Finite Group Databasebeta

← #201461

Subgroups — #201461

Order 512.

1265 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34971C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic, complemented
3#20747C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9482C214yesmaximal, complemented
6#9480C214yesmaximal
7#7940C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920C2212yes—
11#2599—22no—
12#2581C412yescomplemented
13#2536C412yescomplemented
14#2486—21no—
15#2484C2212yes—
16#2409—21no—
17#2406C2212yes—
18#2399C412yescomplemented
19#2398C412yescomplemented
20#2320—24no—
21#2316—21no—
22#2313C2214yes—
23#2310—21no—
24C4×C22⋊Q8—22no—
25C4×#123—22no—
26#2295—24no—
27#2283—22no—
28#2283C2211yescharacteristic
29C4×#117C2212yes—
30#1826C2214yes—
31#1806—24no—
32#1789C2214yes—
33C4×#104—24no—
34#1467—22no—
35#1463C2214yes—
36#1461—22no—
37#1284C2218yes—
38C22×#41C2311yescharacteristic
39C23×Q8D411yescharacteristic
40C2×#123—22no—
41C2×C22⋊Q8—22no—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×#117—22no—
61C2×#117—21no—
62C2×#117C2311yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#432—22no—
69#432—22no—
70#431—22no—
71#431—22no—
72#423—22no—
73C2×#104—28no—
74#407C2314yes—
75C4×M16—22no—
76#400C2×C414yes—
77#398—22no—
78#398C2×C414yes—
79#397—22no—
80#395—22no—
81#395C2×C414yes—
82#394C2×C414yes—
83#393—22no—
84#391—28no—
85#391—22no—
86#389—28no—
87#389—22no—
88#388—24no—
89#387—24no—
90#387—24no—
91#387—216no—
92#386—22no—
93#385—22no—
94#385—24no—
95#383—24no—
96#383—21no—
97#382—24no—
98#382C2×C412yes—
99#381—24no—
100#381—22no—
101#381C2312yes—
102#380—24no—
103#380—21no—
104#380—22no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D414yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—24no—
113C4×C22⋊C4C2314yes—
114#376—24no—
115#376—22no—
116#376C2×C412yes—
117C43D414yes—
118#339—28no—
119#324—28no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—44no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—44no—
126C22×Q8C22⋊C412yes—
127C22×Q8C22⋊C412yes—
128C22×Q8C22×C411yescharacteristic
129C22×D4—21no—
130C22×D4C22×C411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C2×M16—24no—
140#123—48no—
141C22⋊Q8—48no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—42no—
159C4×D4—41no—
160C4×D4—41no—
161C4×D4—42no—
162C4×D4—41no—
163C4×D4—41no—
164#117—22no—
165#117—22no—
166#117—48no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C22⋊C414yes—
206C2×C42C22⋊C414yes—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D418yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216C4⋊C8—42no—
217C4⋊C8—42no—
218#104—416no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221#97—42no—
222C4×C8—42no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—22no—
227#95—24no—
228#95—22no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95C22×C414yes—
236C24—21no—
237C24C22×D411yescharacteristic
238#41—28no—
239#41—44no—
240#41—44no—
241#41—44no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—28no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—44no—
279C22×C4C2×C22⋊C414yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C414yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4#12012yes—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12312yes—
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4112yes—
295M16—48no—
296C2×C8—44no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—88no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—48no—
325C42—48no—
326C42—44no—
327C42—48no—
328C42—42no—
329C42—41no—
330C42—42no—
331C42—41no—
332C42C4×D412yes—
333C42C4×D412yes—
334C42C4×D412yes—
335C42C4×D412yes—
336C42#12112yes—
337C42#12112yes—
338C42#12112yes—
339C42#12112yes—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23C22×C22⋊C411yescharacteristic
349C23C2×#12011yescharacteristic
350C23C2×#12311yescharacteristic
351Q8—24no—
352Q8—48no—
353Q8—42no—
354Q8—48no—
355D4—24no—
356D4—42no—
357D4—44no—
358D4—44no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—48no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4#38611yescharacteristic
399C2×C4#38611yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#38612yes—
402C2×C4#38612yes—
403C2×C4#38611yescharacteristic
404C2×C4#42114yes—
405C2×C4C2×C4×D412yes—
406C2×C4C2×#12112yes—
407C8—84no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—44no—
416C22—44no—
417C22—42no—
418C22—82no—
419C22—81no—
420C22—81no—
421C22C2×#38611yescharacteristic
422C22C2×#42112yes—
423C22#302512yes—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—42no—
431C4—44no—
432C4—44no—
433C4—88no—
434C4—84no—
435C4—81no—
436C4—81no—
437C4#189714yes—
438C4#191212yes—
439C2—22no—
440C2—22no—
441C2—41no—
442C2—82no—
443C2#2095911yescharacteristic
444C2C2×#189711yescharacteristic
445C2C2×#191211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34059C2semidirect
4. #17365C2semidirect
5. #9482C2semidirect
7. #7940C2semidirect
12. #2581C4semidirect
13. #2536C4semidirect
18. #2399C4semidirect
19. #2398C4semidirect

Maximal subgroups

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

subgroupindex
#349712
#340592
#207472
#173652
#94822
#94802
#79402