Finite Group Databasebeta

← #201145

Subgroups — #201145

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2C2×#2545C211yesmaximal, characteristic
3#17666C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9448C218yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic
8C22×#124C411yescharacteristic
9C22×#124C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2984—24no—
12#2983—21no—
13#2983C2214yescomplemented
14#2982—21no—
15Q8×M16—24no—
16#2920—22no—
17#2920C2212yes—
18#2920C2212yes—
19#2545—21no—
20#2545—21no—
21#2545—21no—
22#2545—21no—
23#2545C414yes—
24#2545C2214yes—
25C2×#396—22no—
26C2×#391C411yescharacteristic
27C2×#390C411yescharacteristic
28C2×#388—22no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1533C2218yes—
31#1527—24no—
32#1527—24no—
33#1467C2214yescomplemented
34#1463—22no—
35#1461—21no—
36#1461—21no—
37#1285C2218yescomplemented
38C23×Q8C2×C411yescharacteristic
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—44no—
44C2×C4⋊Q8C2×C414yes—
45C2×#124—21no—
46C2×#124—21no—
47C2×#124—21no—
48C2×#124—21no—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—44no—
54C2×#124C2×C412yes—
55C2×#124C2×C414yes—
56C2×#124C2312yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2312yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D412yes—
78C22×C42C2311yescharacteristic
79#446—28no—
80#446—216no—
81C8×Q8—28no—
82#432—24no—
83#432—24no—
84#432—22no—
85#432D414yes—
86#432C2314yes—
87#432C2314yes—
88#431—24no—
89#431—24no—
90#431—22no—
91#431—22no—
92#431—22no—
93#431—22no—
94#423—24no—
95#423—22no—
96#423—22no—
97#407—22no—
98#407—22no—
99C4×M16—24no—
100C4×M16—22no—
101C4×M16D414yes—
102#396—24no—
103#396—24no—
104#396—24no—
105#396—24no—
106#391—21no—
107#391—21no—
108#391C2×C414yes—
109#390—21no—
110#390—21no—
111#390C2×C414yes—
112#388—24no—
113#388—24no—
114#388—24no—
115#388—24no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2312yes—
120C2×#95—22no—
121C2×#95—21no—
122#329—216no—
123#324—24no—
124#324—24no—
125C25—21no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8C22⋊C411yescharacteristic
136C22×Q8C22⋊C411yescharacteristic
137C22×Q8C22×C412yes—
138C22×Q8C22×C412yes—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4C22×C412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—24no—
154C2×M16—42no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—48no—
158#124—24no—
159#124—24no—
160#124—24no—
161#124—24no—
162#124—44no—
163#124—44no—
164#124—48no—
165#124—42no—
166#124—42no—
167#124—42no—
168#124—42no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—24no—
180C4×Q8—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—48no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42C22⋊C412yes—
223C2×C42C22⋊C412yes—
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236C4⋊C8—28no—
237C4⋊C8—28no—
238C4⋊C8—24no—
239C4⋊C8—24no—
240C4⋊C8—24no—
241C4⋊C8—42no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—42no—
246#97—28no—
247#97—24no—
248#97C2×D418yes—
249C4×C8—28no—
250C4×C8—24no—
251C4×C8—42no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—22no—
256C24—22no—
257C24—42no—
258C24—44no—
259C24—42no—
260C24—44no—
261C24—41no—
262C24—41no—
263C24C2×C22⋊C411yescharacteristic
264C2×Q8—24no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—42no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8#10014yes—
283C2×D4—24no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—84no—
290C2×D4—84no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12311yescharacteristic
329C22×C4#12311yescharacteristic
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333M16—48no—
334M16—84no—
335C2×C8—28no—
336C2×C8—44no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—48no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42C4×D412yes—
371C42C4×D412yes—
372C42#12012yes—
373C42#12112yes—
374C42#12112yes—
375C42#12312yes—
376C42C22×D412yes—
377C42C2×#4112yes—
378C23—21no—
379C23—21no—
380C23—22no—
381C23—22no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—82no—
388C23—82no—
389C23—88no—
390C23—88no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23#38611yescharacteristic
396C23#38611yescharacteristic
397C23C22×C22⋊C411yescharacteristic
398Q8—28no—
399Q8—48no—
400Q8—44no—
401Q8—84no—
402Q8—84no—
403D4—84no—
404D4—84no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#38611yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38612yes—
448C2×C4#38612yes—
449C2×C4C2×#10014yes—
450C2×C4C2×#10012yes—
451C2×C4#41312yes—
452C2×C4C2×C4×D412yes—
453C2×C4C2×#12011yescharacteristic
454C2×C4C2×#12112yes—
455C2×C4C2×#12311yescharacteristic
456C8—48no—
457C8—84no—
458C22—24no—
459C22—24no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—84no—
467C22—84no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—164no—
472C22—164no—
473C22#188412yes—
474C22#191212yes—
475C22C2×#38611yescharacteristic
476C22C22×#10011yescharacteristic
477C22C2×#41311yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—24no—
486C4—44no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—82no—
491C4—82no—
492C4#190014yes—
493C2—21no—
494C2—21no—
495C2—22no—
496C2—22no—
497C2—162no—
498C2C2×#188411yescharacteristic
499C2C2×#190011yescharacteristic
500C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
4. #17365C2semidirect
5. #9448C2semidirect
6. #7950C2semidirect
13. #2983C22semidirect
33. #1467C22semidirect
37. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25452
#176662
#173652
#94482
#79502