Finite Group Databasebeta

← #493717

Subgroups — #493717

Order 512.

1179 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2C2×#2129C212yesmaximal
3#20634C212yesmaximal
4#12800C214yesmaximal, complemented
5#12798C214yesmaximal, complemented
6#12581C212yesmaximal
7C2×#536C2211yescharacteristic
8C2×#520C2211yescharacteristic
9C2×#519C411yescharacteristic, complemented
10C2×#410C2214yes—
11#2362—22no—
12#2360—22no—
13#2359—24no—
14#2327C412yescomplemented
15#2326C2212yes—
16#2317C411yescharacteristic, complemented
17#2316—22no—
18C4×C4⋊Q8C412yescomplemented
19C4×#124C2212yes—
20#2297C2211yescharacteristic
21C2×#391C411yescharacteristic, complemented
22C4×#117C411yescharacteristic, complemented
23#2129—28no—
24C2×#354C2218yes—
25#1799—24no—
26#1799C2218yes—
27#1740—22no—
28#1393C2214yes—
29#1393C2214yes—
30C22×#41C2311yescharacteristic
31C23×Q8C2×C411yescharacteristic
32#536—24no—
33#536D414yes—
34#536Q814yes—
35C2×C4⋊Q8C2×C412yes—
36C2×#124C2312yes—
37C2×#123—22no—
38C2×C22⋊Q8—22no—
39#520—44no—
40#519—44no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×D4D414yes—
47C2×C4×D4C2312yes—
48C2×#117—22no—
49C2×#117—24no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117D412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C42C2×C411yescharacteristic
57#410—28no—
58#410—28no—
59C2×#99—24no—
60C2×#99C2318yes—
61#391—22no—
62#391C2×C411yescharacteristic
63#391C2×C412yes—
64#391C2×C412yes—
65#391C2×C411yescharacteristic
66#390C2311yescharacteristic
67#390C2311yescharacteristic
68#389—24no—
69#389—22no—
70#389—48no—
71#388—24no—
72#388—24no—
73#387—24no—
74#387—24no—
75#386—22no—
76#385—22no—
77#385—24no—
78#384C2×C412yes—
79#382C2×C412yes—
80#380—28no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—24no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87#376C2×C412yes—
88C2×#95—22no—
89C43C2×C412yes—
90#354—216no—
91#342—28no—
92C2×#41—42no—
93C2×#41—42no—
94C2×#41C2×D412yes—
95C2×#41C2×Q812yes—
96C2×#41#4114yes—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8C4⋊C414yescomplemented
102C22×Q8C22×C412yes—
103C22×Q8C22×C411yescharacteristic
104C22×D4C2×D412yes—
105C22×D4C2411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C22⋊C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D414yes—
112C4⋊Q8—24no—
113C4⋊Q8C4⋊C418yescomplemented
114#124—24no—
115#124C2×D414yes—
116#124C2×Q814yes—
117#123—28no—
118#123—28no—
119C22⋊Q8—28no—
120C22⋊Q8—28no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×Q8—44no—
126C4×D4—44no—
127C4×D4—48no—
128#117—22no—
129#117—24no—
130#117—48no—
131#117—44no—
132#117—42no—
133#117—48no—
134#117—48no—
135#117C4⋊C414yescomplemented
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C22×C412yes—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C2×D414yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—42no—
167C2×C42—44no—
168C2×C42C22⋊C414yes—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173#99—28no—
174#99—28no—
175#99—48no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—24no—
182#95—22no—
183#95—48no—
184C24C2×C22⋊C411yescharacteristic
185C24C22×D412yes—
186#41—44no—
187#41—44no—
188#41—48no—
189C2×Q8—22no—
190C2×Q8—24no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—48no—
194C2×Q8—48no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8C2×C4⋊C414yes—
200C2×Q8C2×C4⋊C412yes—
201C2×Q8#11712yes—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4#12114yes—
207C2×D4C22⋊Q814yes—
208C2×D4C22×D411yescharacteristic
209C2×D4C22×Q811yescharacteristic
210C2×D4C2×#4112yes—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—28no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—48no—
229C22×C4—81no—
230C22×C4—84no—
231C22×C4—82no—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C4⋊C412yes—
238C22×C4#11712yes—
239C22×C4#12114yes—
240C22×C4C22⋊Q814yes—
241C22×C4C23×C411yescharacteristic
242C4⋊C4—24no—
243C4⋊C4—216no—
244C4⋊C4—24no—
245C4⋊C4—28no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—88no—
253C4⋊C4—84no—
254C4⋊C4—168no—
255C4⋊C4#9914yes—
256C22⋊C4—28no—
257C22⋊C4—28no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C22⋊C4—82no—
266C42—22no—
267C42—24no—
268C42—44no—
269C42—41no—
270C42—41no—
271C42—44no—
272C42—44no—
273C42—44no—
274C42—42no—
275C42—42no—
276C42—88no—
277C42—82no—
278C42C2×C4⋊C414yes—
279C23—21no—
280C23—22no—
281C23—21no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23#38012yes—
288C23C22×C22⋊C411yescharacteristic
289C23C2×#12112yes—
290C23C2×C22⋊Q812yes—
291Q8—42no—
292Q8—42no—
293Q8—44no—
294Q8—44no—
295Q8—48no—
296D4—44no—
297D4—44no—
298D4—41no—
299D4—41no—
300D4—42no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—28no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—28no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—88no—
332C2×C4—84no—
333C2×C4—88no—
334C2×C4—88no—
335C2×C4—88no—
336C2×C4—162no—
337C2×C4#38014yes—
338C2×C4#38012yes—
339C2×C4C2×#9914yes—
340C2×C4C2×#9912yes—
341C2×C4#41012yes—
342C2×C4C22×C4⋊C411yescharacteristic
343C2×C4C2×#11711yescharacteristic
344C22—21no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—41no—
355C22—42no—
356C22—41no—
357C22—82no—
358C22C2×#38011yescharacteristic
359C22C22×#9911yescharacteristic
360C22C2×#41011yescharacteristic
361C4—24no—
362C4—21no—
363C4—21no—
364C4—24no—
365C4—44no—
366C4—42no—
367C4—42no—
368C4—44no—
369C4—44no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—168no—
374C4#179914yes—
375C2—21no—
376C2—21no—
377C2—42no—
378C2—42no—
379C2—42no—
380C2C2×#179911yescharacteristic
381C2#3336011yescharacteristic
382C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12800C2semidirect
5. #12798C2semidirect
9. C2×#519C4semidirect
14. #2327C4semidirect
16. #2317C4semidirect
18. C4×C4⋊Q8C4semidirect
21. C2×#391C4semidirect
22. C4×#117C4semidirect
101. C22×Q8C4⋊C4semidirect
113. C4⋊Q8C4⋊C4semidirect
135. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342762
C2×#21292
#206342
#128002
#127982
#125812