Finite Group Databasebeta

← #127200

Subgroups — #127200

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20722C211yesmaximal, characteristic
2#17362C211yesmaximal, characteristic
3C4×#390C211yesmaximal, characteristic
4#7213C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#407—21no—
7C2×#390—21no—
8#2280—21no—
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4—21no—
12C2×C43C411yescharacteristic, complemented
13#1819C2212yes—
14#1799—21no—
15#1799C412yescomplemented
16#1781—21no—
17C4×#99—21no—
18C4×#99C412yescomplemented
19#1465—21no—
20#1465C412yescomplemented
21#1463—21no—
22#1462C2212yes—
23#1459—21no—
24#1459C412yescomplemented
25#1437C412yescomplemented
26#1437C2212yes—
27C4⋊#95—21no—
28C23×D4D411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×#117—41no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2311yescharacteristic
51C22×C42—21no—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C412yes—
56C2×#99—22no—
57C2×#99—22no—
58#407—22no—
59#407—22no—
60#407—41no—
61#407—41no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—41no—
64#390—28no—
65#381—41no—
66#380—22no—
67#380—41no—
68#379—22no—
69C4×C4⋊C4—41no—
70C4×C4⋊C4—41no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4—44no—
73C4×C22⋊C4—44no—
74C4×C22⋊C4—41no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95C2×C412yes—
78C43—41no—
79C43—41no—
80#342—22no—
81#342C2×C414yes—
82#324—22no—
83#324—22no—
84#323—22no—
85#323C2×C414yes—
86C25C22⋊C411yescharacteristic
87C25C2×D411yescharacteristic
88C22×D4—24no—
89C22×D4—42no—
90C22×D4—41no—
91C22×D4—41no—
92C22×D4C22⋊C411yescharacteristic
93C22×D4C22⋊C411yescharacteristic
94C22×D4C4⋊C411yescharacteristic
95C22×D4C4⋊C411yescharacteristic
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—42no—
124C2×M16—42no—
125C2×M16—82no—
126C2×M16—82no—
127C22×C8—42no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144#117—82no—
145#117—82no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C4⋊C411yescharacteristic
155C2×C4⋊C4C4⋊C411yescharacteristic
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4C4211yescharacteristic
170C2×C22⋊C4C4211yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C22⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4C22×C411yescharacteristic
174C2×C22⋊C4C22×C411yescharacteristic
175C2×C22⋊C4C2×D411yescharacteristic
176C2×C22⋊C4C2×Q811yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—44no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C4211yescharacteristic
196C2×C42C4211yescharacteristic
197C2×C42C4⋊C411yescharacteristic
198C2×C42C4⋊C411yescharacteristic
199#99—44no—
200#99—44no—
201C22⋊C8—24no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205#95—42no—
206#95—44no—
207#95—44no—
208#95—42no—
209#95—42no—
210C24—22no—
211C24—22no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—42no—
221C24—42no—
222C24—41no—
223C24—41no—
224C24#9511yescharacteristic
225C24#9511yescharacteristic
226C24C2×C22⋊C411yescharacteristic
227C24C2×C22⋊C411yescharacteristic
228C24C2×C4⋊C411yescharacteristic
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—48no—
241C2×D4—48no—
242C2×D4#10414yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4#9511yescharacteristic
312C22×C4#9511yescharacteristic
313C22×C4#9511yescharacteristic
314C22×C4#9511yescharacteristic
315C22×C4#9511yescharacteristic
316C22×C4#9511yescharacteristic
317C22×C4#9911yescharacteristic
318C22×C4#9911yescharacteristic
319C22×C4#9911yescharacteristic
320C22×C4#9911yescharacteristic
321C22×C4C2×C4211yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C4⋊C411yescharacteristic
324C22×C4C2×C4⋊C411yescharacteristic
325M16—82no—
326M16—82no—
327M16—84no—
328C2×C8—44no—
329C2×C8—84no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—48no—
335C4⋊C4—84no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—416no—
345C22⋊C4—416no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—84no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—48no—
360C42—44no—
361C42—44no—
362C42—44no—
363C42—44no—
364C42—82no—
365C42—84no—
366C42—82no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23#33912yes—
413C23C2×#9511yescharacteristic
414C23C2×#9911yescharacteristic
415C23C2×#9911yescharacteristic
416C23C2×#10412yes—
417D4—44no—
418D4—44no—
419D4—44no—
420D4—44no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—28no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4H(3,ℤ/4)12yes—
487C2×C4#33912yes—
488C2×C4#34211yescharacteristic
489C2×C4#34211yescharacteristic
490C2×C4#34211yescharacteristic
491C2×C4#34211yescharacteristic
492C2×C4#35312yes—
493C2×C4#35412yes—
494C8—84no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—81no—
522C22—81no—
523C22—84no—
524C22—84no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—81no—
530C22—81no—
531C22C2×H(3,ℤ/4)11yescharacteristic
532C22C2×#33911yescharacteristic
533C22C2×#34211yescharacteristic
534C22#212012yes—
535C22C2×#35311yescharacteristic
536C22C2×#35411yescharacteristic
537C4—22no—
538C4—22no—
539C4—21no—
540C4—21no—
541C4—48no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—84no—
549C4—164no—
550C4#139312yes—
551C4#139312yes—
552C2—22no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—21no—
557C2—42no—
558C2—42no—
559C2—81no—
560C2—81no—
561C2#794211yescharacteristic
562C2#1257311yescharacteristic
563C2C2×#139311yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
socleC22C2×#342
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7213C2semidirect
9. C2×C4×C4⋊C4C4semidirect
12. C2×C43C4semidirect
15. #1799C4semidirect
18. C4×#99C4semidirect
20. #1465C4semidirect
24. #1459C4semidirect
25. #1437C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#173622
C4×#3902
#72132