Finite Group Databasebeta

← #201308

Subgroups — #201308

Order 512.

1135 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic
3#21829C211yesmaximal, characteristic
4#17590C211yesmaximal, characteristic
5#17562C211yesmaximal, characteristic
6#9460C218yesmaximal
7#7941C211yesmaximal, characteristic
8#7940C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2862C412yescomplemented
14#2862C412yescomplemented
15#2642—22no—
16#2568—22no—
17#2568C414yescomplemented
18#2503—22no—
19#2503C2214yes—
20#2381—22no—
21#2365—22no—
22#2345—22no—
23Q8×C22⋊C4—22no—
24#2331—22no—
25#2327—22no—
26#2313C2214yes—
27#2310—22no—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1997—22no—
32#1997—22no—
33#1821C2214yes—
34#1518—22no—
35#1515—22no—
36#1511C2214yes—
37#1511C2214yes—
38#1506—22no—
39#1503—22no—
40#1284C2214yes—
41#1284C2214yes—
42C23×Q8D412yes—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8C2312yes—
68C2×C4×Q8—24no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C412yes—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C42C2311yescharacteristic
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#423—21no—
85#423—21no—
86#407C2312yes—
87#407C2312yes—
88C4×M16—21no—
89C4×M16—21no—
90#400—28no—
91#400—28no—
92#391—24no—
93#391—28no—
94#391—28no—
95#391—24no—
96#391—24no—
97#391—24no—
98#391C2×C418yes—
99#389—24no—
100#387—28no—
101#387—28no—
102#387—28no—
103#383—21no—
104#383D412yes—
105#382—24no—
106#382—21no—
107#382C2×C418yes—
108#382D412yes—
109#381—28no—
110#381C2312yes—
111#381C2312yes—
112#380—24no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4C2312yes—
125#376—21no—
126#376—21no—
127#376D412yes—
128C43C2×C412yes—
129#343—28no—
130#328—28no—
131#328—28no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—48no—
138C22×Q8C22⋊C414yes—
139C22×Q8C22⋊C414yes—
140C22×Q8C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D414yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C2×M16—22no—
146C2×M16—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171#125—41no—
172#125—41no—
173#125—41no—
174#125—41no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—88no—
192#117—44no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4C22⋊C414yes—
215C2×C4⋊C4C22⋊C414yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—28no—
229C2×C42C22×C414yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C4⋊C8—41no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250#97—41no—
251#97—41no—
252C4×C8—41no—
253C4×C8—41no—
254#95—28no—
255#95—28no—
256#95—22no—
257#95—216no—
258#95—216no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95C2×D414yes—
267#95C2×D414yes—
268C24C22×D411yescharacteristic
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—48no—
276C2×Q8—48no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—48no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—84no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—48no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C418yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315M16—44no—
316M16—44no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—44no—
342C4⋊C4—88no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C42—44no—
349C42—44no—
350C42—44no—
351C42—44no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—88no—
356C42C4×D411yescharacteristic
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42C4×D411yescharacteristic
360C42C4×D411yescharacteristic
361C42C4×D411yescharacteristic
362C42C4×D411yescharacteristic
363C42C4×D411yescharacteristic
364C42#12111yescharacteristic
365C42#12111yescharacteristic
366C42#12111yescharacteristic
367C42#12111yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C23—22no—
373C23—22no—
374C23—41no—
375C23C22×C22⋊C411yescharacteristic
376C23C2×#12011yescharacteristic
377C23C2×#12311yescharacteristic
378Q8—48no—
379Q8—48no—
380Q8—84no—
381Q8—84no—
382Q8—82no—
383Q8—82no—
384Q8—82no—
385Q8—82no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4#38611yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#45514yes—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C8—82no—
434C8—82no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—44no—
443C22C2×#38611yescharacteristic
444C22#288412yes—
445C22C2×#45512yes—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—48no—
456C4—84no—
457C4—84no—
458C4—84no—
459C4—82no—
460C4—82no—
461C4#188512yes—
462C4#188512yes—
463C4#191212yes—
464C2—22no—
465C2—22no—
466C2—41no—
467C2C2×#188511yescharacteristic
468C2#2098611yescharacteristic
469C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2862C4semidirect
14. #2862C4semidirect
17. #2568C4semidirect

Maximal subgroups

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

subgroupindex
#353202
#343712
#218292
#175902
#175622
#94602
#79412
#79402