Finite Group Databasebeta

← #181158

Subgroups — #181158

Order 512.

1509 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#60282C211yesmaximal, characteristic
3C4⋊C4×#41C211yesmaximal, characteristic
4#7953C212yesmaximal, complemented
5#7952C214yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C2×#1285C214yesmaximal, complemented
8C2×C4×#41C412yes—
9C2×C4×#41C2211yescharacteristic
10#2990—24no—
11#2990C418yescomplemented
12#2983—21no—
13#2983—21no—
14#2973—22no—
15#2973—22no—
16#2972—21no—
17#2972—21no—
18#2970—24no—
19#2970C418yescomplemented
20#2920C2212yes—
21#2920C2212yes—
22C2×#432C2214yes—
23C2×#432C2214yes—
24#2351—21no—
25Q8×C4⋊C4—21no—
26#2349—21no—
27#2349—22no—
28D4×C4⋊C4—21no—
29D4×C4⋊C4—22no—
30#2296C412yes—
31#2296C414yes—
32#2285C2212yes—
33#2285C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1285C2214yes—
37#1285C2218yes—
38#1285C2214yes—
39C22×#41C2×C411yescharacteristic
40C4×#41—48no—
41C4×#41C2×C414yes—
42C4×#41C2×C414yes—
43C4×#41D414yes—
44C4×#41Q814yes—
45C2×C4×Q8—22no—
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D412yes—
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D412yes—
54C2×#117C2×C412yes—
55C2×#117C2×C414yes—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C412yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C42C2×C414yes—
61C22×C42C2×C412yes—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64#446—28no—
65#446—28no—
66#435—216no—
67#435—216no—
68C8×D4—28no—
69C8×D4—28no—
70#432—24no—
71#432—24no—
72#432—24no—
73#432—24no—
74#432—22no—
75#432C2×C418yescomplemented
76#432C2×C418yescomplemented
77#432C2×C414yescomplemented
78#432C2314yes—
79#432C2314yes—
80#431—22no—
81#431—22no—
82#431—22no—
83#431C2×C414yescomplemented
84#423—22no—
85#423C2×C414yescomplemented
86C2×C4⋊C8—24no—
87C2×C4⋊C8C2×C418yescomplemented
88#407—22no—
89#407—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C4×M16—22no—
93C4×M16C2×C414yescomplemented
94C2×#97—24no—
95C2×#97C2×C418yescomplemented
96#389—24no—
97#389—22no—
98#389—24no—
99#389—22no—
100#385—24no—
101#385—24no—
102#385—24no—
103#384—21no—
104#384—22no—
105#382—21no—
106#382—22no—
107#380—22no—
108#380—21no—
109#380—22no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C414yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C414yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2312yes—
121C4×C4⋊C4C2311yescharacteristic
122C2×#95—22no—
123C2×#95—21no—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41C4214yes—
127C2×#41C4⋊C414yes—
128C22×Q8C22⋊C411yescharacteristic
129C22×D4C22⋊C412yes—
130C22×D4C22⋊C411yescharacteristic
131C23×C4—22no—
132C23×C4—24no—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—24no—
141C22×C8—28no—
142C22×C8—28no—
143C4×Q8—48no—
144C4×Q8C22⋊C414yes—
145C4×Q8C22⋊C414yes—
146C4×Q8C4⋊C414yes—
147C4×Q8C4⋊C414yes—
148C4×D4—416no—
149C4×D4—48no—
150C4×D4C22⋊C414yes—
151C4×D4C22⋊C414yes—
152C4×D4C22⋊C418yes—
153C4×D4C22⋊C418yes—
154C4×D4C4⋊C418yes—
155C4×D4C4⋊C418yes—
156C4×D4C4⋊C414yes—
157C4×D4C4⋊C414yes—
158#117—44no—
159#117—48no—
160#117C4214yes—
161#117C4218yes—
162#117C4⋊C414yes—
163#117C4⋊C418yes—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—48no—
187C2×C42—44no—
188C2×C42C22×C414yes—
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C414yes—
194C2×C42C22×C412yes—
195C2×C42C22×C414yes—
196C2×C42C22×C412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D414yes—
202C2×C42C2×D412yes—
203C2×C42C2×Q814yes—
204C2×C42C2×Q812yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—28no—
207C4⋊C8—28no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8C22×C418yes—
212C22⋊C8—28no—
