic
454C42C2×#4111yescharacteristic
455C42C2×#4111yescharacteristic
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—88no—
465C23—88no—
466C23—88no—
467C23—82no—
468C23—82no—
469C23—84no—
470C23—84no—
471C23—168no—
472C23—168no—
473C23#38612yes—
474C23#44712yes—
475C23#44812yes—
476C23C2×#12011yescharacteristic
477Q8—22no—
478Q8—22no—
479Q8—22no—
480Q8C2×#10214yescomplemented
481D4—216no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—48no—
486D4—44no—
487D4—44no—
488D4—48no—
489D4—88no—
490D4—84no—
491D4—88no—
492D4—82no—
493D4—82no—
494D4—88no—
495D4—82no—
496D4—82no—
497D4C2×#10214yescomplemented
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—216no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—84no—
527C2×C4—88no—
528C2×C4—816no—
529C2×C4#38612yes—
530C2×C4#38612yes—
531C2×C4#38612yes—
532C2×C4C2×#10214yes—
533C2×C4#41714yes—
534C2×C4#44712yes—
535C2×C4#44812yes—
536C2×C4#48014yes—
537C2×C4#48114yes—
538C2×C4C22×C22⋊C411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12111yescharacteristic
543C2×C4C2×#12311yescharacteristic
544C2×C4C22×D811yescharacteristic
545C2×C4C22×SD1611yescharacteristic
546C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
547C2×C4C2×AGL(1,ℤ/8)12yes—
548C2×C4C2×#13711yescharacteristic
549C2×C4#57512yes—
550C8—44no—
551C8—82no—
552C8—82no—
553C22—24no—
554C22—24no—
555C22—22no—
556C22—22no—
557C22—24no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—1616no—
566C22—164no—
567C22—164no—
568C22#187414yes—
569C22C2×#38611yescharacteristic
570C22C2×#44711yescharacteristic
571C22C2×#44811yescharacteristic
572C22#300612yes—
573C22#302112yes—
574C4—24no—
575C4—22no—
576C4—21no—
577C4—22no—
578C4—21no—
579C4—44no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4—84no—
584C4#187414yes—
585C4C22×#10211yescharacteristic
586C4C2×#41711yescharacteristic
587C4C4×AGL(1,ℤ/8)12yes—
588C4#303811yescharacteristic
589C4#304011yescharacteristic
590C4C2×#48011yescharacteristic
591C4C2×#48111yescharacteristic
592C2—24no—
593C2—44no—
594C2—164no—
595C2#1878612yes—
596C2#1878812yes—
597C2C2×#187411yescharacteristic
598C2#2092811yescharacteristic
599C2#2094511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3038C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×#41C2semidirect
3. #21803C2semidirect
4. C2×#1982C2semidirect
5. C2×#1477C2semidirect
6. #8979C2semidirect
7. #8977C2semidirect
8. #8045C2semidirect
9. #7901C2semidirect
15. #3038C4semidirect
26. Q8×C4⋊C4C22semidirect
30. D4×C4⋊C4C22semidirect
44. C2×Q8⋊C8C22semidirect
45. C2×D4⋊C8C22semidirect
46. #1296C22semidirect
47. #1279C22semidirect
65. C2×#127C2×C4semidirect
94. #460C2×C4semidirect
97. #459C2×C4semidirect
101. #423C2×C4semidirect
314. #41#102semidirect
429. C4⋊C4AGL(1,ℤ/8)semidirect
480. Q8C2×#102semidirect
497. D4C2×#102semidirect

Maximal subgroups

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

subgroupindex
C2×#30382
C4⋊C4×#412
#218032
C2×#19822
C2×#14772
#89792
#89772
#80452
#79012