Finite Group Databasebeta

← #194338

Subgroups — #194338

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic
4C4×#419C211yesmaximal, characteristic, complemented
5#17500C211yesmaximal, characteristic
6#8992C218yesmaximal, complemented
7#8092C211yesmaximal, characteristic, complemented
8#7883C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3103—22no—
12#3103—22no—
13#3103C414yescomplemented
14#3103C2214yes—
15C2×#481—21no—
16C2×#475C411yescharacteristic, complemented
17C2×#475C411yescharacteristic, complemented
18C2×#460C411yescharacteristic, complemented
19C2×#460C411yescharacteristic, complemented
20C2×#448—21no—
21#2984—22no—
22#2984—22no—
23#2983—22no—
24#2983—21no—
25#2982—21no—
26Q8×M16—22no—
27Q8×M16C2214yescomplemented
28C2×C4×SD16—21no—
29C2×C4×SD16—21no—
30#2920—22no—
31#2920C2211yescharacteristic
32#2920C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#419—21no—
35C2×C4×M16C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1980C2214yes—
38#1962—22no—
39#1789—22no—
40#1763—22no—
41C4×#103C2214yescomplemented
42#1736—21no—
43#1492C2214yes—
44#1490—22no—
45#1484—22no—
46#1303C2214yescomplemented
47#1278C2214yescomplemented
48C22×#41—21no—
49C23×Q8D411yescharacteristic
50C22×SD16—21no—
51C22×SD16—21no—
52C22×M16—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532C2×C412yes—
60#532C2312yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#127C2×C411yescharacteristic
63C2×#124—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D412yes—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#481—24no—
88#481—24no—
89#475—22no—
90#475—22no—
91#475C2×C414yes—
92#475C2×C414yes—
93#460—22no—
94#460—22no—
95#460C2×C414yes—
96#460C2×C414yes—
97#448—24no—
98#448—24no—
99#446—28no—
100#446—24no—
101#446—24no—
102#446—28no—
103C8×Q8—24no—
104C8×Q8—24no—
105C4×SD16—24no—
106C4×SD16—24no—
107C4×SD16—24no—
108C4×SD16—24no—
109#432—24no—
110#432—24no—
111#432—21no—
112#432—21no—
113#432—21no—
114#432D412yes—
115#431—24no—
116#431—24no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431—21no—
122#431C2×C412yes—
123#431D412yes—
124#431C2312yes—
125C2×#106—21no—
126C2×#106—21no—
127#423—24no—
128#423D412yes—
129#423C2312yes—
130C2×C4⋊C8C2×C411yescharacteristic
131C2×C4⋊C8C2×C411yescharacteristic
132C2×#104—28no—
133#419—28no—
134C2×#103—21no—
135C2×#103—21no—
136C2×#103—24no—
137C2×#102—21no—
138C2×#102—21no—
139#407—21no—
140#407D412yes—
141C2×C22⋊C8—21no—
142C4×M16—24no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16—21no—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16C2312yes—
149C2×#97—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2312yes—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2312yes—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#376—21no—
157C43—21no—
158#340—24no—
159D4⋊C8—28no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—41no—
163C2×#41—41no—
164C2×#41—84no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—24no—
169C22×Q8—21no—
170C22×Q8—42no—
171C22×Q8C22⋊C411yescharacteristic
172C22×Q8C22⋊C412yes—
173C22×Q8C22⋊C411yescharacteristic
174C22×Q8C2×D412yes—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—41no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—41no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C2×SD16—24no—
186C2×SD16—24no—
187C2×SD16—44no—
188C2×SD16—44no—
189C2×M16—24no—
190C2×M16—22no—
191C2×M16—42no—
192C2×M16—41no—
193C2×M16—41no—
194C2×M16C2×D414yes—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8C22×C412yes—
202#127—21no—
203#127—21no—
204#127—21no—
205#127C22×C412yes—
206#124—22no—
207#124—22no—
208#124—41no—
209#124—41no—
210#121—22no—
211#121—22no—
212#121—22no—
213#121—22no—
214#121—41no—
215#121—41no—
216#121—41no—
217#121—41no—
218C4×Q8—22no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—22no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—22no—
226C4×Q8—21no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8C22⋊C412yes—
235C4×Q8C22⋊C412yes—
236C4×Q8C22⋊C412yes—
237C4×Q8C22⋊C412yes—
238C4×Q8C22×C412yes—
