Finite Group Databasebeta

← #182743

Subgroups — #182743

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34231C211yesmaximal, characteristic
2#20480C211yesmaximal, characteristic
3#20367C211yesmaximal, characteristic
4C2×#1531C211yesmaximal, characteristic
5C2×#1526C211yesmaximal, characteristic
6#8227C214yesmaximal, complemented
7#8225C214yesmaximal, complemented
8#8035C211yesmaximal, characteristic
9#8029C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C22×#97C2211yescharacteristic
15#2369C412yes—
16#2365C412yes—
17#2362—22no—
18#2356—22no—
19#2347C412yes—
20#2346C412yes—
21#2332—22no—
22#2320—21no—
23C4×#125C2214yes—
24#2283C2211yescharacteristic
25#2276—21no—
26C4×#117C2211yescharacteristic
27#1770—21no—
28C8×C4⋊C4—21no—
29#1753—22no—
30#1746—22no—
31#1745C2212yes—
32#1745C2214yes—
33#1729—22no—
34C4×#97C2212yes—
35#1531C418yescomplemented
36#1526C418yescomplemented
37C2×#330C2212yes—
38C2×#330C2214yes—
39C2×#330C2212yes—
40#1295C2212yes—
41#1295C2214yes—
42#1295C2212yes—
43C22×M16—21no—
44C22×M16—21no—
45C23×C8—21no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533C2×C414yes—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125C2312yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×#117C2×C412yes—
59C2×#117D412yes—
60C22×C42C2311yescharacteristic
61#432—22no—
62#432C2×C414yes—
63C2×C4⋊C8—21no—
64C2×C22⋊C8—21no—
65C4×M16—22no—
66C4×M16—22no—
67C4×M16C2×C414yes—
68C4×M16C2×C414yes—
69C2×#97—21no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97C2×C414yes—
73C2×#97C2312yes—
74C2×#97C2311yescharacteristic
75C2×#97C2311yescharacteristic
76C2×#97C2312yes—
77C2×#97C2312yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81#391—22no—
82#391—24no—
83#389—22no—
84#389C2×C414yes—
85#389C2×C414yes—
86#388—24no—
87#388—24no—
88#387—24no—
89#387—24no—
90#387—24no—
91#383—21no—
92#383—22no—
93#382—22no—
94#382C2312yes—
95#381—21no—
96#380C2×C411yescharacteristic
97#380C2×C411yescharacteristic
98C4×C4⋊C4—22no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C22⋊C4—21no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111#376—21no—
112#376—22no—
113#376D412yes—
114C43C2312yes—
115#336—28no—
116#336—22no—
117#336—22no—
118#330C2×C418yes—
119#330C2×C418yes—
120#330C2×C4116yes—
121C22×Q8—21no—
122C22×Q8—21no—
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—44no—
131C22×C8—21no—
132C22×C8—24no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—44no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8C4214yes—
141C4⋊Q8C4214yes—
142#125—22no—
143#125—22no—
144#125C22×C414yes—
145#125C22×C414yes—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—48no—
155#117—24no—
156#117—44no—
157#117C4218yes—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4C22⋊C414yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42C22×C414yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C412yes—
181C2×C42C22×C412yes—
182C2×C42C2×D414yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42#4112yes—
188C2×C42#4112yes—
189C2×C42C2411yescharacteristic
190C4⋊C8—24no—
191C22⋊C8—24no—
192#97—24no—
193#97—22no—
194#97—21no—
195#97—21no—
196#97—22no—
197#97—22no—
198#97—44no—
199#97C22×C414yes—
200#97C22×C412yes—
201#97C22×C412yes—
202#97C22×C414yes—
203#97C22×C414yes—
204C4×C8—24no—
205C4×C8—24no—
206C4×C8—44no—
207#95—21no—
208#95—21no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95—24no—
213#95—24no—
214#95—22no—
215#95—22no—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C414yes—
219C24C2×C22⋊C411yescharacteristic
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—44no—
223C2×Q8—81no—
224C2×Q8—81no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4C2×C4212yes—
234C22×C4C2×C22⋊C414yes—
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4#11712yes—
240C22×C4C4×D412yes—
241C22×C4C4×D412yes—
242C22×C4C4×D412yes—
243C22×C4C4×D412yes—
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247M16—48no—
248M16—48no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—48no—
260C2×C8—44no—
261C2×C8—41no—
262C2×C8—41no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—48no—
276C4⋊C4—84no—
277C4⋊C4C2×C4214yes—
278C4⋊C4C2×C4214yes—
279C22⋊C4—28no—
280C22⋊C4—44no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C42—28no—
284C42—42no—
285C42—42no—
286C42—44no—
287C42—44no—
288C42—48no—
289C42—44no—
290C42—84no—
291C42C2×C4212yes—
292C42#11712yes—
293C42C4×D412yes—
294C42C4×D412yes—
295C42C4×D412yes—
296C42C4×D412yes—
297C42C23×C411yescharacteristic
298C42C23×C411yescharacteristic
299C42C2×#4111yescharacteristic
300C42C2×#4111yescharacteristic
301C23—22no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23C4×C22⋊C412yes—
306C23C22×C22⋊C411yescharacteristic
307Q8—48no—
308Q8—84no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4C4×C22⋊C412yes—
340C2×C4C4×C22⋊C412yes—
341C2×C4C4×C22⋊C412yes—
342C2×C4#42114yes—
343C2×C4C22×C4211yescharacteristic
344C2×C4C2×#11711yescharacteristic
345C2×C4C2×C4×D411yescharacteristic
346C2×C4C2×C4×D411yescharacteristic
347C2×C4C2×C4×D411yescharacteristic
348C2×C4C2×C4×D411yescharacteristic
349C8—24no—
350C8—44no—
351C8—44no—
352C8—44no—
353C8—42no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22C2×C4×C22⋊C411yescharacteristic
360C22#288912yes—
361C22C2×#42112yes—
362C22#290111yescharacteristic
363C22#290111yescharacteristic
364C4—21no—
365C4—22no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—44no—
373C4—44no—
374C4—44no—
375C4—44no—
376C4—82no—
377C4—82no—
378C4#175914yes—
379C2—42no—
380C2#1768212yes—
381C2#1768312yes—
382C2#2039911yescharacteristic
383C2C2×#175911yescharacteristic
384C2#2040511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8227C2semidirect
7. #8225C2semidirect
35. #1531C4semidirect
36. #1526C4semidirect

Maximal subgroups

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

subgroupindex
#342312
#204802
#203672
C2×#15312
C2×#15262
#82272
#82252
#80352
#80292