Finite Group Databasebeta

← #189379

Subgroups — #189379

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60236C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3C2×#1574C211yesmaximal, characteristic, complemented
4#17745C211yesmaximal, characteristic
5#17704C211yesmaximal, characteristic
6#8688C212yesmaximal, complemented
7#8686C212yesmaximal, complemented
8#8684C212yesmaximal, complemented
9#8682C212yesmaximal, complemented
10#8004C211yesmaximal, characteristic, complemented
11#8000C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2981C414yes—
14#2979—21no—
15#2979—21no—
16#2975C412yes—
17#2974—22no—
18#2972C412yes—
19#2935—21no—
20#2935—21no—
21#2931C2212yes—
22#2931C2212yes—
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#423C2211yescharacteristic
29#2880—22no—
30#1950—21no—
31#1950—21no—
32#1855—22no—
33#1848C2212yescomplemented
34#1837C2212yescomplemented
35#1574—22no—
36#1574—21no—
37#1574—21no—
38C2×#335C2212yescomplemented
39C2×#334C2212yescomplemented
40#1557—22no—
41#1549—21no—
42#1547C2212yes—
43#1547C2212yes—
44#1547C2212yes—
45#1547C2212yes—
46#1546—21no—
47#1545—21no—
48#1545—21no—
49#1540—22no—
50#1292C2211yescharacteristic, complemented
51#1292C2211yescharacteristic, complemented
52#1292C2211yescharacteristic, complemented
53#1292C2211yescharacteristic, complemented
54#1292C2212yescomplemented
55#1292C2212yescomplemented
56C22×#41—21no—
57C2×#131—22no—
58C2×#131—22no—
59C22×M16—21no—
60C22×M16—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532C2×C412yes—
66#532D412yes—
67#532Q812yes—
68#532C2312yes—
69C2×C4⋊Q8D411yescharacteristic
70C2×#127D411yescharacteristic
71C2×#124—21no—
72C2×#121—21no—
73C2×#121Q812yes—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C22×C42C2311yescharacteristic
77#436—22no—
78#436—22no—
79#436—22no—
80#436—22no—
81#436C2×C414yes—
82#436C2×C414yes—
83#435—24no—
84#435—24no—
85C8×D4—22no—
86C8×D4—22no—
87C8×D4—22no—
88C8×D4—22no—
89C8×D4C2×C414yes—
90C8×D4C2×C414yes—
91#432—21no—
92#432—21no—
93#432—21no—
94#432C2×C412yes—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431C2×C412yes—
103#431C2×C411yescharacteristic
104#431C2×C411yescharacteristic
105#431C2×C412yes—
106#431D412yes—
107#431D411yescharacteristic
108#431D411yescharacteristic
109C2×#108—22no—
110C2×#108—22no—
111#423—21no—
112#423—21no—
113#423—21no—
114#423—21no—
115#423—21no—
116#423—21no—
117#423—21no—
118#423C2311yescharacteristic
119#423C2312yes—
120#423C2311yescharacteristic
121#423C2311yescharacteristic
122#423C2311yescharacteristic
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8D411yescharacteristic
127C2×C4⋊C8D411yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131#408—22no—
132#408—22no—
133#408—24no—
134#408—24no—
135#407—21no—
136#407—24no—
137#407—21no—
138C2×C22⋊C8—24no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8D412yes—
141C4×M16—21no—
142C4×M16—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2312yes—
145#341—22no—
146#341—22no—
147#335—22no—
148#335—22no—
149#334—22no—
150#334—22no—
151#332—22no—
152#332—22no—
153#332—22no—
154#332—22no—
155#331—22no—
156#331—22no—
157#331—22no—
158#331—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—81no—
164C2×#41—81no—
165C2×#41—81no—
166C2×#41—81no—
167C22×Q8—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C4⋊C412yes—
172C23×C4—22no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×Q811yescharacteristic
175C23×C4#4111yescharacteristic
176#131—82no—
177#131—82no—
178#131—82no—
179#131—82no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—24no—
183C2×M16—24no—
184C2×M16—22no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—41no—
188C2×M16—41no—
189C2×M16—41no—
190C2×M16—81no—
191C2×M16—81no—
192C2×M16—82no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8—22no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—41no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C4⋊C412yes—
204C4⋊Q8C2×D412yes—
205#127—21no—
206#127—21no—
207#127C4⋊C412yes—
208#127C2×D412yes—
209#124—22no—
210#124—22no—
211#124—22no—
212#124—22no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121C4⋊C414yes—
220#121C2×Q814yes—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4C4⋊C414yes—
228C4×D4C22×C414yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C4⋊C412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×Q811yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#108—82no—
246#108—82no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—22no—
257C4⋊C8—21no—
258C4⋊C8—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8C22⋊C411yescharacteristic
266C4⋊C8C22⋊C411yescharacteristic
267C4⋊C8C22⋊C412yes—
