Finite Group Databasebeta

← #181734

Subgroups — #181734

Order 512.

1419 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic
2#60125C211yesmaximal, characteristic
3#34121C211yesmaximal, characteristic
4#8096C214yesmaximal, complemented
5#8095C212yesmaximal, complemented
6#8092C214yesmaximal, complemented
7C2×#1303C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2988C418yescomplemented
10#2982—21no—
11#2972—22no—
12#2972—21no—
13C8×#41—24no—
14#2925—24no—
15#2925C418yescomplemented
16#2921—22no—
17#2920C2214yes—
18C2×#431C2212yes—
19C2×#431C2214yes—
20C2×#423C2212yes—
21#2879—21no—
22#2873—21no—
23#2357C412yes—
24#2356C412yes—
25#2355—21no—
26#2354—21no—
27#2353C414yes—
28#2352—22no—
29#2298—22no—
30#2297—21no—
31#2285C2211yescharacteristic
32#2283C2212yes—
33C4×#117—21no—
34C4×#117C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1303C2214yes—
37#1303C2214yes—
38#1303C2218yes—
39C22×#41D411yescharacteristic
40C22×M16—22no—
41C4×#41C2×C414yes—
42C4×#41C2×C414yes—
43C4×#41D414yes—
44C4×#41Q814yes—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8D411yescharacteristic
47C2×C4×D4C2×C412yes—
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D411yescharacteristic
51C2×#117—22no—
52C2×#117—21no—
53C2×#117D411yescharacteristic
54C2×#117D412yes—
55C22×C4⋊C4—21no—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59C8×Q8—28no—
60C8×D4—28no—
61C8×D4—216no—
62#433—216no—
63#432—24no—
64#432C2×C418yescomplemented
65#431—22no—
66#431—24no—
67#431—22no—
68#431—24no—
69#431—24no—
70#431C2×C414yes—
71#431C2×C418yescomplemented
72#431C2×C418yescomplemented
73#431C2314yes—
74#423—22no—
75#423—22no—
76#423C2×C414yes—
77#423C2314yes—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2×C414yes—
80C2×C4⋊C8C2×C414yes—
81C2×C4⋊C8C2×C418yescomplemented
82#408—28no—
83#407—24no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—24no—
86#405—28no—
87C4×M16—24no—
88C2×C4×C8—22no—
89C2×C4×C8—24no—
90#391—22no—
91#390—22no—
92#388—24no—
93#388—24no—
94#387—24no—
95#387—24no—
96#386—24no—
97#384—21no—
98#383C2×C414yes—
99#382—22no—
100#382C2×C412yes—
101#381C2×C414yes—
102#380—21no—
103#380—22no—
104#380C2×C412yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2311yescharacteristic
113C4×C4⋊C4C2311yescharacteristic
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4C2×C414yes—
119#376—22no—
120#376C2×C414yes—
121C2×#95C2×C412yes—
122C43—22no—
123C2×#41—24no—
124C2×#41—24no—
125C2×#41C22⋊C414yes—
126C2×#41C4⋊C414yes—
127C22×Q8C22⋊C411yescharacteristic
128C22×D4C22⋊C412yes—
129C22×D4C22⋊C411yescharacteristic
130C23×C4—22no—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—44no—
140C2×M16—44no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—28no—
144C4×Q8C4214yes—
145C4×Q8C22⋊C414yes—
146C4×Q8C4⋊C414yes—
147C4×Q8C4⋊C414yes—
148C4×D4C4218yes—
149C4×D4C4214yes—
150C4×D4C22⋊C418yes—
151C4×D4C22⋊C414yes—
152C4×D4C4⋊C418yes—
153C4×D4C4⋊C414yes—
154C4×D4C4⋊C418yes—
155C4×D4C4⋊C414yes—
156#117—48no—
157#117—44no—
158#117C22⋊C414yes—
159#117C22⋊C418yes—
160#117C4⋊C418yes—
161#117C4⋊C414yes—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42—44no—
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C412yes—
189C2×C42C22×C412yes—
190C2×C42C22×C414yes—
191C2×C42C22×C414yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D414yes—
198C2×C42C2×Q814yes—
199C2×C42C2×Q812yes—
200C2×C42C2411yescharacteristic
201C4⋊C8—24no—
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8—28no—
205C4⋊C8—24no—
206C4⋊C8C22×C418yes—
207C4⋊C8C22×C418yes—
