Finite Group Databasebeta

← #189407

Subgroups — #189407

Order 512.

819 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60214C211yesmaximal, characteristic
2#20827C211yesmaximal, characteristic, complemented
3#17797C211yesmaximal, characteristic, complemented
4#17670C211yesmaximal, characteristic
5#17669C211yesmaximal, characteristic
6#8693C212yesmaximal, complemented
7#8692C212yesmaximal, complemented
8#8691C212yesmaximal, complemented
9#8690C212yesmaximal, complemented
10#7998C211yesmaximal, characteristic, complemented
11#7996C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2969C412yes—
14#2969C412yes—
15#2968—21no—
16#2968—21no—
17#2967—22no—
18#2966C2212yes—
19#2966C2212yes—
20Q8×M16C412yes—
21D4×M16—22no—
22D4×M16C412yes—
23#2921—21no—
24#2920C2211yescharacteristic
25#2920C2211yescharacteristic
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29#2873—21no—
30C2×C4×M16C2211yescharacteristic
31#1849—21no—
32#1838—21no—
33#1811—22no—
34C4×#108C2212yescomplemented
35C4×#108C2212yescomplemented
36#1569—21no—
37#1569—21no—
38#1568C2212yescomplemented
39#1568C2212yescomplemented
40#1565—22no—
41#1535—22no—
42#1534—21no—
43#1533—21no—
44#1532—22no—
45#1529C2212yes—
46#1529C2212yes—
47#1529C2212yes—
48#1529C2212yes—
49#1528—21no—
50#1527—21no—
51#1291C2212yescomplemented
52#1291C2212yescomplemented
53#1291C2211yescharacteristic, complemented
54#1291C2211yescharacteristic, complemented
55#1291C2211yescharacteristic, complemented
56#1291C2211yescharacteristic, complemented
57C22×M16—22no—
58C22×M16D412yes—
59#529—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529C2×C412yes—
64#529D412yes—
65#529Q812yes—
66#529C2312yes—
67C2×#126—21no—
68C2×#125D411yescharacteristic
69C2×#124D411yescharacteristic
70C2×#123—21no—
71C2×C22⋊Q8Q811yescharacteristic
72C2×#121Q811yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C22×C42C2311yescharacteristic
78#446—22no—
79#446—22no—
80#446C2×C412yes—
81#446C2×C412yes—
82C8×Q8C2×C412yes—
83C8×Q8C2×C412yes—
84#436—22no—
85#436—22no—
86#436C2×C412yes—
87#436C2×C412yes—
88#435—22no—
89#435—22no—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94C8×D4—22no—
95C8×D4—22no—
96C8×D4C2×C412yes—
97C8×D4C2×C412yes—
98#433—22no—
99#433—22no—
100#433—22no—
101#433—22no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432—21no—
111#432C2311yescharacteristic
112#432C2311yescharacteristic
113#432C2312yes—
114#432C2311yescharacteristic
115#432C2311yescharacteristic
116#431—21no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431—21no—
122#431—21no—
123#431C2×C412yes—
124C2×#108—22no—
125C2×#108—22no—
126#423—22no—
127#423—21no—
128#423—21no—
129#423—21no—
130#423C2×C412yes—
131C2×C4⋊C8—21no—
132#407—21no—
133#407—22no—
134#407—21no—
135#407—21no—
136#407—21no—
137C2×C22⋊C8—21no—
138#405—22no—
139#405—22no—
140#405—22no—
141#405—22no—
142C4×M16—22no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16C2×C412yes—
146C4×M16C2×C411yescharacteristic
147C4×M16C2×C411yescharacteristic
148C4×M16D412yes—
149C4×M16D411yescharacteristic
150C4×M16D411yescharacteristic
151C4×M16C2312yes—
152C4×M16C2311yescharacteristic
153C4×M16C2311yescharacteristic
154C2×#97—21no—
155C2×#97D411yescharacteristic
156C2×#97D411yescharacteristic
157C2×C4×C8C2312yes—
158#341—22no—
159#341—22no—
160#335—22no—
161#335—22no—
162#334—22no—
163#334—22no—
164#330—22no—
165#330—22no—
166#330—22no—
167#330—22no—
168#329—22no—
169#329—22no—
170#329—22no—
171#329—22no—
172C22×Q8C4⋊C411yescharacteristic
173C22×D4—21no—
174C22×D4C4⋊C411yescharacteristic
175C23×C4—21no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×Q811yescharacteristic
178C23×C4#4111yescharacteristic
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—44no—
186C2×M16—48no—
187C2×M16—41no—
188C2×M16C22⋊C412yes—
189C2×M16C22⋊C412yes—
190C2×M16C22⋊C412yes—
191C2×M16C22⋊C412yes—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—41no—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197#126—22no—
198#126—22no—
199#126—22no—
200#126—22no—
201#125—21no—
202#125—21no—
203#125C4⋊C412yes—
204#125C2×D412yes—
205#124—21no—
206#124—21no—
207#124C4⋊C412yes—
208#124C2×D412yes—
209#123—22no—
210#123—22no—
211#123—22no—
212#123—22no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8C4⋊C412yes—
216C22⋊Q8C2×Q812yes—
217#121—21no—
218#121—21no—
219#121C4⋊C412yes—
220#121C2×Q812yes—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8C4⋊C412yes—
224C4×Q8C22×C412yes—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4C4⋊C412yes—
