Finite Group Databasebeta

← #190413

Subgroups — #190413

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2283C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4#8711C214yesmaximal, complemented
5#8707C212yesmaximal, complemented
6#8706C212yesmaximal, complemented
7#7895C214yesmaximal, complemented
8C23×C42C2211yescharacteristic
9C22×C4⋊C8C2212yes—
10C2×#407—21no—
11C22×C22⋊C8—21no—
12C2×C4×M16C2212yes—
13#2283—21no—
14#2283—21no—
15#2283—21no—
16#2283—21no—
17#2283C412yes—
18#2283C412yes—
19#2283C2212yes—
20#2283C2212yes—
21C2×#383—21no—
22C2×#382C411yescharacteristic
23C2×#382C411yescharacteristic
24C2×#381—21no—
25C2×#380C411yescharacteristic
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—21no—
30C2×#376—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1749—21no—
36#1749—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1728—21no—
40C2×#336—21no—
41#1279C2218yescomplemented
42#1279C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—22no—
47C23×C8—22no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42—28no—
55C22×C42—21no—
56C22×C42—21no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—28no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8C2314yes—
72#407—28no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—44no—
78C4×M16—28no—
79C2×#97—22no—
80C2×C4×C8C2314yes—
81#383—22no—
82#383—22no—
83#383—22no—
84#383—22no—
85#382—21no—
86#382—21no—
87#382—21no—
88#382—21no—
89#382C2×C412yes—
90#382C2×C412yes—
91#382C2×C412yes—
92#382C2×C412yes—
93#381—22no—
94#381—22no—
95#381—22no—
96#381—22no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380C2×C412yes—
102#380C2×C412yes—
103#380C2×C412yes—
104#380C2×C412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117#376—22no—
118#376—22no—
119#376—22no—
120#376—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C25C22⋊C411yescharacteristic
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—28no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8—24no—
170C22×C8—24no—
171C22×C8—44no—
172C22×C8—44no—
173C22×C8—44no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—44no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4118yes—
226C2×C42C2411yescharacteristic
227C4⋊C8—28no—
228C4⋊C8—44no—
229C22⋊C8—28no—
230C22⋊C8—28no—
231C22⋊C8—44no—
232C22⋊C8—44no—
233#97—44no—
234C4×C8—28no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24#10414yes—
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C411yescharacteristic
260C24C2×C22⋊C411yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—416no—
319C22×C4—416no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4#11714yes—
342C22×C4#12314yes—
343C22×C4#12314yes—
344C22×C4#13114yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4#13712yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350M16—416no—
351C2×C8—28no—
352C2×C8—44no—
353C2×C8—44no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—88no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—44no—
383C42—42no—
384C42—42no—
385C42—44no—
386C42—44no—
387C42#11714yes—
388C42#12314yes—
389C42#12314yes—
390C42C2×#4114yes—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—28no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—28no—
406C23—21no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23#38112yes—
413C23#40812yes—
414C23#41712yes—
415C23C2×#10412yes—
416C23C2×#10412yes—
417C23C2×#10412yes—
418C23#42112yes—
419C23C22×C22⋊C411yescharacteristic
420C2×C4—28no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4#38112yes—
482C2×C4#38112yes—
483C2×C4#38112yes—
484C2×C4#40812yes—
485C2×C4#41712yes—
486C2×C4C2×#11712yes—
487C2×C4C2×#12312yes—
488C2×C4C2×#12312yes—
489C2×C4C2×#13112yes—
490C2×C4#56812yes—
491C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
492C2×C4C2×#13711yescharacteristic
493C2×C4#57512yes—
494C8—48no—
495C8—84no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—416no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22#148314yes—
517C22#182614yes—
518C22C2×#38111yescharacteristic
519C22C2×#40811yescharacteristic
520C22#287911yescharacteristic
521C22C2×#41711yescharacteristic
522C22#289611yescharacteristic
523C22C22×#10411yescharacteristic
524C22C2×#42111yescharacteristic
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4#292414yes—
540C4#309412yes—
541C4#309512yes—
542C2—24no—
543C2—24no—
544C2—42no—
545C2C2×#148311yescharacteristic
546C2#1746511yescharacteristic
547C2#1749211yescharacteristic
548C2#1750211yescharacteristic
549C2#2068511yescharacteristic
550C2#2070211yescharacteristic
551C2C2×#182611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8711C2semidirect
5. #8707C2semidirect
6. #8706C2semidirect
7. #7895C2semidirect
41. #1279C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22832
C2×#17972
C4×#4072
#87112
#87072
#87062
#78952