Finite Group Databasebeta

← #193054

Subgroups — #193054

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34399C211yesmaximal, characteristic
2#21839C211yesmaximal, characteristic
3#20819C211yesmaximal, characteristic
4#20765C211yesmaximal, characteristic
5C2×#1526C211yesmaximal, characteristic
6#8915C214yesmaximal, complemented
7#8912C214yesmaximal, complemented
8#8035C211yesmaximal, characteristic
9#8033C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2382C2214yes—
15#2360—22no—
16#2356—21no—
17#2351—21no—
18#2343—22no—
19#2331—21no—
20#2329—22no—
21#2327—21no—
22#2320—22no—
23#2285C2211yescharacteristic
24#2283C2211yescharacteristic
25C2×#382C2211yescharacteristic
26#1992C2214yes—
27#1988—21no—
28#1987—21no—
29#1847C2212yes—
30#1836C2214yes—
31#1836C2212yes—
32#1835—22no—
33#1825—22no—
34#1809—22no—
35C4×#107—21no—
36C4×#106—21no—
37#1736—22no—
38#1526C418yescomplemented
39C2×#330C2212yes—
40C2×#330C2212yes—
41#1295C2212yes—
42#1295C2212yes—
43#1295C2214yes—
44C22×M16—21no—
45C22×M16—21no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125C2312yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×#117—21no—
59C2×#117D412yes—
60C22×C4⋊C4D412yes—
61C22×C42C2311yescharacteristic
62#432—22no—
63#432—22no—
64C2×#107—22no—
65C2×#106—22no—
66C2×C4⋊C8—21no—
67C2×C22⋊C8—21no—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16C2×C414yes—
71C4×M16C2×C414yes—
72C2×#97C2312yes—
73C2×#97C2311yescharacteristic
74C2×#97C2312yes—
75C2×#97C2311yescharacteristic
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#391—24no—
79#391—22no—
80#389—22no—
81#389—24no—
82#389—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#387—24no—
87#387—24no—
88#385—24no—
89#384C2312yes—
90#383—22no—
91#383—21no—
92#382—22no—
93#382—21no—
94#382—21no—
95#382—21no—
96#382D412yes—
97#382C2312yes—
98#382C2312yes—
99#381—21no—
100#380—21no—
101#380—21no—
102#380D412yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C22⋊C4—24no—
110C4×C22⋊C4—21no—
111#376—22no—
112#376—21no—
113C2×#95—21no—
114C2×#95—21no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—28no—
120#330C2×C418yescomplemented
121#330C2×C418yescomplemented
122C22×Q8—21no—
123C22×Q8—21no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—44no—
131C2×M16—42no—
132C2×M16—42no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8C4⋊C414yes—
140C4⋊Q8C4⋊C414yes—
141#125—22no—
142#125—22no—
143#125C22×C414yes—
144#125C2×Q814yes—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—84no—
154#117—24no—
155#117—44no—
156#117C4⋊C418yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—41no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—41no—
182C2×C42C22×C412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×Q812yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189C2×C42C2411yescharacteristic
190#107—44no—
191#106—44no—
192C4⋊C8—42no—
193C22⋊C8—42no—
194#97—21no—
195#97—21no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97C22×C412yes—
200#97C22×C414yes—
201#97C22×C412yes—
202C4×C8—24no—
203C4×C8—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—41no—
218#95—41no—
219C24#12011yescharacteristic
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4C2×C4⋊C412yes—
239C22×C4C4×D412yes—
240C22×C4C4×D412yes—
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4C22⋊Q812yes—
245C22×C4C22⋊Q812yes—
246C22×C4#12312yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250M16—48no—
251M16—48no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—22no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—44no—
260C2×C8—42no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C4⋊C4C2×C4⋊C414yes—
280C4⋊C4C2×C4⋊C414yes—
281C22⋊C4—28no—
282C22⋊C4—44no—
283C22⋊C4—82no—
284C42—28no—
285C42—42no—
286C42—42no—
287C42—84no—
288C42—84no—
289C42—82no—
290C42C2×C4⋊C412yes—
291C42C4×D412yes—
292C42C4×D412yes—
293C42C22⋊Q812yes—
294C42C22⋊Q812yes—
295C42#12312yes—
296C42C23×C411yescharacteristic
297C42C22×Q811yescharacteristic
298C42C2×#4111yescharacteristic
299C42C2×#4111yescharacteristic
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23#38512yes—
305C23C2×#12011yescharacteristic
306Q8—48no—
307Q8—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—164no—
338C2×C4#38512yes—
339C2×C4#38512yes—
340C2×C4#38512yes—
341C2×C4#45412yes—
342C2×C4#45512yes—
343C2×C4C22×C4⋊C411yescharacteristic
344C2×C4C2×C4×D411yescharacteristic
345C2×C4C2×C4×D411yescharacteristic
346C2×C4C2×C22⋊Q811yescharacteristic
347C2×C4C2×C22⋊Q811yescharacteristic
348C2×C4C2×#12311yescharacteristic
349C8—24no—
350C8—44no—
351C8—42no—
352C8—82no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22C2×#38511yescharacteristic
359C22C2×#45511yescharacteristic
360C22C2×#45411yescharacteristic
361C22#301812yes—
362C22#302012yes—
363C4—21no—
364C4—21no—
365C4—22no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—44no—
372C4—44no—
373C4—82no—
374C4—84no—
375C4—82no—
376C4—162no—
377C4#186014yes—
378C2—42no—
379C2#1902512yes—
380C2#1903112yes—
381C2#2085711yescharacteristic
382C2C2×#186011yescharacteristic
383C2#2086311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8915C2semidirect
7. #8912C2semidirect
38. #1526C4semidirect
120. #330C2×C4semidirect
121. #330C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343992
#218392
#208192
#207652
C2×#15262
#89152
#89122
#80352
#80332