Finite Group Databasebeta

← #205518

Subgroups — #205518

Order 512.

911 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60846C211yesmaximal, characteristic
2#20547C211yesmaximal, characteristic, complemented
3#17806C211yesmaximal, characteristic, complemented
4#17545C211yesmaximal, characteristic, complemented
5#17533C211yesmaximal, characteristic
6#17466C211yesmaximal, characteristic
7C2×#1481C211yesmaximal, characteristic, complemented
8#9754C214yesmaximal, complemented
9#9752C214yesmaximal, 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
27C22×C4⋊C8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1770C2214yes—
30#1767—22no—
31#1752—22no—
32#1574—22no—
33#1573—22no—
34#1564C2214yes—
35#1508—22no—
36#1506—22no—
37#1503C2212yes—
38#1503C2214yes—
39#1502C2212yes—
40#1501—22no—
41#1500—22no—
42#1483—22no—
43#1481—22no—
44#1481C2212yes—
45#1481C2214yes—
46#1480C2212yes—
47#1477—22no—
48C2×Q8⋊C8—22no—
49C23×C8—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yes—
57#533C2×C411yescharacteristic
58#533C2×C411yescharacteristic
59#533C2312yes—
60#533C2311yescharacteristic
61#532C2×C411yescharacteristic
62#532C2×C411yescharacteristic
63#532C2311yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72C2×#125—21no—
73#525—21no—
74#525—21no—
75#525—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C4×#41—21no—
79C4×#41—21no—
80C4×#41—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×#117—21no—
84C2×#117—21no—
85C22×C42C2311yescharacteristic
86#500C2×C412yes—
87#500C2×C412yes—
88#499—24no—
89#498C2×C412yes—
90#498C2×C412yes—
91#488—24no—
92#487C2×C412yes—
93#487C2×C412yes—
94#486C2×C412yes—
95#486C2×C412yes—
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#431C2×C411yescharacteristic
115#431C2×C411yescharacteristic
116#431C2×C412yes—
117#431D412yes—
118#431D412yes—
119#431C2312yes—
120#431C2312yes—
121#427—24no—
122#423—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8D412yes—
128C2×C4⋊C8D412yes—
129C2×C4⋊C8D412yes—
130C2×C4⋊C8C2312yes—
131#416—24no—
132#407—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8D412yes—
135C4×M16—21no—
136C2×#97—24no—
137C2×C4×C8—21no—
138C2×C4×C8C2312yes—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2312yes—
141C4×C22⋊C4—21no—
142C4×C22⋊C4D412yes—
143#376D412yes—
144C43—21no—
145#336—24no—
146#335—24no—
147#334—24no—
148#328—24no—
149#328—28no—
150#327—24no—
151Q8⋊C8—28no—
152Q8⋊C8—24no—
153D4⋊C8—24no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C2×M16—41no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—41no—
167C22×C8—82no—
168C22×C8C2×D414yes—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C22×C412yes—
177C4⋊Q8C22×C411yescharacteristic
178C4⋊Q8C22×C411yescharacteristic
179C4⋊Q8C22×C411yescharacteristic
180#127C22×C411yescharacteristic
181#126—41no—
182#126—41no—
183#125—22no—
184#125—21no—
185#125—21no—
186#125—22no—
187#125—21no—
188#125—41no—
189#125—41no—
190#124—21no—
191#124—21no—
192#124—21no—
193#123—41no—
194#123—41no—
195#123—41no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—22no—
203C22⋊Q8—21no—
204C22⋊Q8—22no—
205C22⋊Q8—21no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210#121—21no—
211#121—21no—
212#121—21no—
213#121—21no—
214#121—21no—
215#121—21no—
216#120—41no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—22no—
226C4×Q8—21no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243#117—22no—
244#117—21no—
245#117—21no—
246#117—22no—
247#117—21no—
248#117—44no—
249#117—41no—
250#117—41no—
251#117—41no—
252#117—41no—
253#117—41no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—41no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42C2411yescharacteristic
280#107—24no—
281#107—24no—
282#106—24no—
283#106—24no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—41no—
290C4⋊C8—41no—
291C4⋊C8C22⋊C412yes—
292C4⋊C8C22⋊C412yes—
293C4⋊C8C22⋊C414yes—
294C4⋊C8C2×D414yes—
295#103—24no—
296#103—24no—
297#102—24no—
298#102—24no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8C22⋊C412yes—
304C22⋊C8C22⋊C412yes—
305C22⋊C8C22⋊C414yes—
306#97—41no—
307#97—84no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—41no—
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—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—81no—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12411yescharacteristic
358C22×C4#12411yescharacteristic
359C22×C4#12711yescharacteristic
360C22×C4#12711yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364M16—82no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—44no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—82no—
372C2×C8—82no—
373C2×C8—82no—
374C2×C8C2×C22⋊C414yes—
375C2×C8C2×C22⋊C414yes—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—41no—
404C22⋊C4—42no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C42—44no—
414C42—41no—
415C42—42no—
416C42—42no—
417C42—42no—
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—22no—
440C23—41no—
441C23—41no—
442C23#39011yescharacteristic
443C23#39011yescharacteristic
444C23C2×#12111yescharacteristic
445Q8—42no—
446Q8—44no—
447Q8—42no—
448Q8—82no—
449Q8—81no—
450Q8—82no—
451Q8—82no—
452D4—42no—
453D4—42no—
454D4—81no—
455D4—82no—
456D4—82no—
457D4—82no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4#39011yescharacteristic
491C2×C4#39011yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4#39011yescharacteristic
494C2×C4#39011yescharacteristic
495C2×C4#39011yescharacteristic
496C2×C4#44414yes—
497C2×C4C22×C22⋊C411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×#12411yescharacteristic
502C2×C4C2×#12711yescharacteristic
503C2×C4#57712yes—
504C2×C4#57812yes—
505C8—24no—
506C8—24no—
507C8—44no—
508C8—82no—
509C8—82no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—44no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22#197512yes—
522C22C2×#39011yescharacteristic
523C22#304612yes—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—84no—
533C4—81no—
534C4—81no—
535C4—82no—
536C4—81no—
537C4—82no—
538C4#197512yes—
539C4#289812yes—
540C4C2×#44412yes—
541C4#304211yescharacteristic
542C4#304411yescharacteristic
543C2—22no—
544C2—22no—
545C2—41no—
546C2—161no—
547C2C2×#197511yescharacteristic
548C2#2124411yescharacteristic
549C2#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. #20547C2semidirect
3. #17806C2semidirect
4. #17545C2semidirect
7. C2×#1481C2semidirect
8. #9754C2semidirect
9. #9752C2semidirect
14. #3231C4semidirect
15. #3230C4semidirect
16. #3229C4semidirect
17. #3228C4semidirect
18. #3227C4semidirect
19. #3226C4semidirect

Maximal subgroups

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

subgroupindex
#608462
#205472
#178062
#175452
#175332
#174662
C2×#14812
#97542
#97522