213C22⋊C8—28no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218#97—28no—
219#97—28no—
220#97—24no—
221#97C22×C418yes—
222C4×C8—24no—
223C4×C8—24no—
224#95—22no—
225#95—24no—
226#95—22no—
227#95—24no—
228#95—21no—
229#95—22no—
230#95—21no—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C412yes—
233#41—216no—
234#41—216no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8#9514yes—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4#9518yes—
241C2×D4#9514yes—
242C2×D4#10014yes—
243C2×D4#10114yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—82no—
258C22×C4—81no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4C2×C4214yes—
262C22×C4C2×C4212yes—
263C22×C4C2×C22⋊C414yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C4⋊C414yes—
274C22×C4C2×C4⋊C412yes—
275C22×C4C2×C4⋊C412yes—
276C22×C4C2×C4⋊C414yes—
277C22×C4C2×C4⋊C412yes—
278C22×C4C2×C4⋊C414yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—84no—
283M16—84no—
284C2×C8—28no—
285C2×C8—28no—
286C2×C8—88no—
287C2×C8—88no—
288C4⋊C4—24no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—48no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—44no—
295C4⋊C4—84no—
296C4⋊C4—82no—
297C4⋊C4#9518yes—
298C4⋊C4#9518yes—
299C4⋊C4#9514yes—
300C4⋊C4#9514yes—
301C22⋊C4—44no—
302C22⋊C4—48no—
303C22⋊C4#9518yes—
304C22⋊C4#9518yes—
305C22⋊C4#9518yes—
306C42—42no—
307C42—44no—
308C42—42no—
309C42—48no—
310C42—44no—
311C42C2×C4214yes—
312C42C2×C4212yes—
313C42C2×C22⋊C412yes—
314C42C2×C22⋊C412yes—
315C42C2×C22⋊C414yes—
316C42C2×C22⋊C414yes—
317C42C2×C4⋊C414yes—
318C42C2×C4⋊C412yes—
319C42C2×C4⋊C414yes—
320C42C2×C4⋊C414yes—
321C42C2×C4⋊C412yes—
322C42C2×C4⋊C412yes—
323C42C23×C411yescharacteristic
324C42C23×C411yescharacteristic
325C42C22×D411yescharacteristic
326C42C22×Q811yescharacteristic
327C23—21no—
328C23—21no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23—41no—
333C23C2×#9512yes—
334C23C2×#9514yes—
335C23C2×#10014yes—
336C23C2×#10114yes—
337C23C22×C22⋊C411yescharacteristic
338Q8—28no—
339D4—216no—
340D4#343116yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—22no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—816no—
373C2×C4—88no—
374C2×C4C2×#9512yes—
375C2×C4C2×#9514yes—
376C2×C4C2×#9512yes—
377C2×C4C2×#9514yes—
378C2×C4C2×#9514yes—
379C2×C4C2×#9512yes—
380C2×C4C2×#10012yes—
381C2×C4C2×#10112yes—
382C2×C4C22×C4211yescharacteristic
383C2×C4C22×C22⋊C411yescharacteristic
384C2×C4C22×C22⋊C411yescharacteristic
385C2×C4C22×C4⋊C411yescharacteristic
386C2×C4C22×C4⋊C411yescharacteristic
387C2×C4C22×C4⋊C411yescharacteristic
388C8—48no—
389C8—48no—
390C22—21no—
391C22—24no—
392C22—21no—
393C22—24no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—44no—
398C22—44no—
399C22—42no—
400C22—42no—
401C22C2×#34318yes—
402C22C22×#9511yescharacteristic
403C22C22×#10011yescharacteristic
404C22C22×#10111yescharacteristic
405C4—21no—
406C4—21no—
407C4—24no—
408C4—22no—
409C4—21no—
410C4—24no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—24no—
415C4—44no—
416C4—84no—
417C4—82no—
418C4C2×#34314yes—
419C4#290112yes—
420C2—21no—
421C2—21no—
422C2—28no—
423C2—42no—
424C2#2033811yescharacteristic
425C2#2034311yescharacteristic
426C2C22×#34311yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7953C2semidirect
5. #7952C2semidirect
6. #7950C2semidirect
7. C2×#1285C2semidirect
11. #2990C4semidirect
19. #2970C4semidirect
75. #432C2×C4semidirect
76. #432C2×C4semidirect
77. #432C2×C4semidirect
83. #431C2×C4semidirect
85. #423C2×C4semidirect
87. C2×C4⋊C8C2×C4semidirect
93. C4×M16C2×C4semidirect
95. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602822
C4⋊C4×#412
#79532
#79522
#79502
C2×#12852