Finite Group Databasebeta

← #127199

Subgroups — #127199

Order 512.

1109 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#20722C211yesmaximal, characteristic
2#17364C211yesmaximal, characteristic
3C4×#386C211yesmaximal, characteristic
4#7213C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C23×C42C411yescharacteristic, complemented
7#2920—21no—
8C2×#386—21no—
9#2280—21no—
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4C411yescharacteristic, complemented
12#1819C2212yes—
13#1799—21no—
14#1799C412yescomplemented
15#1781—21no—
16C4×#99—21no—
17C4×#99C412yescomplemented
18#1466—21no—
19#1466C412yescomplemented
20#1462—21no—
21#1462C2212yes—
22#1460—21no—
23#1460C412yescomplemented
24#1437C2212yes—
25#1436C412yescomplemented
26C22⋊#95—21no—
27C4×#95—21no—
28C23×D4D411yescharacteristic
29C24×C4—21no—
30C24×C4C2×C411yescharacteristic
31C24×C4D411yescharacteristic
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⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4D411yescharacteristic
49C22×C22⋊C4C2311yescharacteristic
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—44no—
53C22×C42—41no—
54C22×C42—41no—
55C22×C42C2×C412yes—
56C22×C42D411yescharacteristic
57#432—22no—
58#432—41no—
59#431—22no—
60#431—41no—
61#423—41no—
62C2×#99—22no—
63C2×#99—22no—
64#407—22no—
65C4×M16—41no—
66#386—28no—
67#381—41no—
68#380—22no—
69#380—41no—
70#379—22no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4—44no—
73C4×C22⋊C4—41no—
74C4×C22⋊C4—41no—
75C4×C22⋊C4—41no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95C2×C412yes—
80#342—22no—
81#342C2×C414yescomplemented
82#323—22no—
83#323—22no—
84#323—22no—
85#323C2×C414yescomplemented
86C25C22⋊C411yescharacteristic
87C25C2×D411yescharacteristic
88C22×D4—24no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4—42no—
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—24no—
106C23×C4—22no—
107C23×C4—44no—
108C23×C4—44no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4C4211yescharacteristic
116C23×C4C4211yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C4⋊C411yescharacteristic
123C23×C4C4⋊C411yescharacteristic
124C23×C4C4⋊C411yescharacteristic
125C23×C4C4⋊C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—42no—
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—
144C4×D4—41no—
145#117—82no—
146#117—82no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C4211yescharacteristic
151C2×C4⋊C4C4211yescharacteristic
152C2×C4⋊C4C4⋊C411yescharacteristic
153C2×C4⋊C4C4⋊C411yescharacteristic
154C2×C22⋊C4—28no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4C22⋊C411yescharacteristic
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C4⋊C411yescharacteristic
170C2×C22⋊C4C4⋊C411yescharacteristic
171C2×C22⋊C4C22×C411yescharacteristic
172C2×C22⋊C4C22×C411yescharacteristic
173C2×C22⋊C4C2×D411yescharacteristic
174C2×C22⋊C4C2×Q811yescharacteristic
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—48no—
183C2×C42—44no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—44no—
187C2×C42—44no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C4⋊C411yescharacteristic
196C2×C42C4⋊C411yescharacteristic
197C4⋊C8—42no—
198C4⋊C8—42no—
199#99—44no—
200#99—44no—
201C22⋊C8—24no—
202C22⋊C8—42no—
203#97—42no—
204C4×C8—42no—
205#95—28no—
206#95—42no—
207#95—44no—
208#95—44no—
209#95—42no—
210#95—42no—
211C24—22no—
212C24—22no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—21no—
217C24—21no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—42no—
222C24—41no—
223C24—42no—
224C24—41no—
225C24#9511yescharacteristic
226C24#9511yescharacteristic
227C24C2×C22⋊C411yescharacteristic
228C24C2×C22⋊C411yescharacteristic
229C24C2×C4⋊C411yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—48no—
242C2×D4—48no—
243C2×D4#10414yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
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—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—416no—
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—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—416no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4#9511yescharacteristic
318C22×C4#9511yescharacteristic
319C22×C4#9511yescharacteristic
320C22×C4#9511yescharacteristic
321C22×C4#9511yescharacteristic
322C22×C4#9511yescharacteristic
323C22×C4#9911yescharacteristic
324C22×C4#9911yescharacteristic
325C22×C4#9911yescharacteristic
326C22×C4#9911yescharacteristic
327C22×C4C2×C4211yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C4⋊C411yescharacteristic
330C22×C4C2×C4⋊C411yescharacteristic
331M16—84no—
332C2×C8—44no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—84no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—416no—
347C22⋊C4—48no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—84no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—44no—
362C42—44no—
363C42—44no—
364C42—44no—
365C42—82no—
366C42—82no—
367C42—84no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—21no—
373C23—24no—
374C23—24no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—42no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—44no—
401C23—44no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23—82no—
412C23H(3,ℤ/4)12yes—
413C23#33912yes—
414C23C2×#9511yescharacteristic
415C23C2×#9911yescharacteristic
416C23C2×#9911yescharacteristic
417C23C2×#10412yes—
418D4—44no—
419D4—44no—
420D4—44no—
421D4—44no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—28no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—432no—
453C2×C4—41no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4#33912yes—
490C2×C4#34211yescharacteristic
491C2×C4#34211yescharacteristic
492C2×C4#34211yescharacteristic
493C2×C4#34211yescharacteristic
494C2×C4AΣL(1,𝔽16)12yes—
495C2×C4#35212yes—
496C8—84no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—48no—
515C22—42no—
516C22—44no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22—82no—
530C22—84no—
531C22—82no—
532C22—81no—
533C22C2×H(3,ℤ/4)11yescharacteristic
534C22C2×#33911yescharacteristic
535C22C2×#34211yescharacteristic
536C22C2×AΣL(1,𝔽16)11yescharacteristic
537C22C2×#35211yescharacteristic
538C22#212612yes—
539C4—22no—
540C4—21no—
541C4—21no—
542C4—48no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—84no—
551C4—164no—
552C4#139012yes—
553C4#139012yes—
554C2—21no—
555C2—22no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—42no—
560C2—42no—
561C2—42no—
562C2—81no—
563C2#794211yescharacteristic
564C2C2×#139011yescharacteristic
565C2#1258111yescharacteristic
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
6. C23×C42C4semidirect
11. C2×C4×C22⋊C4C4semidirect
14. #1799C4semidirect
17. C4×#99C4semidirect
19. #1466C4semidirect
23. #1460C4semidirect
25. #1436C4semidirect
81. #342C2×C4semidirect
85. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#173642
C4×#3862
#72132