Finite Group Databasebeta

← #203127

Subgroups — #203127

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic, complemented
2#21899C211yesmaximal, characteristic
3#17719C211yesmaximal, characteristic
4#17473C211yesmaximal, characteristic
5#17323C211yesmaximal, characteristic, complemented
6#9592C212yesmaximal, complemented
7#9591C212yesmaximal, complemented
8#9590C212yesmaximal, complemented
9#9589C212yesmaximal, complemented
10#8098C211yesmaximal, characteristic, complemented
11C2×#1279C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15#2920C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1986C2212yes—
20#1980C2212yes—
21#1950—21no—
22#1950—21no—
23#1950C2212yes—
24#1950C2212yes—
25#1896—22no—
26#1810—22no—
27#1806—21no—
28#1806—21no—
29#1552C412yescomplemented
30#1551C412yescomplemented
31#1550C411yescharacteristic, complemented
32#1550C411yescharacteristic, complemented
33#1547C411yescharacteristic, complemented
34#1547C411yescharacteristic, complemented
35#1542C2212yes—
36#1540C2212yes—
37#1494—21no—
38#1494—21no—
39#1493C2211yescharacteristic
40#1493C2211yescharacteristic
41#1492C2211yescharacteristic
42#1492C2211yescharacteristic
43#1476—21no—
44#1476—21no—
45#1452—21no—
46#1451—21no—
47#1451C2212yes—
48#1450—21no—
49C8×M16—21no—
50C8×M16C2212yes—
51#1303C2212yes—
52#1303C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C23×C8—21no—
56C23×C8—21no—
57#533C2×C411yescharacteristic
58#533C2×C411yescharacteristic
59#533C2311yescharacteristic
60#532C2×C411yescharacteristic
61#532C2×C411yescharacteristic
62#532C2311yescharacteristic
63#529—21no—
64#529—21no—
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#525—21no—
70#525—21no—
71#525—21no—
72C4×#41—21no—
73C4×#41—21no—
74C4×#41—21no—
75C22×C4⋊C4—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#432—21no—
79#432—21no—
80#432—21no—
81#432—21no—
82#432—21no—
83#432C2×C412yes—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431C2312yes—
90C2×#108—22no—
91C2×#108—22no—
92#423C2×C411yescharacteristic
93#423C2×C411yescharacteristic
94#423D411yescharacteristic
95#423D411yescharacteristic
96#423C2311yescharacteristic
97#423C2311yescharacteristic
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2×C411yescharacteristic
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C412yes—
104C2×C4⋊C8C2312yes—
105C2×#104—22no—
106C2×#104—22no—
107#407—21no—
108#407—21no—
109#407D412yes—
110C4×M16—21no—
111C4×M16—21no—
112C4×M16C2312yes—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2312yes—
126C2×#95—21no—
127#341—22no—
128#341—22no—
129#339—22no—
130#339—22no—
131#333C2×C412yescomplemented
132#333C2×C412yescomplemented
133#332C2×C412yescomplemented
134#332C2×C412yescomplemented
135#332C2×C412yescomplemented
136#332C2×C412yescomplemented
137#331C2×C412yescomplemented
138#331C2×C412yescomplemented
139Q8⋊C8—21no—
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143D4⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147#322—22no—
148#322—22no—
149#322—22no—
150C82—22no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16C2×D412yes—
163C2×M16C2×D412yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—44no—
175C22×C8—81no—
176C22×C8—81no—
177C4⋊Q8C22×C411yescharacteristic
178C4⋊Q8C22×C411yescharacteristic
179C4⋊Q8C22×C411yescharacteristic
180#127C22×C411yescharacteristic
181#126—41no—
182#126—41no—
183#125—21no—
184#125—21no—
185#125—21no—
186#124—21no—
187#124—21no—
188#124—21no—
189#123—41no—
190#123—41no—
191#123—41no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204#120—41no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—41no—
227#117—41no—
228#117—41no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C42—22no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—81no—
237C2×C42—81no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42C2411yescharacteristic
250#108—82no—
251#108—82no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—22no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8C22×C412yes—