268C4⋊C8C22⋊C412yes—
269C4⋊C8C22⋊C411yescharacteristic
270C4⋊C8C22⋊C411yescharacteristic
271C4⋊C8C22×C412yes—
272C4⋊C8C22×C412yes—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C22⋊C8—24no—
278C22⋊C8—24no—
279C22⋊C8—41no—
280C22⋊C8—41no—
281C22⋊C8—41no—
282C22⋊C8—41no—
283C22⋊C8C22⋊C412yes—
284C22⋊C8C22⋊C412yes—
285C22⋊C8C22⋊C412yes—
286C22⋊C8C22⋊C412yes—
287#97—22no—
288#97—41no—
289C4×C8—22no—
290C4×C8—41no—
291C4×C8—41no—
292C4×C8—41no—
293C4×C8C22×C412yes—
294C4×C8C22×C412yes—
295C4×C8C2×D412yes—
296C4×C8C2×D412yes—
297C24—22no—
298C24C22⋊Q811yescharacteristic
299#41—42no—
300#41—42no—
301#41—42no—
302#41—42no—
303#41—82no—
304#41—82no—
305#41—82no—
306#41—82no—
307#41—82no—
308#41—82no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—81no—
312C2×Q8—81no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—44no—
321C2×D4—81no—
322C2×D4—81no—
323C2×D4—81no—
324C2×D4—81no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4C2×C4⋊C414yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C4⋊C411yescharacteristic
343C22×C4C2×C4⋊C411yescharacteristic
344C22×C4#11711yescharacteristic
345C22×C4#11711yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4C22⋊Q811yescharacteristic
351C22×C4C22⋊Q811yescharacteristic
352C22×C4C22⋊Q811yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4C22⋊Q811yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—42no—
365M16—42no—
366M16—41no—
367M16—41no—
368M16—84no—
369M16—84no—
370M16—82no—
371M16—82no—
372M16—82no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—84no—
386C2×C8—84no—
387C2×C8—82no—
388C2×C8—81no—
389C2×C8—81no—
390C2×C8—82no—
391C2×C8C2×C22⋊C412yes—
392C2×C8C2×C22⋊C412yes—
393C2×C8C2×C22⋊C412yes—
394C2×C8C2×C22⋊C412yes—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4C2×C4⋊C414yes—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C42C2×C22⋊C411yescharacteristic
404C42C2×C22⋊C411yescharacteristic
405C42C2×C4⋊C411yescharacteristic
406C42C2×C4⋊C411yescharacteristic
407C42#11711yescharacteristic
408C42#11711yescharacteristic
409C42#12111yescharacteristic
410C42#12111yescharacteristic
411C42#12111yescharacteristic
412C42#12111yescharacteristic
413C42C22⋊Q811yescharacteristic
414C42C22⋊Q811yescharacteristic
415C42C23×C411yescharacteristic
416C42C22×D411yescharacteristic
417C42C22×D411yescharacteristic
418C42C2×#4111yescharacteristic
419C23—21no—
420C23—21no—
421C23—24no—
422C23—41no—
423C23—41no—
424C23—44no—
425C23—82no—
426C23—82no—
427C23#38011yescharacteristic
428C23#38011yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430Q8—42no—
431Q8—42no—
432Q8—82no—
433D4—44no—
434D4—44no—
435D4—44no—
436D4—44no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—82no—
442D4—82no—
443D4—84no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—84no—
473C2×C4—82no—
474C2×C4#38011yescharacteristic
475C2×C4#38011yescharacteristic
476C2×C4#38011yescharacteristic
477C2×C4#38011yescharacteristic
478C2×C4#38011yescharacteristic
479C2×C4#38011yescharacteristic
480C2×C4C22×C22⋊C411yescharacteristic
481C2×C4C22×C4⋊C411yescharacteristic
482C2×C4C2×#11711yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4#57611yescharacteristic
487C2×C4#57611yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57711yescharacteristic
490C8—22no—
491C8—22no—
492C8—22no—
493C8—22no—
494C8—42no—
495C8—42no—
496C8—82no—
497C8—81no—
498C8—81no—
499C8—81no—
500C8—82no—
501C8—81no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—81no—
514C22—81no—
515C22—164no—
516C22C2×#38011yescharacteristic
517C22#308111yescharacteristic
518C22#308111yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—44no—
528C4#181312yes—
529C4#289711yescharacteristic
530C4#289711yescharacteristic
531C4#304111yescharacteristic
532C4#304211yescharacteristic
533C4#306811yescharacteristic
534C4#306911yescharacteristic
535C2—21no—
536C2—21no—
537C2—41no—
538C2—41no—
539C2—84no—
540C2#2065011yescharacteristic
541C2C2×#181311yescharacteristic
542C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
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. #21993C2semidirect
3. C2×#1574C2semidirect
6. #8688C2semidirect
7. #8686C2semidirect
8. #8684C2semidirect
9. #8682C2semidirect
10. #8004C2semidirect
11. #8000C2semidirect
33. #1848C22semidirect
34. #1837C22semidirect
38. C2×#335C22semidirect
39. C2×#334C22semidirect
50. #1292C22semidirect
51. #1292C22semidirect
52. #1292C22semidirect
53. #1292C22semidirect
54. #1292C22semidirect
55. #1292C22semidirect

Maximal subgroups

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

subgroupindex
#602362
#219932
C2×#15742
#177452
#177042
#86882
#86862
#86842
#86822
#80042
#80002