Finite Group Databasebeta

← #493039

Subgroups — #493039

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34208C211yesmaximal, characteristic
2C2×#2132C212yesmaximal
3#20397C212yesmaximal, complemented
4#12631C214yesmaximal, complemented
5C4×#355C214yesmaximal, complemented
6#12587C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#520C411yescharacteristic
9C2×#520C2211yescharacteristic
10C2×#413C2214yes—
11#2873—22no—
12#2364—24no—
13#2352—22no—
14#2352—22no—
15#2328C412yes—
16#2327C2212yes—
17#2311C411yescharacteristic
18#2311C2211yescharacteristic
19C4×#124C412yes—
20C4×#124C2212yes—
21C2×#382C411yescharacteristic
22C4×#117C411yescharacteristic
23#2132C4116yescomplemented
24C2×#356C2214yes—
25C2×#355C2214yes—
26C4×#101C2218yes—
27C4×#100—24no—
28C2×#343—22no—
29#1394C2218yes—
30C22×#41C2×C411yescharacteristic
31C22×#41C2311yescharacteristic
32C22×M16—24no—
33#532—24no—
34#532C2×C418yes—
35C2×C4⋊Q8C2311yescharacteristic
36C2×#127C2×C411yescharacteristic
37C2×#124C2×C411yescharacteristic
38C2×#124C2311yescharacteristic
39C2×#121—22no—
40#520—44no—
41#520—44no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8C2312yes—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D414yes—
49C2×#117—21no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117D412yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4D412yes—
57C22×C42C2×C411yescharacteristic
58#413—28no—
59#413C2×C4116yescomplemented
60C2×#101C2318yes—
61C2×#100—24no—
62#405—48no—
63C4×M16—28no—
64#391C2×C412yes—
65#391C2312yes—
66#390C2×C412yes—
67#390C2×C412yes—
68#388—24no—
69#388—24no—
70#387—24no—
71#387—24no—
72#386—22no—
73#386—22no—
74#386—24no—
75#386—24no—
76#382—22no—
77#382C2×C412yes—
78#382C2×C412yes—
79#380—22no—
80#380—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—24no—
86#376C2×C412yes—
87C2×#95—22no—
88C43C2×C412yes—
89#356C2×C4116yescomplemented
90#355C2×C4116yescomplemented
91#343—28no—
92C2×#41—22no—
93C2×#41—42no—
94C2×#41—42no—
95C2×#41—42no—
96C2×#41—42no—
97C2×#41C4214yes—
98C2×#41C22×C414yes—
99C2×#41#4114yes—
100C22×Q8C22×C411yescharacteristic
101C22×Q8C2411yescharacteristic
102C22×D4—21no—
103C22×D4—22no—
104C22×D4C22×C411yescharacteristic
105C22×D4C22×C411yescharacteristic
106C22×D4C2×D412yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C414yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D412yes—
115C2×M16—216no—
116C2×M16—28no—
117C2×M16—48no—
118C2×M16—44no—
119C22×C8—44no—
120C4⋊Q8—22no—
121C4⋊Q8C22×C414yes—
122#127—22no—
123#127C4214yes—
124#124—22no—
125#124—22no—
126#124C4214yes—
127#124C22×C414yes—
128#121—216no—
129C4×Q8—44no—
130C4×Q8—44no—
131C4×D4—216no—
132C4×D4—44no—
133C4×D4—48no—
134C4×D4—44no—
135C4×D4—48no—
136#117—24no—
137#117—28no—
138#117—44no—
139#117—42no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C2×D414yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C414yes—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42C4214yes—
168C2×C42C22×C412yes—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C412yes—
172C2×C42C2×D414yes—
173#101—28no—
174#101C22×C4116yes—
175#100—216no—
176#97—48no—
177C4×C8—48no—
178#95—24no—
179#95—24no—
180#95—22no—
181#95—22no—
182#95—24no—
183#95—24no—
184C24—21no—
185C24C2×C22⋊C412yes—
186C24C2×C22⋊C411yescharacteristic
187#41—44no—
188#41—44no—
189#41—48no—
190#41—48no—
191#41—48no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8C2×C4212yes—
198C2×Q8C23×C412yes—
199C2×Q8C2×#4112yes—
200C2×D4—24no—
201C2×D4—28no—
202C2×D4—21no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—82no—
212C2×D4—82no—
213C2×D4C2×C4212yes—
214C2×D4C2×C4212yes—
215C2×D4#11712yes—
216C2×D4C4×D418yes—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—84no—
237C22×C4—82no—
238C22×C4—84no—
239C22×C4—82no—
240C22×C4C2×C4212yes—
241C22×C4C2×C22⋊C414yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4#11712yes—
246C22×C4C4×D418yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D412yes—
249M16—216no—
250M16—48no—
251M16—416no—
252C2×C8—48no—
253C2×C8—416no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4#9914yes—
264C22⋊C4—28no—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—24no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C42—21no—
273C42—21no—
274C42—24no—
275C42—28no—
276C42—24no—
277C42—42no—
278C42—44no—
279C42—82no—
280C42—82no—
281C42C2×C4212yes—
282C42C2×C4212yes—
283C23—21no—
284C23—22no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23C4×C22⋊C414yes—
297C23C4×C22⋊C412yes—
298C23C22×C22⋊C411yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—42no—
302Q8—42no—
303D4—42no—
304D4—42no—
305D4—48no—
306D4—42no—
307D4—42no—
308D4—48no—
309D4—88no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—48no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—84no—
339C2×C4—164no—
340C2×C4—164no—
341C2×C4—162no—
342C2×C4—162no—
343C2×C4C4×C22⋊C412yes—
344C2×C4C2×#9914yes—
345C2×C4C2×#9912yes—
346C2×C4#41012yes—
347C2×C4C22×C4211yescharacteristic
348C2×C4C2×#11711yescharacteristic
349C2×C4C2×C4×D414yes—
350C8—416no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—24no—
355C22—42no—
356C22—44no—
357C22—42no—
358C22—44no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—82no—
365C22—82no—
366C22—81no—
367C22—81no—
368C22C2×C4×C22⋊C411yescharacteristic
369C22C22×#9911yescharacteristic
370C22C2×#41011yescharacteristic
371C4—24no—
372C4—24no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—44no—
379C4—42no—
380C4—82no—
381C4—82no—
382C4C4×#9914yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—44no—
387C2—82no—
388C2C2×C4×#9911yescharacteristic
389C2#3337011yescharacteristic
390C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C22×C42
Frattini, cosocleC22×Q8C24
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20397C2semidirect
4. #12631C2semidirect
5. C4×#355C2semidirect
6. #12587C2semidirect
23. #2132C4semidirect
59. #413C2×C4semidirect
89. #356C2×C4semidirect
90. #355C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342082
C2×#21322
#203972
#126312
C4×#3552
#125872