Finite Group Databasebeta

← #203128

Subgroups — #203128

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21382C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3#17719C211yesmaximal, characteristic
4#17481C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic, complemented
6#9592C212yesmaximal, complemented
7#9591C212yesmaximal, complemented
8#9590C212yesmaximal, complemented
9#9589C212yesmaximal, complemented
10#8097C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2920C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×#108—21no—
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C4×#117C2211yescharacteristic
20#1986C2211yescharacteristic
21#1986C2211yescharacteristic
22#1980C2211yescharacteristic
23#1980C2211yescharacteristic
24#1950—21no—
25#1950—21no—
26#1950C2212yes—
27#1950C2212yes—
28#1871—21no—
29#1871—21no—
30C4×#104—21no—
31C4×#104—21no—
32C2×#341—21no—
33#1552C412yescomplemented
34#1551C412yescomplemented
35#1550C411yescharacteristic, complemented
36#1550C411yescharacteristic, complemented
37#1547C411yescharacteristic, complemented
38#1547C411yescharacteristic, complemented
39#1542C2212yes—
40#1540C2212yes—
41#1495—22no—
42#1493C2212yes—
43#1492C2212yes—
44#1477—21no—
45#1477—21no—
46#1453—22no—
47#1451—21no—
48#1451C2212yes—
49C8×M16—21no—
50C8×M16C2212yes—
51#1303C2212yes—
52#1303C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×M16—21no—
56C22×M16D411yescharacteristic
57C22×M16D411yescharacteristic
58C23×C8—21no—
59#533C2×C411yescharacteristic
60#533C2×C411yescharacteristic
61#533C2311yescharacteristic
62#532C2×C411yescharacteristic
63#532C2×C411yescharacteristic
64#532C2311yescharacteristic
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71#525—21no—
72#525—21no—
73#525—21no—
74C4×#41—21no—
75C4×#41—21no—
76C4×#41—21no—
77C2×#117—21no—
78C22×C42C2311yescharacteristic
79#432—21no—
80#432C2×C412yes—
81#431—21no—
82#431—21no—
83#431—21no—
84#431D411yescharacteristic
85#431D411yescharacteristic
86#431D412yes—
87#431C2312yes—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—44no—
91#423—21no—
92#423—21no—
93#423—21no—
94#423C2×C411yescharacteristic
95#423C2×C411yescharacteristic
96#423C2312yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8C2×C411yescharacteristic
105C2×C4⋊C8C2×C412yes—
106C2×C4⋊C8C2312yes—
107C2×#104—21no—
108C2×#104—21no—
109C2×#104—21no—
110C2×#104—21no—
111#407—21no—
112C2×C22⋊C8—21no—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16—21no—
116C4×M16D411yescharacteristic
117C4×M16D411yescharacteristic
118C4×M16C2311yescharacteristic
119C4×M16C2311yescharacteristic
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2312yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—21no—
129C43—21no—
130#341—22no—
131#341—22no—
132#339—21no—
133#339—21no—
134#339—21no—
135#339—21no—
136#333C2×C412yes—
137#333C2×C412yes—
138#332C2×C412yes—
139#332C2×C412yes—
140#332C2×C412yes—
141#332C2×C412yes—
142#331C2×C412yes—
143#331C2×C412yes—
144Q8⋊C8—22no—
145Q8⋊C8—22no—
146D4⋊C8—22no—
147D4⋊C8—22no—
148#322—22no—
149#322—22no—
150#322—22no—
151C82—22no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—41no—
166C2×M16—41no—
167C2×M16—41no—
168C2×M16—42no—
169C2×M16—42no—
170C2×M16C2×D411yescharacteristic
171C2×M16C2×D411yescharacteristic
172C2×M16C2×D411yescharacteristic
173C2×M16C2×D411yescharacteristic
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—41no—
184C4⋊Q8C22×C411yescharacteristic
185C4⋊Q8C22×C411yescharacteristic
186C4⋊Q8C22×C411yescharacteristic
187#127C22×C411yescharacteristic
188#126—41no—
189#126—41no—
190#125—21no—
191#125—21no—
192#125—21no—
193#124—21no—
194#124—21no—
195#124—21no—
196#123—41no—
197#123—41no—
198#123—41no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#121—21no—
210#121—21no—
211#120—41no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230#117—24no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—42no—
235#117—41no—
236#117—41no—
237#117—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42C2411yescharacteristic
258#108—44no—