232C4×D4C22×C412yes—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C4⋊C411yescharacteristic
242C2×C4⋊C4C4⋊C411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C42—21no—
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×Q811yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42C2411yescharacteristic
261#108—44no—
262#108—44no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—22no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C22×C412yes—
278C4⋊C8C22×C412yes—
279C22⋊C8—22no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—22no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295#97—22no—
296#97—22no—
297#97—21no—
298#97—21no—
299#97—21no—
300#97—21no—
301#97—22no—
302#97—21no—
303#97—21no—
304#97—22no—
305#97—41no—
306#97C22⋊C411yescharacteristic
307#97C22⋊C411yescharacteristic
308#97C22⋊C411yescharacteristic
309#97C22⋊C411yescharacteristic
310#97C22⋊C412yes—
311#97C22⋊C412yes—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—41no—
319C4×C8C22×C412yes—
320C4×C8C22×C412yes—
321C4×C8C2×D412yes—
322C4×C8C2×D412yes—
323C24—21no—
324C24C22⋊Q811yescharacteristic
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8C2×C4⋊C412yes—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—81no—
332C2×D4—82no—
333C2×D4C2×C4⋊C412yes—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C4⋊C411yescharacteristic
346C22×C4C2×C4⋊C411yescharacteristic
347C22×C4#11711yescharacteristic
348C22×C4#11711yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4C22⋊Q811yescharacteristic
355C22×C4C22⋊Q811yescharacteristic
356C22×C4C22⋊Q811yescharacteristic
357C22×C4C22⋊Q811yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×Q811yescharacteristic
360C22×C4C2×#4111yescharacteristic
361M16—24no—
362M16—24no—
363M16—42no—
364M16—42no—
365M16—44no—
366M16—44no—
367M16—44no—
368M16—42no—
369M16—42no—
370M16—42no—
371M16—42no—
372M16—82no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—42no—
377C2×C8—44no—
378C2×C8—44no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—82no—
386C2×C8C2×C22⋊C412yes—
387C2×C8C2×C22⋊C412yes—
388C2×C8C2×C22⋊C412yes—
389C2×C8C2×C22⋊C412yes—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4C2×C4⋊C412yes—
401C4⋊C4C2×C4⋊C412yes—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C42—22no—
413C42—22no—
414C42C2×C22⋊C411yescharacteristic
415C42C2×C22⋊C411yescharacteristic
416C42C2×C4⋊C411yescharacteristic
417C42C2×C4⋊C411yescharacteristic
418C42#11711yescharacteristic
419C42#11711yescharacteristic
420C42#12111yescharacteristic
421C42#12111yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42C22⋊Q811yescharacteristic
425C42C22⋊Q811yescharacteristic
426C42C23×C411yescharacteristic
427C42C22×D411yescharacteristic
428C42C22×D411yescharacteristic
429C42C2×#4111yescharacteristic
430C23—22no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—81no—
436C23—81no—
437C23#38011yescharacteristic
438C23#38011yescharacteristic
439C23C2×C22⋊Q811yescharacteristic
440Q8—42no—
441Q8—81no—
442D4—42no—
443D4—44no—
444D4—81no—
445D4—82no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4#38011yescharacteristic
474C2×C4#38011yescharacteristic
475C2×C4#38011yescharacteristic
476C2×C4#38011yescharacteristic
477C2×C4#38011yescharacteristic
478C2×C4#38011yescharacteristic
479C2×C4C22×C22⋊C411yescharacteristic
480C2×C4C22×C4⋊C411yescharacteristic
481C2×C4C2×#11711yescharacteristic
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×C22⋊Q811yescharacteristic
485C8—22no—
486C8—22no—
487C8—22no—
488C8—42no—
489C8—42no—
490C8—81no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—82no—
495C8—82no—
496C8#42114yes—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—82no—
510C22—162no—
511C22C2×#38011yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4C2×#42112yes—
527C2—41no—
528C2—41no—
529C2—41no—
530C2—82no—
531C2#2065411yescharacteristic
532C2#2067311yescharacteristic
533C2#2067511yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121Q8
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. #20827C2semidirect
3. #17797C2semidirect
6. #8693C2semidirect
7. #8692C2semidirect
8. #8691C2semidirect
9. #8690C2semidirect
10. #7998C2semidirect
11. #7996C2semidirect
34. C4×#108C22semidirect
35. C4×#108C22semidirect
38. #1568C22semidirect
39. #1568C22semidirect
51. #1291C22semidirect
52. #1291C22semidirect
53. #1291C22semidirect
54. #1291C22semidirect
55. #1291C22semidirect
56. #1291C22semidirect

Maximal subgroups

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

subgroupindex
#602142
#208272
#177972
#176702
#176692
#86932
#86922
#86912
#86902
#79982
#79962