Finite Group Databasebeta

← #200723

Subgroups — #200723

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3182C211yesmaximal, characteristic
2#60251C211yesmaximal, characteristic, complemented
3#17663C211yesmaximal, characteristic
4#17595C211yesmaximal, characteristic
5#17584C211yesmaximal, characteristic, complemented
6#9418C218yesmaximal, complemented
7#7994C211yesmaximal, characteristic, complemented
8#7973C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3182—22no—
12#3182—22no—
13#3182C414yescomplemented
14#3182C2214yes—
15C2×#500C411yescharacteristic, complemented
16C2×#492C411yescharacteristic, complemented
17C2×#487C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#468—22no—
20#2987C2214yescomplemented
21#2985—22no—
22#2984—22no—
23#2983—22no—
24#2962—22no—
25#2961—22no—
26C2×#440—22no—
27#2924—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2283C2211yescharacteristic
33#1536C2214yes—
34#1533—22no—
35#1525—22no—
36#1519—22no—
37#1518—22no—
38#1515C2214yescomplemented
39#1514C2214yes—
40#1508—22no—
41#1505—22no—
42#1291C2214yescomplemented
43#1288C2214yescomplemented
44C22×#41—21no—
45C22×SD16—22no—
46C22×M16—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533C2×C412yes—
52#533D412yes—
53#533D412yes—
54#533C2312yes—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532C2×C412yes—
62#532C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×#127C2×C411yescharacteristic
67C2×#125—21no—
68C2×#124—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8D412yes—
71C2×#121—21no—
72C2×#121—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8D412yes—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117—21no—
78C22×C42C2311yescharacteristic
79#500—22no—
80#500C2×C414yes—
81#492—22no—
82#492C2×C414yes—
83#487—22no—
84#487C2×C414yes—
85#486—22no—
86#486C2×C414yes—
87#468—28no—
88#468—28no—
89#446—28no—
90#446—24no—
91#446—28no—
92#446—28no—
93C8×Q8—24no—
94#440—28no—
95#440—28no—
96#433—28no—
97#432—24no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432D412yes—
102#432D412yes—
103#432C2312yes—
104#432C2312yes—
105#432C2312yes—
106#431—24no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111C2×#107—22no—
112#424—24no—
113#423—21no—
114#423—21no—
115C2×C4⋊C8—21no—
116C2×#103—22no—
117C2×#102—22no—
118#407—21no—
119#407—21no—
120C2×C22⋊C8D412yes—
121#405—24no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16—21no—
125C4×M16C2×C412yes—
126C4×M16D412yes—
127C4×M16C2312yes—
128C2×#97—21no—
129C2×#97—21no—
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C411yescharacteristic
132#383C2312yes—
133#382—21no—
134#381—21no—
135#380—21no—
136C4×C4⋊C4—21no—
137C4×C22⋊C4—21no—
138#376—21no—
139#329—28no—
140#328—28no—
141#327—28no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—41no—
145C2×#41—41no—
146C2×#41—84no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8C22⋊C412yes—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C2×SD16—28no—
159C2×SD16—48no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—44no—
164C2×M16—41no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C4⋊Q8—21no—
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×C412yes—
178C4⋊Q8C22×C412yes—
179C4⋊Q8C22×C412yes—
180#127—21no—
181#127—21no—
182#127—21no—
183#127C22×C412yes—
184#125—22no—
185#125—22no—
186#125—22no—
187#125—22no—
188#124—22no—
189#124—22no—
190#124—41no—
191#124—41no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8C22⋊C414yes—
199C22⋊Q8C2×D414yes—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—41no—
205#121—41no—
206#121—41no—
207#121—41no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8C22⋊C414yes—
215C4×Q8C2×D414yes—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4C22⋊C412yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—22no—
242C2×C42—41no—
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42C2411yescharacteristic
255#107—28no—
256C4⋊C8—28no—
257C4⋊C8—28no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—41no—
264#103—28no—
265#102—28no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—41no—
271C22⋊C8C2×D414yes—
272#97—28no—
273#97—22no—
274#97—22no—
275#97—22no—
276#97—22no—
277#97—22no—
278#97C2×D414yes—
279C4×C8—28no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—22no—
283C4×C8—41no—
284C4×C8C22×C412yes—
285C4×C8C22×C412yes—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—41no—
290C24—41no—
291C24#12111yescharacteristic
292#41—84no—
293#41—84no—
294#41—84no—
295#41—84no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—44no—
301C2×Q8—41no—
302C2×Q8—82no—
303C2×Q8—82no—
304C2×Q8C2×C22⋊C414yes—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—82no—
315C2×D4—82no—
316C2×D4—84no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—84no—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4#12311yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345SD16—416no—
346M16—44no—
347M16—44no—
348M16—44no—
349M16—82no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—44no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—82no—
374C4⋊C4C2×C22⋊C414yes—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—82no—
382C42—24no—
383C42—24no—
384C42—82no—
385C42C2×C22⋊C411yescharacteristic
386C42C2×C22⋊C411yescharacteristic
387C42C4×D411yescharacteristic
388C42C4×D411yescharacteristic
389C42C4×D411yescharacteristic
390C42C4×D411yescharacteristic
391C42#12011yescharacteristic
392C42#12011yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42#12311yescharacteristic
396C42#12311yescharacteristic
397C42C23×C411yescharacteristic
398C42C22×D411yescharacteristic
399C42C22×D411yescharacteristic
400C42C2×#4111yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23#38611yescharacteristic
409C23#38611yescharacteristic
410C23C2×#12111yescharacteristic
411Q8—44no—
412Q8—44no—
413Q8—82no—
414Q8—82no—
415Q8—82no—
416Q8—82no—
417D4—44no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—84no—
422D4—84no—
423D4—82no—
424D4—82no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#38611yescharacteristic
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4C22×C22⋊C411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×#12011yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C8—24no—
467C8—44no—
468C8—42no—
469C8—44no—
470C8—82no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—164no—
484C22#190312yes—
485C22C2×#38611yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—44no—
497C4—44no—
498C4—82no—
499C4—162no—
500C4#188412yes—
501C4#190512yes—
502C2—22no—
503C2—41no—
504C2—41no—
505C2—162no—
506C2C2×#188411yescharacteristic
507C2C2×#190311yescharacteristic
508C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60251C2semidirect
5. #17584C2semidirect
6. #9418C2semidirect
7. #7994C2semidirect
8. #7973C2semidirect
13. #3182C4semidirect
15. C2×#500C4semidirect
16. C2×#492C4semidirect
17. C2×#487C4semidirect
18. C2×#486C4semidirect
20. #2987C22semidirect
38. #1515C22semidirect
42. #1291C22semidirect
43. #1288C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31822
#602512
#176632
#175952
#175842
#94182
#79942
#79732