Finite Group Databasebeta

← #205512

Subgroups — #205512

Order 512.

943 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60846C211yesmaximal, characteristic
2#20533C211yesmaximal, characteristic, complemented
3#17806C211yesmaximal, characteristic, complemented
4#17546C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6#17464C211yesmaximal, characteristic, complemented
7#17458C211yesmaximal, characteristic
8#9751C214yesmaximal, complemented
9#9749C214yesmaximal, complemented
10#3444C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3233—22no—
13#3232—22no—
14#3231C412yescomplemented
15#3230C411yescharacteristic, complemented
16#3229C411yescharacteristic, complemented
17#3228C412yescomplemented
18#3227C411yescharacteristic, complemented
19#3226C411yescharacteristic, complemented
20#3160C2212yes—
21#3131C2212yes—
22#3091—22no—
23#3076—22no—
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C8×C4⋊C4—22no—
30C8×C4⋊C4C2214yes—
31C4×C22⋊C8—22no—
32#1574—22no—
33#1573—22no—
34#1564C2214yes—
35#1509—22no—
36#1504—22no—
37#1503—22no—
38#1503C2212yes—
39#1503C2214yes—
40#1502C2212yes—
41C2×#328—22no—
42#1485—22no—
43#1482—22no—
44#1481C2212yes—
45#1481C2214yes—
46#1480C2212yes—
47#1479—22no—
48#1476—22no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533C2×C412yes—
56#533C2×C411yescharacteristic
57#533C2×C411yescharacteristic
58#533C2312yes—
59#533C2311yescharacteristic
60#532C2×C411yescharacteristic
61#532C2×C411yescharacteristic
62#532C2311yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71C2×#125—21no—
72#525—21no—
73#525—21no—
74#525—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C4×#41—21no—
78C4×#41—21no—
79C4×#41—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#500C2×C412yescomplemented
87#500C2×C412yescomplemented
88#499—24no—
89#498C2×C412yescomplemented
90#498C2×C412yescomplemented
91#488—24no—
92#487C2×C412yescomplemented
93#487C2×C412yescomplemented
94#486C2×C412yescomplemented
95#486C2×C412yescomplemented
96#485—24no—
97#484—22no—
98#483—22no—
99#482—24no—
100#481—22no—
101#480—22no—
102#479—24no—
103#478—24no—
104#477—22no—
105#476—22no—
106#475—22no—
107#474—22no—
108#432—21no—
109#432—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431C2×C411yescharacteristic
117#431C2×C411yescharacteristic
118#431C2×C412yes—
119#431D412yes—
120#431D412yes—
121#431D412yes—
122#431D412yes—
123#431C2312yes—
124#431C2312yes—
125#427—24no—
126#423—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8D412yes—
129C2×C4⋊C8C2312yes—
130#416—24no—
131#407—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8D412yes—
135C4×M16—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8—24no—
139C2×C4×C8C2312yes—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2312yes—
144C2×#95D412yes—
145#336—24no—
146#335—24no—
147#334—24no—
148#328—28no—
149#328—24no—
150#327—24no—
151Q8⋊C8—24no—
152Q8⋊C8—28no—
153D4⋊C8—24no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C23×C4—22no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×M16—41no—
164C22×C8—22no—
165C22×C8—41no—
166C22×C8C2×D414yes—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C411yescharacteristic
176C4⋊Q8C22×C411yescharacteristic
177C4⋊Q8C22×C411yescharacteristic
178#127C22×C411yescharacteristic
179#126—41no—
180#126—41no—
181#125—22no—
182#125—21no—
183#125—21no—
184#125—22no—
185#125—21no—
186#125—41no—
187#125—41no—
188#124—21no—
189#124—21no—
190#124—21no—
191#123—41no—
192#123—41no—
193#123—41no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—22no—
201C22⋊Q8—21no—
202C22⋊Q8—22no—
203C22⋊Q8—21no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208#121—21no—
209#121—21no—
210#121—21no—
211#121—21no—
212#121—21no—
213#121—21no—
214#120—41no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—22no—
222C4×Q8—21no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241#117—22no—
242#117—21no—
243#117—21no—
244#117—22no—
245#117—21no—
246#117—41no—
247#117—41no—
248#117—41no—
249#117—41no—
