Finite Group Databasebeta

← #197516

Subgroups — #197516

Order 512.

1411 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3C2×#1520C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9202C214yesmaximal, complemented
6#9201C214yesmaximal, complemented
7C2×#1289C212yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2855—21no—
12#2848—21no—
13#2627C2212yes—
14#2626C2212yes—
15#2579—22no—
16#2564C414yescomplemented
17#2487C412yescomplemented
18#2417—22no—
19#2409C412yescomplemented
20#2353—24no—
21#2352—24no—
22#2315—21no—
23#2313C2212yes—
24#2312C2212yes—
25#2308—21no—
26#2308—21no—
27#2283—21no—
28#2280—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31#2273—21no—
32#1520—24no—
33#1520C2218yes—
34C2×#328—24no—
35#1459—22no—
36#1459—22no—
37C2×#324C2212yes—
38C2×#323C2212yes—
39#1289C2218yes—
40C23×Q8C2×C411yescharacteristic
41C23×D4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×#117—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4C2311yescharacteristic
73C22×C42D411yescharacteristic
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80#400C2×C414yescomplemented
81#399C2×C414yescomplemented
82#398—24no—
83#396—24no—
84#395C2×C414yescomplemented
85#393C2×C414yescomplemented
86#391—22no—
87#389—22no—
88#388—28no—
89#387—24no—
90#387—28no—
91#386—28no—
92#386—24no—
93#386—22no—
94#386—22no—
95#385—24no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#383—22no—
101#382—22no—
102#381—22no—
103#381—22no—
104#381—22no—
105#380—22no—
106#380—22no—
107#380—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380—24no—
112#380C2×C412yes—
113#380D414yes—
114#380C2312yes—
115#380C2314yes—
116#379—22no—
117#379—22no—
118C4×C4⋊C4—22no—
119C4×C22⋊C4—28no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123#376—22no—
124C2×#95C2×C412yes—
125C2×#95D414yes—
126#328—216no—
127#324—24no—
128#323—24no—
129C25C2×D411yescharacteristic
130C25C2×D411yescharacteristic
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—82no—
135C22×Q8C22×C412yes—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4C22⋊C412yes—
141C22×D4C22⋊C412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C22×C8—24no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—84no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—84no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190#117—84no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—84no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22×C412yes—
233C2×C22⋊C4C2×D412yes—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42C22⋊C412yes—
241C2×C42C22⋊C412yes—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95C22×C414yes—
257#95C2×D418yes—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—81no—
263C24—81no—
264C24—81no—
265C24—81no—
266C24C2×C22⋊C412yes—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269C24#12012yes—
270C24#12312yes—
271C24#13514yes—
272C24#13714yes—
273C24C22×D411yescharacteristic
274C2×Q8—24no—
275C2×Q8—44no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—84no—
279C2×Q8—82no—
280C2×Q8—84no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—48no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—28no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—48no—
317C22×C4—82no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—81no—
323C22×C4—82no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—84no—
331C22×C4—82no—
332C22×C4—84no—
333C22×C4—82no—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4#12311yescharacteristic
350C22×C4#12311yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4112yes—
355C2×C8—44no—
356C2×C8—84no—
357C2×C8—84no—
358C2×C8—84no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—48no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—82no—
372C4⋊C4—84no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—48no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C22⋊C4—84no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—84no—
394C42—84no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—41no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—81no—
415C23—81no—
416C23—84no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23#38612yes—
421C23#38612yes—
422C23#41612yes—
423C23#41612yes—
424C23#41912yes—
425C23#41912yes—
426C23#45214yes—
427C23#45214yes—
428C23#48518yes—
429C23C22×C22⋊C411yescharacteristic
430C23C2×#12011yescharacteristic
431C23C2×#12311yescharacteristic
432C23C2×#13512yes—
433C23C2×#13712yes—
434Q8—84no—
435Q8—82no—
436Q8—82no—
437D4—48no—
438D4—48no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—164no—
487C2×C4—168no—
488C2×C4#38611yescharacteristic
489C2×C4#38611yescharacteristic
490C2×C4#38611yescharacteristic
491C2×C4#38611yescharacteristic
492C2×C4C2×C4×D412yes—
493C2×C4C2×#12112yes—
494C8—84no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—21no—
504C22—22no—
505C22—42no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—48no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—82no—
519C22—84no—
520C22—84no—
521C22#187314yes—
522C22#187314yes—
523C22C2×#38611yescharacteristic
524C22C2×#41611yescharacteristic
525C22C2×#41911yescharacteristic
526C22C2×#45212yes—
527C22C2×#48512yes—
528C4—22no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—84no—
537C4—88no—
538C4—82no—
539C4—82no—
540C4#191312yes—
541C2—21no—
542C2—22no—
543C2—21no—
544C2—22no—
545C2—24no—
546C2—24no—
547C2—82no—
548C2C2×#187311yescharacteristic
549C2#2101211yescharacteristic
550C2C2×#191311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1520C2semidirect
5. #9202C2semidirect
6. #9201C2semidirect
7. C2×#1289C2semidirect
16. #2564C4semidirect
17. #2487C4semidirect
19. #2409C4semidirect
80. #400C2×C4semidirect
81. #399C2×C4semidirect
84. #395C2×C4semidirect
85. #393C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
#341852
C2×#15202
C2×#14592
#92022
#92012
C2×#12892