267C4⋊C8C22×C412yes—
268C4⋊C8C22×C412yes—
269C4⋊C8C22×C412yes—
270#104—82no—
271#104—82no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—41no—
277C22⋊C8—41no—
278#97—22no—
279#97—22no—
280#97—41no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—41no—
290C4×C8C2×D411yescharacteristic
291C4×C8C2×D411yescharacteristic
292C4×C8C2×D411yescharacteristic
293C4×C8C2×D411yescharacteristic
294#95—22no—
295#95—41no—
296C24#12411yescharacteristic
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—42no—
302#41—42no—
303#41—82no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4#11711yescharacteristic
328C22×C4#11711yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4#12311yescharacteristic
337C22×C4#12411yescharacteristic
338C22×C4#12411yescharacteristic
339C22×C4#12411yescharacteristic
340C22×C4#12611yescharacteristic
341C22×C4#12611yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—42no—
350M16—42no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355C2×C8—24no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—81no—
376C2×C8—81no—
377C2×C8—82no—
378C4⋊C4—24no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—42no—
406C42—42no—
407C42—82no—
408C42#11711yescharacteristic
409C42#11711yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42C4×D411yescharacteristic
413C42C4×D411yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12311yescharacteristic
417C42#12311yescharacteristic
418C42#12611yescharacteristic
419C42#12611yescharacteristic
420C42C23×C411yescharacteristic
421C42C22×D411yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—41no—
429C23#38811yescharacteristic
430C23#38811yescharacteristic
431C23C2×#12411yescharacteristic
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—81no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—82no—
445D4—82no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—88no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#38811yescharacteristic
481C2×C4#38811yescharacteristic
482C2×C4#38811yescharacteristic
483C2×C4#38811yescharacteristic
484C2×C4#38811yescharacteristic
485C2×C4#38811yescharacteristic
486C2×C4#44312yes—
487C2×C4#44412yes—
488C2×C4C2×#11711yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4C2×#12611yescharacteristic
494C2×C4#57611yescharacteristic
495C2×C4#57711yescharacteristic
496C2×C4#57711yescharacteristic
497C2×C4#57811yescharacteristic
498C8—22no—
499C8—22no—
500C8—41no—
501C8—41no—
502C8—42no—
503C8—42no—
504C8—42no—
505C8—42no—
506C8—42no—
507C8—42no—
508C8—81no—
509C8—81no—
510C22—24no—
511C22—24no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22#192912yes—
522C22#193212yes—
523C22C2×#38811yescharacteristic
524C22#313611yescharacteristic
525C22#313711yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—84no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4#192912yes—
539C4#294611yescharacteristic
540C4#294711yescharacteristic
541C4C2×#44311yescharacteristic
542C4C2×#44411yescharacteristic
543C4#306011yescharacteristic
544C4#306111yescharacteristic
545C2—21no—
546C2—21no—
547C2—22no—
548C2—22no—
549C2—161no—
550C2C2×#192911yescharacteristic
551C2C2×#193211yescharacteristic
552C2#2109211yescharacteristic
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
1. #21969C2semidirect
5. #17323C2semidirect
6. #9592C2semidirect
7. #9591C2semidirect
8. #9590C2semidirect
9. #9589C2semidirect
10. #8098C2semidirect
11. C2×#1279C2semidirect
29. #1552C4semidirect
30. #1551C4semidirect
31. #1550C4semidirect
32. #1550C4semidirect
33. #1547C4semidirect
34. #1547C4semidirect
131. #333C2×C4semidirect
132. #333C2×C4semidirect
133. #332C2×C4semidirect
134. #332C2×C4semidirect
135. #332C2×C4semidirect
136. #332C2×C4semidirect
137. #331C2×C4semidirect
138. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219692
#218992
#177192
#174732
#173232
#95922
#95912
#95902
#95892
#80982
C2×#12792