259#108—44no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—22no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8C22×C412yes—
274C4⋊C8C22×C412yes—
275C4⋊C8C22×C412yes—
276C4⋊C8C22×C412yes—
277#104—42no—
278#104—42no—
279#104—42no—
280#104—42no—
281C22⋊C8—22no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286#97—22no—
287#97—21no—
288#97—21no—
289#97—21no—
290#97—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—22no—
295C4×C8—21no—
296C4×C8—22no—
297C4×C8C2×D412yes—
298C4×C8C2×D412yes—
299#95—21no—
300#95—21no—
301#95—41no—
302C24#12411yescharacteristic
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—42no—
308#41—42no—
309#41—82no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—81no—
331C22×C4#11711yescharacteristic
332C22×C4#11711yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12311yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4#12411yescharacteristic
342C22×C4#12411yescharacteristic
343C22×C4#12411yescharacteristic
344C22×C4#12611yescharacteristic
345C22×C4#12611yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—42no—
356M16—42no—
357M16—42no—
358M16—42no—
359M16—42no—
360M16—42no—
361M16—82no—
362M16AGL(1,ℤ/8)12yes—
363M16#13712yes—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—44no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—82no—
383C2×C8—82no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C22⋊C4—44no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C42—44no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—82no—
419C42#11711yescharacteristic
420C42#11711yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42C4×D411yescharacteristic
424C42C4×D411yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12311yescharacteristic
428C42#12311yescharacteristic
429C42#12611yescharacteristic
430C42#12611yescharacteristic
431C42C23×C411yescharacteristic
432C42C22×D411yescharacteristic
433C42C2×#4111yescharacteristic
434C42C2×#4111yescharacteristic
435C23—21no—
436C23—21no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23#38811yescharacteristic
441C23#38811yescharacteristic
442C23C2×#12411yescharacteristic
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—41no—
447Q8—81no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—81no—
453D4—82no—
454D4—81no—
455D4—81no—
456D4—82no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4#38811yescharacteristic
494C2×C4#38811yescharacteristic
495C2×C4#38811yescharacteristic
496C2×C4#38811yescharacteristic
497C2×C4#38811yescharacteristic
498C2×C4#38811yescharacteristic
499C2×C4#44312yes—
500C2×C4#44412yes—
501C2×C4C2×#11711yescharacteristic
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12311yescharacteristic
506C2×C4C2×#12611yescharacteristic
507C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
508C2×C4C2×#13711yescharacteristic
509C2×C4#57511yescharacteristic
510C2×C4#57511yescharacteristic
511C8—22no—
512C8—22no—
513C8—42no—
514C8—42no—
515C8—42no—
516C8—42no—
517C8—42no—
518C8—42no—
519C8—41no—
520C8—41no—
521C8—82no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22#192912yes—
535C22C2×#38811yescharacteristic
536C22#314111yescharacteristic
537C22#314211yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—82no—
547C4—82no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4#192912yes—
552C4C4×AGL(1,ℤ/8)11yescharacteristic
553C4C4×#13711yescharacteristic
554C4C2×#44311yescharacteristic
555C4C2×#44411yescharacteristic
556C4#305511yescharacteristic
557C4#305611yescharacteristic
558C2—21no—
559C2—21no—
560C2—22no—
561C2—41no—
562C2—161no—
563C2C2×#192911yescharacteristic
564C2#2108011yescharacteristic
565C2#2109311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
5. #17327C2semidirect
6. #9592C2semidirect
7. #9591C2semidirect
8. #9590C2semidirect
9. #9589C2semidirect
10. #8097C2semidirect
11. #7891C2semidirect
33. #1552C4semidirect
34. #1551C4semidirect
35. #1550C4semidirect
36. #1550C4semidirect
37. #1547C4semidirect
38. #1547C4semidirect

Maximal subgroups

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

subgroupindex
#213822
C2×#19502
#177192
#174812
#173272
#95922
#95912
#95902
#95892
#80972
#78912