Finite Group Databasebeta

← #189414

Subgroups — #189414

Order 512.

802 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60290C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic, complemented
3#17819C211yesmaximal, characteristic, complemented
4#17670C211yesmaximal, characteristic
5#17642C211yesmaximal, characteristic
6#8693C212yesmaximal, complemented
7#8692C212yesmaximal, complemented
8#8691C212yesmaximal, complemented
9#8690C212yesmaximal, complemented
10#7996C211yesmaximal, characteristic, complemented
11#7992C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2994—21no—
14#2994—21no—
15#2993C412yes—
16#2993C412yes—
17#2983C411yescharacteristic
18#2982C411yescharacteristic
19#2975C411yescharacteristic
20#2974—22no—
21#2972C411yescharacteristic
22#2969—21no—
23#2969—21no—
24#2966C2212yes—
25#2966C2212yes—
26#2924—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×C4×M16C2211yescharacteristic
33#1950—21no—
34#1950—21no—
35#1855—22no—
36C4×#108C2212yescomplemented
37C4×#108C2212yescomplemented
38#1574—22no—
39#1574—21no—
40#1574—21no—
41#1568C2212yescomplemented
42#1568C2212yescomplemented
43#1535—22no—
44#1529C2212yes—
45#1529C2212yes—
46#1529C2212yes—
47#1529C2212yes—
48#1528—21no—
49#1528—21no—
50#1527—21no—
51#1527—21no—
52#1524—22no—
53#1291C2211yescharacteristic, complemented
54#1291C2211yescharacteristic, complemented
55#1291C2211yescharacteristic, complemented
56#1291C2211yescharacteristic, complemented
57#1291C2212yescomplemented
58#1291C2212yescomplemented
59C22×M16—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529—21no—
64#529C2×C412yes—
65#529D412yes—
66#529Q812yes—
67#529C2312yes—
68C2×#126—21no—
69C2×#125D411yescharacteristic
70C2×#124D411yescharacteristic
71C2×#123—21no—
72C2×C22⋊Q8Q811yescharacteristic
73C2×#121Q811yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C22×C42C2311yescharacteristic
79#446—21no—
80#446—21no—
81#446C2×C412yes—
82#446C2×C412yes—
83C8×Q8—21no—
84C8×Q8—21no—
85C8×Q8C2×C412yes—
86C8×Q8C2×C412yes—
87#436—21no—
88#436—21no—
89#436—22no—
90#436—22no—
91#436C2×C412yes—
92#436C2×C412yes—
93#435—22no—
94#435—22no—
95#435—22no—
96#435—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4C2×C412yes—
102C8×D4C2×C412yes—
103#433—22no—
104#433—22no—
105#433—22no—
106#433—22no—
107#432—21no—
108#432—22no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432C2311yescharacteristic
114#432C2311yescharacteristic
115#432C2312yes—
116#432C2311yescharacteristic
117#432C2311yescharacteristic
118#431—21no—
119#431—22no—
120#431—21no—
121#431—21no—
122#431—21no—
123#431—21no—
124#431—21no—
125#431—21no—
126#431C2×C411yescharacteristic
127#431C2×C411yescharacteristic
128#431C2×C411yescharacteristic
129#431C2×C411yescharacteristic
130#431C2×C412yes—
131#431D412yes—
132#431D411yescharacteristic
133#431D411yescharacteristic
134C2×#108—22no—
135C2×#108—22no—
136#424—22no—
137#424—22no—
138#423—21no—
139#423—21no—
140#423—21no—
141#423C2×C411yescharacteristic
142#423C2×C411yescharacteristic
143C2×C4⋊C8D411yescharacteristic
144C2×C4⋊C8D411yescharacteristic
145#407—21no—
146#407—22no—
147#407—21no—
148#407—21no—
149#407—21no—
150C2×C22⋊C8—22no—
151C2×C22⋊C8D412yes—
152#405—22no—
153#405—22no—
154C4×M16—21no—
155C4×M16—21no—
156C4×M16—21no—
157C4×M16—21no—
158C4×M16—21no—
159C4×M16—21no—
160C4×M16C2311yescharacteristic
161C4×M16C2311yescharacteristic
162C4×M16C2312yes—
163C2×#97—21no—
164C2×C4×C8—21no—
165C2×C4×C8C2312yes—
166#341—22no—
167#341—22no—
168#335—22no—
169#335—22no—
170#334—22no—
171#334—22no—
172#330—22no—
173#330—22no—
174#330—22no—
175#330—22no—
176#329—22no—
177#329—22no—
178#329—22no—
179#329—22no—
180C22×Q8C4⋊C411yescharacteristic
181C22×D4—21no—
182C22×D4C4⋊C411yescharacteristic
183C23×C4—21no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×Q811yescharacteristic
186C23×C4#4111yescharacteristic
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—41no—
197C2×M16—41no—
198C2×M16—41no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—41no—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204#126—22no—
205#126—22no—
206#126—22no—
207#126—22no—
208#125—21no—
209#125—21no—
210#125C4⋊C412yes—
211#125C2×D412yes—
212#124—21no—
213#124—21no—
214#124C4⋊C412yes—
215#124C2×D412yes—
216#123—22no—
217#123—22no—
218#123—22no—
219#123—22no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8C4⋊C412yes—
223C22⋊Q8C2×Q812yes—
224#121—21no—
225#121—21no—
226#121C4⋊C412yes—
227#121C2×Q812yes—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8C4⋊C412yes—
231C4×Q8C22×C412yes—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C4⋊C412yes—
