Finite Group Databasebeta

← #480395

Subgroups — #480395

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#17365C212yesmaximal, complemented
3#11536C212yesmaximal, complemented
4#11534C212yesmaximal, complemented
5#11514C212yesmaximal, complemented
6#11508C212yesmaximal
7#11478C212yesmaximal, complemented
8#11476C212yesmaximal, complemented
9C2×#536C411yescharacteristic
10C2×C4×#41C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2920—22no—
13#2327C411yescharacteristic
14#2327C2211yescharacteristic
15#2326C411yescharacteristic
16#2326C2211yescharacteristic
17#2323—22no—
18#2322—22no—
19#2317C2211yescharacteristic
20#2311C2211yescharacteristic
21#2310C411yescharacteristic
22C4×C4⋊Q8C411yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×C22⋊Q8—22no—
27C4×#123—22no—
28C4×#117C411yescharacteristic
29#1467—22no—
30#1467C2214yes—
31#1463—22no—
32#1463C2214yes—
33#1461—22no—
34#1461C2214yes—
35#1325C2214yes—
36#1324C2214yes—
37#1319C2214yes—
38#1318C2214yes—
39C22×#41C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41#536—24no—
42#536—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49C4×#41—42no—
50C4×#41—42no—
51C2×C4×Q8—42no—
52C2×C4×Q8—42no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×#117—21no—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42C2×C411yescharacteristic
73C22×C42D412yes—
74#432—24no—
75#432—42no—
76#431—24no—
77#431—42no—
78#423—42no—
79#407—24no—
80C4×M16—42no—
81#391—22no—
82#391C2×C411yescharacteristic
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2311yescharacteristic
88#390C2×C411yescharacteristic
89#390C2311yescharacteristic
90#389—22no—
91#389—22no—
92#389—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#384C2×C411yescharacteristic
106#384C2311yescharacteristic
107#382C2×C411yescharacteristic
108#382C2311yescharacteristic
109#381—22no—
110#380—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123#324—24no—
124#324—24no—
125#324—24no—
126#324C2318yes—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—42no—
130C2×#41—42no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8C22⋊C412yes—
136C22×Q8C22×C411yescharacteristic
137C22×Q8C22×C411yescharacteristic
138C22×Q8C22×C411yescharacteristic
139C22×D4—21no—
140C22×D4C22×C411yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C2×M16—44no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8M1614yescomplemented
154#124—22no—
155#124—22no—
156#124—22no—
157#124M1614yescomplemented
158#123—216no—
159C22⋊Q8—216no—
160C4×Q8—41no—
161C4×Q8—42no—
162C4×Q8—41no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—41no—
169C4×Q8—41no—
170C4×Q8—42no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—42no—
176C4×D4—44no—
177C4×D4—42no—
178C4×D4—42no—
179#117—22no—
180#117—22no—
181#117—44no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C412yes—
230C2×C42C22⋊C412yes—
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C4⋊C8—44no—
241C4⋊C8—44no—
242C22⋊C8—28no—
243C22⋊C8—44no—
244#97—44no—
245C4×C8—44no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258C24—21no—
259C24C2×C22⋊C411yescharacteristic
260#41—48no—
261#41—48no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8C2×M1612yes—
277C2×Q8C2×M1612yes—
278C2×Q8C2×M1612yes—
279C2×D4—24no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4C2×M1612yes—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—81no—
315C22×C4#9912yes—
316C22×C4#9912yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D412yes—
327M16—88no—
328C2×C8—48no—
329C4⋊C4—24no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—24no—
347C22⋊C4—28no—
348C22⋊C4—28no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C42—21no—
356C42—21no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—21no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—21no—
367C42—22no—
368C42—21no—
369C42—22no—
370C42—21no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—41no—
378C42—41no—
379C42—81no—
380C42—81no—
381C42C2×M1612yes—
382C42C2×M1612yes—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—41no—
391C23—41no—
392C23C2×#9912yes—
393C23C22×C22⋊C411yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—48no—
397Q8—42no—
398Q8—42no—
399Q8—42no—
400Q8—42no—
401D4—42no—
402D4—42no—
403D4—48no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—84no—
454C2×C4—82no—
455C2×C4#40714yes—
456C2×C4C2×#9911yescharacteristic
457C2×C4C2×#9912yes—
458C2×C4C2×#9911yescharacteristic
459C2×C4#41312yes—
460C2×C4C22×M1611yescharacteristic
461C2×C4C22×M1611yescharacteristic
462C8—88no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—42no—
469C22—42no—
470C22—44no—
471C22—42no—
472C22—82no—
473C22—81no—
474C22—81no—
475C22C2×#40711yescharacteristic
476C22C22×#9911yescharacteristic
477C22C2×#41311yescharacteristic
478C4—21no—
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—24no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—42no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4#213312yes—
497C2—21no—
498C2—21no—
499C2—42no—
500C2—82no—
501C2#1736211yescharacteristic
502C2#3335811yescharacteristic
503C2C2×#213311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17365C2semidirect
3. #11536C2semidirect
4. #11534C2semidirect
5. #11514C2semidirect
7. #11478C2semidirect
8. #11476C2semidirect
153. C4⋊Q8M16semidirect
157. #124M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#173652
#115362
#115342
#115142
#115082
#114782
#114762