250#117—41no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—82no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42C2411yescharacteristic
278#107—24no—
279#107—24no—
280#106—24no—
281#106—24no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—41no—
286C4⋊C8—41no—
287C4⋊C8C22⋊C412yes—
288C4⋊C8C22⋊C412yes—
289C4⋊C8C22⋊C414yes—
290C4⋊C8C2×D414yes—
291#103—24no—
292#103—24no—
293#102—24no—
294#102—24no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8C22⋊C412yes—
302C22⋊C8C22⋊C412yes—
303C22⋊C8C22⋊C414yes—
304#97—41no—
305C4×C8—22no—
306C4×C8—22no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—41no—
310C4×C8—84no—
311C4×C8C22×C412yes—
312C4×C8C22×C412yes—
313C4×C8C2×D414yes—
314#95—22no—
315#95C2×D414yes—
316C24#12111yescharacteristic
317#41—42no—
318#41—42no—
319#41—44no—
320#41—82no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—44no—
329C2×Q8—82no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12411yescharacteristic
361C22×C4#12411yescharacteristic
362C22×C4#12711yescharacteristic
363C22×C4#12711yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367M16—82no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—82no—
376C2×C8C2×C22⋊C414yes—
377C2×C8C2×C22⋊C414yes—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C22⋊C4—42no—
405C22⋊C4—41no—
406C22⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C42—41no—
416C42—42no—
417C42—44no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—82no—
422C42C2×C22⋊C411yescharacteristic
423C42C2×C22⋊C411yescharacteristic
424C42C4×D411yescharacteristic
425C42C4×D411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42#12111yescharacteristic
429C42#12111yescharacteristic
430C42#12411yescharacteristic
431C42#12411yescharacteristic
432C42#12711yescharacteristic
433C42#12711yescharacteristic
434C42C23×C411yescharacteristic
435C42C22×D411yescharacteristic
436C42C22×D411yescharacteristic
437C42C2×#4111yescharacteristic
438C23—22no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23#39011yescharacteristic
444C23#39011yescharacteristic
445C23C2×#12111yescharacteristic
446Q8—42no—
447Q8—44no—
448Q8—42no—
449Q8—82no—
450Q8—81no—
451Q8—82no—
452Q8—82no—
453D4—42no—
454D4—42no—
455D4—81no—
456D4—82no—
457D4—82no—
458D4—81no—
459D4—81no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—28no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—88no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#39011yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4#39011yescharacteristic
494C2×C4#39011yescharacteristic
495C2×C4#39011yescharacteristic
496C2×C4#39011yescharacteristic
497C2×C4#44314yes—
498C2×C4C22×C22⋊C411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×#12111yescharacteristic
502C2×C4C2×#12411yescharacteristic
503C2×C4C2×#12711yescharacteristic
504C2×C4#57712yes—
505C2×C4#57812yes—
506C8—24no—
507C8—24no—
508C8—44no—
509C8—82no—
510C8—82no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—42no—
519C22—42no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22#197412yes—
524C22C2×#39011yescharacteristic
525C22#304211yescharacteristic
526C22#304411yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—48no—
536C4—81no—
537C4—81no—
538C4—82no—
539C4—81no—
540C4—82no—
541C4#197412yes—
542C4#289812yes—
543C4C2×#44312yes—
544C4#304612yes—
545C2—21no—
546C2—22no—
547C2—21no—
548C2—41no—
549C2—161no—
550C2C2×#197411yescharacteristic
551C2#2124411yescharacteristic
552C2#2126211yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#3444C22
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. #20533C2semidirect
3. #17806C2semidirect
5. C2×#1503C2semidirect
6. #17464C2semidirect
8. #9751C2semidirect
9. #9749C2semidirect
14. #3231C4semidirect
15. #3230C4semidirect
16. #3229C4semidirect
17. #3228C4semidirect
18. #3227C4semidirect
19. #3226C4semidirect
86. #500C2×C4semidirect
87. #500C2×C4semidirect
89. #498C2×C4semidirect
90. #498C2×C4semidirect
92. #487C2×C4semidirect
93. #487C2×C4semidirect
94. #486C2×C4semidirect
95. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#608462
#205332
#178062
#175462
C2×#15032
#174642
#174582
#97512
#97492