239C4×D4C22×C412yes—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C4⋊C411yescharacteristic
249C2×C4⋊C4C4⋊C411yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×Q811yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268#108—44no—
269#108—44no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—22no—
278C4⋊C8—21no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283C4⋊C8C22⋊C412yes—
284C4⋊C8C22⋊C412yes—
285C4⋊C8C22⋊C411yescharacteristic
286C4⋊C8C22⋊C411yescharacteristic
287C4⋊C8C22⋊C411yescharacteristic
288C4⋊C8C22⋊C411yescharacteristic
289C4⋊C8C22×C411yescharacteristic
290C4⋊C8C22×C411yescharacteristic
291C4⋊C8C22×C411yescharacteristic
292C4⋊C8C22×C411yescharacteristic
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—22no—
304C22⋊C8—41no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8C22⋊C412yes—
309C22⋊C8C22⋊C412yes—
310C22⋊C8C22⋊C412yes—
311C22⋊C8C22⋊C412yes—
312#97—22no—
313#97—22no—
314#97—21no—
315#97—21no—
316#97—21no—
317#97—21no—
318#97—22no—
319#97—21no—
320#97—21no—
321#97—22no—
322#97—41no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—22no—
330C4×C8—41no—
331C4×C8—41no—
332C4×C8—41no—
333C4×C8C22×C412yes—
334C4×C8C22×C412yes—
335C4×C8C2×D412yes—
336C4×C8C2×D412yes—
337C24—21no—
338C24C22⋊Q811yescharacteristic
339C2×Q8—21no—
340C2×Q8—42no—
341C2×Q8C2×C4⋊C412yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—42no—
346C2×D4—82no—
347C2×D4C2×C4⋊C412yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C4⋊C411yescharacteristic
362C22×C4C2×C4⋊C411yescharacteristic
363C22×C4#11711yescharacteristic
364C22×C4#11711yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4C22⋊Q811yescharacteristic
370C22×C4C22⋊Q811yescharacteristic
371C22×C4C22⋊Q811yescharacteristic
372C22×C4C22⋊Q811yescharacteristic
373C22×C4C22⋊Q811yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377M16—22no—
378M16—22no—
379M16—41no—
380M16—41no—
381M16—44no—
382M16—42no—
383M16—42no—
384M16—82no—
385M16—82no—
386M16—81no—
387M16—81no—
388M16—82no—
389M16—82no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—44no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—82no—
404C2×C8—82no—
405C2×C8C2×C22⋊C412yes—
406C2×C8C2×C22⋊C412yes—
407C2×C8C2×C22⋊C412yes—
408C2×C8C2×C22⋊C412yes—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4C2×C4⋊C412yes—
420C4⋊C4C2×C4⋊C412yes—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C42—22no—
432C42—22no—
433C42C2×C22⋊C411yescharacteristic
434C42C2×C22⋊C411yescharacteristic
435C42C2×C4⋊C411yescharacteristic
436C42C2×C4⋊C411yescharacteristic
437C42#11711yescharacteristic
438C42#11711yescharacteristic
439C42#12111yescharacteristic
440C42#12111yescharacteristic
441C42#12111yescharacteristic
442C42#12111yescharacteristic
443C42C22⋊Q811yescharacteristic
444C42C22⋊Q811yescharacteristic
445C42C23×C411yescharacteristic
446C42C22×D411yescharacteristic
447C42C22×D411yescharacteristic
448C42C2×#4111yescharacteristic
449C23—22no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—81no—
455C23—81no—
456C23#38011yescharacteristic
457C23#38011yescharacteristic
458C23C2×C22⋊Q811yescharacteristic
459Q8—42no—
460Q8—42no—
461D4—42no—
462D4—44no—
463D4—42no—
464D4—82no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4#38011yescharacteristic
496C2×C4#38011yescharacteristic
497C2×C4#38011yescharacteristic
498C2×C4#38011yescharacteristic
499C2×C4#38011yescharacteristic
500C2×C4#38011yescharacteristic
501C2×C4C22×C22⋊C411yescharacteristic
502C2×C4C22×C4⋊C411yescharacteristic
503C2×C4C2×#11711yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C8—22no—
508C8—22no—
509C8—22no—
510C8—22no—
511C8—42no—
512C8—42no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—82no—
518C8—82no—
519C8—81no—
520C8—81no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22—162no—
535C22C2×#38011yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4#181312yes—
550C4#290111yescharacteristic
551C4#290111yescharacteristic
552C2—41no—
553C2—41no—
554C2—41no—
555C2—82no—
556C2#2065511yescharacteristic
557C2C2×#181311yescharacteristic
558C2#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. #21370C2semidirect
3. #17819C2semidirect
6. #8693C2semidirect
7. #8692C2semidirect
8. #8691C2semidirect
9. #8690C2semidirect
10. #7996C2semidirect
11. #7992C2semidirect
36. C4×#108C22semidirect
37. C4×#108C22semidirect
41. #1568C22semidirect
42. #1568C22semidirect
53. #1291C22semidirect
54. #1291C22semidirect
55. #1291C22semidirect
56. #1291C22semidirect
57. #1291C22semidirect
58. #1291C22semidirect

Maximal subgroups

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

subgroupindex
#602902
#213702
#178192
#176702
#176422
#86932
#86922
#86912
#86902
#79962
#79922