239C4×Q8C22×C412yes—
240C4×Q8C2×D412yes—
241C4×Q8C2×D412yes—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C4×D4—41no—
250#117—24no—
251#117—42no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C2×C42#4111yescharacteristic
295C2×C42C2411yescharacteristic
296#106—24no—
297#106—24no—
298C4⋊C8—28no—
299C4⋊C8—28no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8C22×C412yes—
306C4⋊C8C22×C412yes—
307C4⋊C8C2×D414yes—
308#104—88no—
309#103—216no—
310#103—24no—
311#103—24no—
312#102—24no—
313#102—24no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320#97—28no—
321#97—22no—
322#97—22no—
323#97—41no—
324#97—41no—
325C4×C8—28no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—22no—
332#95—22no—
333#95—41no—
334C24—41no—
335C24#12111yescharacteristic
336#41—48no—
337#41—84no—
338#41—84no—
339#41—84no—
340C2×Q8—22no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—44no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—41no—
356C2×Q8—82no—
357C2×Q8#10414yescomplemented
358C2×Q8C2×C22⋊C412yes—
359C2×Q8C2×C22⋊C412yes—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—84no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—84no—
390C22×C4—81no—
391C22×C4—82no—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4#12011yescharacteristic
399C22×C4#12011yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12311yescharacteristic
406C22×C4#12311yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410SD16—48no—
411SD16—48no—
412M16—28no—
413M16—44no—
414M16—44no—
415M16—44no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4C2×C22⋊C412yes—
444C4⋊C4C2×C22⋊C412yes—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—44no—
459C42—44no—
460C42—82no—
461C42C2×C22⋊C411yescharacteristic
462C42C2×C22⋊C411yescharacteristic
463C42C4×D411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42#12011yescharacteristic
468C42#12011yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12311yescharacteristic
472C42#12311yescharacteristic
473C42C23×C411yescharacteristic
474C42C22×D411yescharacteristic
475C42C22×D411yescharacteristic
476C42C2×#4111yescharacteristic
477C23—22no—
478C23—22no—
479C23—41no—
480C23—42no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23#38611yescharacteristic
485C23#38611yescharacteristic
486C23C2×#12111yescharacteristic
487Q8—24no—
488Q8—22no—
489Q8—22no—
490Q8—22no—
491Q8—44no—
492Q8—44no—
493Q8—42no—
494Q8—44no—
495Q8—82no—
496Q8C2×#10414yes—
497D4—44no—
498D4—42no—
499D4—42no—
500D4—44no—
501D4—82no—
502D4—82no—
503D4—84no—
504D4—84no—
505D4—82no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—84no—
543C2×C4#38611yescharacteristic
544C2×C4#38611yescharacteristic
545C2×C4#38611yescharacteristic
546C2×C4#38611yescharacteristic
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4C2×#10412yes—
550C2×C4#42112yes—
551C2×C4C22×C22⋊C411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×#12011yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12311yescharacteristic
557C2×C4#57714yes—
558C8—44no—
559C8—42no—
560C8—42no—
561C8—44no—
562C8—44no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—164no—
575C22#187112yes—
576C22C2×#38611yescharacteristic
577C22#304312yes—
578C4—21no—
579C4—21no—
580C4—24no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—84no—
591C4—82no—
592C4#187112yes—
593C4C22×#10411yescharacteristic
594C4C2×#42111yescharacteristic
595C4#294912yes—
596C4#301012yes—
597C2—22no—
598C2—22no—
599C2—41no—
600C2—162no—
601C2C2×#187111yescharacteristic
602C2#2093611yescharacteristic
603C2#2093911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60248C2semidirect
4. C4×#419C2semidirect
6. #8992C2semidirect
7. #8092C2semidirect
8. #7883C2semidirect
13. #3103C4semidirect
16. C2×#475C4semidirect
17. C2×#475C4semidirect
18. C2×#460C4semidirect
19. C2×#460C4semidirect
27. Q8×M16C22semidirect
41. C4×#103C22semidirect
46. #1303C22semidirect
47. #1278C22semidirect
357. C2×Q8#104semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
#602482
#218152
C4×#4192
#175002
#89922
#80922
#78832