Finite Group Databasebeta

← #374480

Subgroups — #374480

Order 512.

1327 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60125C211yesmaximal, characteristic
2#34947C211yesmaximal, characteristic
3#21329C212yesmaximal, complemented
4C4×#410C211yesmaximal, characteristic
5#11160C214yesmaximal
6#10893C214yesmaximal, complemented
7#10852C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2925—24no—
11#2925—24no—
12#2921—22no—
13#2920—22no—
14C2×#423C2212yes—
15C2×#410C2212yes—
16#2879C2212yes—
17#2873C2212yes—
18#2568C411yescharacteristic, complemented
19#2567C411yescharacteristic, complemented
20#2566C412yescomplemented
21#2565C412yescomplemented
22#2564—22no—
23#2563C411yescharacteristic, complemented
24#2542C411yescharacteristic, complemented
25#2417—22no—
26#2396—22no—
27C4×C22⋊Q8—22no—
28#2297C2212yes—
29#2285C2212yes—
30C4×#117C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1995—24no—
33#1994—24no—
34C4×#99—22no—
35C4×#99—22no—
36#1740C2212yes—
37#1733C2218yes—
38C2×#339C2218yes—
39C22×#41C2311yescharacteristic
40C22×M16C2314yes—
41#533—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125C2×C412yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×#117D412yes—
64C2×#117C2314yes—
65C2×#117C2312yes—
66C2×#117C2311yescharacteristic
67C22×C4⋊C4—21no—
68C22×C42D412yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#433—216no—
72#432—24no—
73#432—24no—
74#431—24no—
75#431—24no—
76#423—22no—
77#423—22no—
78#423D414yes—
79#423D414yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82#410—28no—
83C2×#99—24no—
84#408—28no—
85#407—24no—
86#405—28no—
87C4×M16—24no—
88#400—22no—
89#399—22no—
90#398—24no—
91#396—24no—
92#395—22no—
93#393—22no—
94#391—22no—
95#391C2×C412yes—
96#391C2×C412yes—
97#390C2×C412yes—
98#390C2×C412yes—
99#389—22no—
100#387—24no—
101#386—24no—
102#385—24no—
103#384C2×C412yes—
104#384C2×C412yes—
105#382C2×C412yes—
106#382C2×C412yes—
107#380—22no—
108#380—24no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C22⋊C4—24no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120#376—22no—
121C2×#95—21no—
122C43—22no—
123#342—24no—
124#339—216no—
125#339—216no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—24no—
129C2×#41—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8C22×C411yescharacteristic
133C22×D4C22×C411yescharacteristic
134C22×D4C2×D412yes—
135C23×C4—22no—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C2×M16—24no—
143C2×M16—28no—
144C2×M16—28no—
145C2×M16C2×D418yes—
146C22×C8—24no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151#125—24no—
152#125—24no—
153C22⋊Q8—48no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—42no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—42no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176#117—22no—
177#117—22no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117—24no—
183#117—24no—
184#117—24no—
185#117#4118yes—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C2×D412yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—28no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C22⋊C412yes—
221C2×C42C22⋊C412yes—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237#99—216no—
238C22⋊C8—28no—
239C22⋊C8—28no—
240#97—28no—
241C4×C8—28no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—42no—
250#95—42no—
251C24C2×C22⋊C412yes—
252C24C22×D411yescharacteristic
253#41—44no—
254#41—44no—
255#41—44no—
256#41—44no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—82no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—81no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4C2×#4114yes—
307C22×C42−1+414yes—
308M16—216no—
309M16—44no—
310M16—44no—
311C2×C8—28no—
312C2×C8—48no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—84no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—28no—
336C22⋊C4—48no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—84no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—24no—
347C42—24no—
348C42—28no—
349C42—42no—
350C42—42no—
351C42—44no—
352C42—44no—
353C42—84no—
354C42—84no—
355C42C4×D412yes—
356C42C4×D412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12011yescharacteristic
360C42#12011yescharacteristic
361C42#12012yes—
362C42#12311yescharacteristic
363C42#12311yescharacteristic
364C42#12312yes—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23C22×C22⋊C411yescharacteristic
373C23C2×#12112yes—
374C23#52512yes—
375C23#53612yes—
376Q8—44no—
377Q8—82no—
378Q8—84no—
379Q8—82no—
380D4—48no—
381D4—44no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—88no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4#38611yescharacteristic
418C2×C4#38611yescharacteristic
419C2×C4#38611yescharacteristic
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#38611yescharacteristic
423C2×C4C2×C4×D412yes—
424C2×C4C4×#4112yes—
425C2×C4#51912yes—
426C2×C4C2×#12011yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C2×C4#53212yes—
429C2×C4#53612yes—
430C2×C4#54714yes—
431C2×C4#54814yes—
432C2×C4#55214yes—
433C2×C4#55614yes—
434C8—48no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—22no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—82no—
446C22C2×#38611yescharacteristic
447C22#229711yescharacteristic
448C22#231011yescharacteristic
449C22#241514yes—
450C4—21no—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—22no—
455C4—44no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—82no—
460C4—88no—
461C4—82no—
462C4—82no—
463C4#188412yes—
464C4#236214yes—
465C4#243212yes—
466C4#249912yes—
467C2—21no—
468C2—24no—
469C2—21no—
470C2—82no—
471C2#1413811yescharacteristic
472C2C2×#188411yescharacteristic
473C2#2094711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21329C2semidirect
6. #10893C2semidirect
7. #10852C2semidirect
18. #2568C4semidirect
19. #2567C4semidirect
20. #2566C4semidirect
21. #2565C4semidirect
23. #2563C4semidirect
24. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#601252
#349472
#213292
C4×#4102
#111602
#108932
#108522