Finite Group Databasebeta

← #214520

Subgroups — #214520

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#9137C214yesmaximal, complemented
5#9131C214yesmaximal, complemented
6#8725C212yesmaximal, complemented
7#8340C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C23×C42C2211yescharacteristic
11C2×#423C2212yes—
12C22×C4⋊C8C2212yes—
13C2×#407—21no—
14C22×C22⋊C8—21no—
15#2857—21no—
16#2857—21no—
17#2857—21no—
18#2857—21no—
19#2857C412yes—
20#2857C412yes—
21#2857C2212yes—
22#2857C2212yes—
23C2×#391—21no—
24C2×#389—21no—
25C2×#389C412yes—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380C412yes—
29#1797—21no—
30#1797—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36#1797C2212yes—
37#1728—21no—
38C2×#336—21no—
39C2×#332C2218yescomplemented
40#1296C2218yescomplemented
41C23×Q8—21no—
42C24×C4C2×C411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C22×M16—22no—
46C23×C8—22no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—44no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2312yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—48no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C22⋊C4—22no—
64C22×C42—44no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#423—44no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—44no—
81C2×C4⋊C8C2314yes—
82C2×C4⋊C8C2314yes—
83#407—44no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—44no—
89#391—22no—
90#391—22no—
91#391—22no—
92#391—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389C2×C414yes—
100#389C2×C414yes—
101#384—21no—
102#384—21no—
103#384C2×C412yes—
104#384C2312yes—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380C2×C414yes—
112#380C2×C414yes—
113C2×#95—22no—
114C2×#95—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#332—48no—
120C25C22⋊C411yescharacteristic
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—42no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4SD1614yes—
148C23×C4Q1614yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—44no—
161C22×C8—24no—
162C22×C8—24no—
163C22×C8—44no—
164C22×C8—44no—
165C22×C8—44no—
166C4⋊Q8—28no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C22⋊Q8—48no—
171C22⋊Q8—48no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C22×C414yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—48no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—28no—
224C4⋊C8—28no—
225C4⋊C8—44no—
226C4⋊C8—44no—
227C22⋊C8—44no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230C22⋊C8—44no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24#10314yes—
242C24C2×C22⋊C411yescharacteristic
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—44no—
252C2×Q8—88no—
253C2×Q8—88no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—416no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4#10114yes—
315C22×C4#10314yes—
316C22×C4#10314yes—
317C22×C4#10314yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×SD1612yes—
328C22×C4C2×SD1612yes—
329C22×C4C2×SD1612yes—
330C22×C4C2×Q1612yes—
331C22×C4C2×Q1612yes—
332C22×C4C2×Q1612yes—
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4AGL(1,ℤ/8)14yes—
335C22×C4#13712yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339M16—88no—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—44no—
344C2×C8—44no—
345C2×C8—44no—
346C2×C8—88no—
347C4⋊C4—28no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42H(5,𝔽2)14yes—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—28no—
387C23—28no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23C2×#10112yes—
399C23#41312yes—
400C23C2×#10312yes—
401C23C2×#10312yes—
402C23C2×#10312yes—
403C23#41712yes—
404C23#41812yes—
405C23C22×C22⋊C411yescharacteristic
406Q8—88no—
407Q8—84no—
408Q8—84no—
409C2×C4—28no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—88no—
462C2×C4—88no—
463C2×C4C2×#10112yes—
464C2×C4C2×#10112yes—
465C2×C4C2×#10312yes—
466C2×C4C2×#10312yes—
467C2×C4C2×#10312yes—
468C2×C4#41712yes—
469C2×C4#41812yes—
470C2×C4#51812yes—
471C2×C4#53412yes—
472C2×C4#53412yes—
473C2×C4C22×SD1611yescharacteristic
474C2×C4C22×Q1611yescharacteristic
475C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)12yes—
477C2×C4C2×#13711yescharacteristic
478C2×C4#57512yes—
479C8—84no—
480C8—84no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—416no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22#154914yes—
502C22#155014yes—
503C22#230811yescharacteristic
504C22C22×#10111yescharacteristic
505C22C2×#41311yescharacteristic
506C22C22×#10311yescharacteristic
507C22C2×#41811yescharacteristic
508C22C2×#41711yescharacteristic
509C22#289611yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—84no—
521C4—84no—
522C4#318712yes—
523C4#318912yes—
524C4#319212yes—
525C4#319312yes—
526C2—24no—
527C2—24no—
528C2—42no—
529C2C2×#154911yescharacteristic
530C2C2×#155011yescharacteristic
531C2#1773111yescharacteristic
532C2#1775111yescharacteristic
533C2#2175311yescharacteristic
534C2#2175911yescharacteristic
535C2#2176411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9137C2semidirect
5. #9131C2semidirect
6. #8725C2semidirect
7. #8340C2semidirect
39. C2×#332C22semidirect
40. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#206262
C2×#17972
#91372
#91312
#87252
#83402