Finite Group Databasebeta

← #188420

Subgroups — #188420

Order 512.

1028 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34151C211yesmaximal, characteristic
2#21349C211yesmaximal, characteristic, complemented
3#20815C211yesmaximal, characteristic, complemented
4C2×#1494C211yesmaximal, characteristic
5C2×#1482C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8568C212yesmaximal, complemented
8#8566C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7902C211yesmaximal, characteristic, complemented
11#7890C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2374—22no—
18#2373—22no—
19#2370C412yes—
20#2367C412yes—
21#2363C412yes—
22#2362C412yes—
23#2298—21no—
24#2295C2211yescharacteristic
25#2293—21no—
26#2293C2211yescharacteristic
27#2290—21no—
28C42×Q8C2211yescharacteristic
29C42×D4C2211yescharacteristic
30C4×#117—21no—
31C4×#117C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1939—22no—
34#1850—22no—
35#1842C2212yes—
36#1831C2212yes—
37#1798—21no—
38#1798—21no—
39#1768—21no—
40#1768—21no—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1494—24no—
44#1482—24no—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C23×C8—21no—
54#529—22no—
55#529—22no—
56#529D414yes—
57#529Q814yes—
58C2×#126—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#123—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×#121C2×C411yescharacteristic
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C2×#117C2×C412yes—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×#97—21no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98C2×C4×C8C2312yes—
99#391C2×C412yes—
100#390—22no—
101#390C2×C412yes—
102#389—22no—
103#389C2×C412yes—
104#389C2×C412yes—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#386—22no—
116#385—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D412yes—
138#376—21no—
139#376—21no—
140#376D411yescharacteristic
141#376D411yescharacteristic
142C43—21no—
143C43—21no—
144C43C2311yescharacteristic
145C43C2311yescharacteristic
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150Q8⋊C8—24no—
151Q8⋊C8—24no—
152D4⋊C8—24no—
153D4⋊C8—24no—
154C22×Q8C22×C411yescharacteristic
155C22×D4—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—44no—
167C22×C8—42no—
168#126—24no—
169#126—24no—
170#125—22no—
171#125C4⋊C414yes—
172#124—22no—
173#124C4⋊C414yes—
174#123—24no—
175#123—24no—
176C22⋊Q8—22no—
177C22⋊Q8C4⋊C414yes—
178#121—22no—
179#121C4⋊C414yes—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—24no—
183C4×Q8—42no—
184C4×Q8C2×D412yes—
185C4×Q8C2×Q812yes—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—42no—
192C4×D4C2×D412yes—
193C4×D4C2×Q812yes—
194#117—24no—
195#117—24no—
196#117—24no—
197#117—24no—
198#117—42no—
199#117—44no—
200#117—42no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—44no—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—44no—
255C4⋊C8—44no—
256C4⋊C8—44no—
257C4⋊C8—42no—
258C4⋊C8—44no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262#97—42no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—42no—
268C4×C8—42no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281#95C22⋊C412yes—
282C24—21no—
283C24C2×C22⋊C411yescharacteristic
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8C2×C4⋊C412yes—
288C2×D4—24no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—82no—
293C2×D4C2×C4⋊C412yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—82no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C4⋊C412yes—
319C22×C4#11712yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C2×C8—24no—
328C2×C8—42no—
329C2×C8—48no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4C2×C4⋊C412yes—
356C4⋊C4C2×C4⋊C412yes—
357C4⋊C4#13114yes—
358C22⋊C4—24no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—42no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—82no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—42no—
380C42—42no—
381C42—44no—
382C42—44no—
383C42—42no—
384C42—42no—
385C42—44no—
386C42—82no—
387C42—82no—
388C42C2×C4⋊C412yes—
389C42#11712yes—
390C42#12112yes—
391C42#12112yes—
392C42C22⋊Q812yes—
393C42C22⋊Q812yes—
394C42#13514yes—
395C42C22×D411yescharacteristic
396C42C22×Q811yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C23—22no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—22no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23#38012yes—
409C23C22×C22⋊C411yescharacteristic
410Q8—24no—
411Q8—41no—
412Q8—41no—
413D4—24no—
414D4—41no—
415D4—41no—
416D4—84no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—88no—
456C2×C4#38012yes—
457C2×C4#38012yes—
458C2×C4#38012yes—
459C2×C4#41614yes—
460C2×C4C22×C4⋊C411yescharacteristic
461C2×C4C2×#11711yescharacteristic
462C2×C4C2×#12111yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4C2×#13112yes—
467C2×C4#56812yes—
468C2×C4C2×#13512yes—
469C2×C4#57512yes—
470C8—44no—
471C8—82no—
472C8—82no—
473C8—82no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—82no—
483C22—82no—
484C22C2×#38011yescharacteristic
485C22#288012yes—
486C22C2×#41611yescharacteristic
487C22#289611yescharacteristic
488C22#290112yes—
489C4—24no—
490C4—22no—
491C4—22no—
492C4—21no—
493C4—21no—
494C4—22no—
495C4—42no—
496C4—44no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4#290612yes—
504C4#292211yescharacteristic
505C4#292311yescharacteristic
506C4#303512yes—
507C4#307612yes—
508C2—42no—
509C2—82no—
510C2#1750812yes—
511C2#1751612yes—
512C2#2063011yescharacteristic
513C2#2064511yescharacteristic
514C2#2065511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21349C2semidirect
3. #20815C2semidirect
6. #8569C2semidirect
7. #8568C2semidirect
8. #8566C2semidirect
9. #8563C2semidirect
10. #7902C2semidirect
11. #7890C2semidirect

Maximal subgroups

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

subgroupindex
#341512
#213492
#208152
C2×#14942
C2×#14822
#85692
#85682
#85662
#85632
#79022
#78902