208C22⋊C8—24no—
209C22⋊C8—28no—
210C22⋊C8—28no—
211C22⋊C8—28no—
212C22⋊C8—24no—
213#97—28no—
214C4×C8—24no—
215C4×C8—28no—
216C4×C8—28no—
217C4×C8—24no—
218#95—22no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95C22×C412yes—
225#95C22×C412yes—
226#95C22×C414yes—
227C24C2×C22⋊C412yes—
228C24C2×C22⋊C411yescharacteristic
229#41—28no—
230#41—28no—
231#41—216no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8#9514yes—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4#9518yes—
240C2×D4#9514yes—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—82no—
251C22×C4—81no—
252C22×C4—82no—
253C22×C4C2×C4214yes—
254C22×C4C2×C4212yes—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C4⋊C414yes—
266C22×C4C2×C4⋊C414yes—
267C22×C4C2×C4⋊C412yes—
268C22×C4C2×C4⋊C414yes—
269C22×C4C2×C4⋊C412yes—
270C22×C4C2×C4⋊C412yes—
271C22×C4#13514yes—
272C22×C4#13514yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—44no—
277M16—44no—
278M16—48no—
279C2×C8—28no—
280C2×C8—28no—
281C2×C8—48no—
282C2×C8—84no—
283C2×C8—88no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—82no—
290C4⋊C4#9518yes—
291C4⋊C4#9514yes—
292C4⋊C4#9518yes—
293C4⋊C4#9514yes—
294C22⋊C4—48no—
295C22⋊C4—44no—
296C22⋊C4—84no—
297C22⋊C4#9518yes—
298C22⋊C4#9518yes—
299C22⋊C4#9518yes—
300C42—48no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—44no—
305C42—84no—
306C42C2×C4214yes—
307C42C2×C4212yes—
308C42C2×C22⋊C414yes—
309C42C2×C22⋊C412yes—
310C42C2×C22⋊C414yes—
311C42C2×C22⋊C412yes—
312C42C2×C4⋊C412yes—
313C42C2×C4⋊C414yes—
314C42C2×C4⋊C414yes—
315C42C2×C4⋊C412yes—
316C42C2×C4⋊C414yes—
317C42C2×C4⋊C412yes—
318C42C23×C411yescharacteristic
319C42C23×C411yescharacteristic
320C42C22×D411yescharacteristic
321C42C22×Q811yescharacteristic
322C23—22no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23C2×#9514yes—
329C23C2×#9512yes—
330C23#41612yes—
331C23#41612yes—
332C23C22×C22⋊C411yescharacteristic
333Q8—28no—
334D4—216no—
335D4—28no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—88no—
364C2×C4C2×#9514yes—
365C2×C4C2×#9512yes—
366C2×C4C2×#9514yes—
367C2×C4C2×#9514yes—
368C2×C4C2×#9512yes—
369C2×C4C2×#9512yes—
370C2×C4#41614yes—
371C2×C4#41614yes—
372C2×C4#41612yes—
373C2×C4#41612yes—
374C2×C4#42718yes—
375C2×C4C22×C4211yescharacteristic
376C2×C4C22×C22⋊C411yescharacteristic
377C2×C4C22×C22⋊C411yescharacteristic
378C2×C4C22×C4⋊C411yescharacteristic
379C2×C4C22×C4⋊C411yescharacteristic
380C2×C4C22×C4⋊C411yescharacteristic
381C2×C4C2×#13512yes—
382C2×C4C2×#13512yes—
383C8—48no—
384C8—48no—
385C22—21no—
386C22—22no—
387C22—21no—
388C22—22no—
389C22—24no—
390C22—24no—
391C22—24no—
392C22—42no—
393C22—42no—
394C22—82no—
395C22—82no—
396C22#173514yes—
397C22C22×#9511yescharacteristic
398C22C2×#41611yescharacteristic
399C22C2×#41611yescharacteristic
400C4—21no—
401C4—24no—
402C4—22no—
403C4—24no—
404C4—21no—
405C4—21no—
406C4—24no—
407C4—22no—
408C4—44no—
409C4—84no—
410C4—82no—
411C4#173514yes—
412C4C2×#41612yes—
413C4#290112yes—
414C4C2×#42712yes—
415C2—21no—
416C2—21no—
417C2—24no—
418C2—44no—
419C2#2034311yescharacteristic
420C2C2×#173511yescharacteristic
421C2#2036011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C22×#95
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8096C2semidirect
5. #8095C2semidirect
6. #8092C2semidirect
7. C2×#1303C2semidirect
9. #2988C4semidirect
15. #2925C4semidirect
64. #432C2×C4semidirect
71. #431C2×C4semidirect
72. #431C2×C4semidirect
81. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602762
#601252
#341212
#80962
#80952
#80922
C2×#13032