Finite Group Databasebeta

← #201488

Subgroups — #201488

Order 512.

1567 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35249C211yesmaximal, characteristic
2C4×#519C211yesmaximal, characteristic, complemented
3#21815C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9476C218yesmaximal, complemented
6#7941C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10#2920C2212yes—
11#2854—22no—
12#2632C414yescomplemented
13#2580—22no—
14#2567C412yescomplemented
15#2548C412yescomplemented
16#2501—21no—
17#2436—21no—
18#2406C2214yes—
19#2326—22no—
20#2283C2211yescharacteristic
21#2276—24no—
22#2274—21no—
23#2274C2214yes—
24C42×Q8—24no—
25C4×#117—22no—
26C4×#117C2212yes—
27#1980—24no—
28#1962—24no—
29#1789C2218yes—
30#1467—22no—
31#1463C2214yes—
32#1461—22no—
33#1284C2218yes—
34C22×#41C2312yes—
35C23×Q8D411yescharacteristic
36#532—22no—
37#532—22no—
38#532—22no—
39#532—22no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#127C2×C411yescharacteristic
42C2×#124—22no—
43C2×#124C2×C412yes—
44C2×#123—24no—
45C2×#121—21no—
46C2×#121—21no—
47#519—216no—
48C2×C4×Q8—28no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×#117—24no—
60C2×#117—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#432—22no—
66#432—22no—
67#431—22no—
68#431—22no—
69#423—22no—
70C2×#104—216no—
71#407C2314yes—
72C4×M16—22no—
73#399—24no—
74#395C2×C418yes—
75#394C2×C418yes—
76#393—24no—
77#391—22no—
78#390—22no—
79#390—24no—
80#389—24no—
81#388—28no—
82#386—24no—
83#385—28no—
84#384—22no—
85#383C2×C412yes—
86#382—21no—
87#381C2312yes—
88#380—21no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—28no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—216no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4D414yes—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4C2314yes—
100#376—24no—
101#376—21no—
102#376—22no—
103C43—24no—
104C43D414yes—
105#324—28no—
106C2×#41—24no—
107C2×#41—24no—
108C2×#41—84no—
109C22×Q8—28no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8C22⋊C412yes—
113C22×Q8C22⋊C412yes—
114C22×Q8C22×C412yes—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4C22×C412yes—
119C23×C4—22no—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C2×M16—24no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128#127—22no—
129#127—22no—
130#124—24no—
131#124—24no—
132#124—48no—
133#123—88no—
134#121—41no—
135#121—42no—
136#121—41no—
137#121—41no—
138#121—42no—
139#121—41no—
140C4×Q8—264no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153#117—232no—
154#117—216no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22×C414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C42—28no—
176C2×C42—24no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—28no—
184C2×C42C22⋊C414yes—
185C2×C42C22⋊C414yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D418yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4112yes—
194C2×C42#4112yes—
195C4⋊C8—42no—
196C4⋊C8—42no—
197#104—816no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200#97—42no—
201C4×C8—42no—
202#95—28no—
203#95—22no—
204#95—22no—
205#95—28no—
206#95—24no—
207#95—24no—
208#95—22no—
209#95C22×C414yes—
210C24—22no—
211C24C22×D411yescharacteristic
212#41—416no—
213#41—88no—
214C2×Q8—216no—
215C2×Q8—216no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—28no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—82no—
222C2×Q8#10418yescomplemented
223C2×D4—24no—
224C2×D4—44no—
225C2×D4—48no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—44no—
232C2×D4—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—28no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—84no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C414yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12012yes—
255C22×C4#12011yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12312yes—
258C22×C4#12311yescharacteristic
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D412yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4112yes—
263M16—48no—
264C2×C8—44no—
265C4⋊C4—232no—
266C4⋊C4—216no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—44no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—88no—
276C22⋊C4—216no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—416no—
282C22⋊C4—88no—
283C42—232no—
284C42—216no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—48no—
290C42—48no—
291C42—44no—
292C42—88no—
293C42C4×D412yes—
294C42C4×D412yes—
295C42C4×D412yes—
296C42C4×D412yes—
297C42#12112yes—
298C42#12112yes—
299C42#12112yes—
300C42#12112yes—
301C23—22no—
302C23—22no—
303C23—41no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—42no—
308C23C22×C22⋊C411yescharacteristic
309C23C2×#12011yescharacteristic
310C23C2×#12311yescharacteristic
311Q8—216no—
312Q8—216no—
313Q8—48no—
314D4—48no—
315D4—82no—
316D4—84no—
317D4—84no—
318D4—84no—
319D4—82no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—216no—
332C2×C4—24no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—88no—
349C2×C4—82no—
350C2×C4#38611yescharacteristic
351C2×C4#38611yescharacteristic
352C2×C4#38611yescharacteristic
353C2×C4#38612yes—
354C2×C4#38612yes—
355C2×C4#38611yescharacteristic
356C2×C4C2×#10418yes—
357C2×C4C2×#10414yes—
358C2×C4C2×C4×D412yes—
359C2×C4C2×#12112yes—
360C8—84no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—88no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22C2×#38611yescharacteristic
374C22C22×#10412yes—
375C22#302712yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—28no—
383C4—416no—
384C4—44no—
385C4—84no—
386C4—82no—
387C4#189614yes—
388C4#191212yes—
389C2—22no—
390C2—22no—
391C2—41no—
392C2—162no—
393C2#2096111yescharacteristic
394C2C2×#189611yescharacteristic
395C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#519C2semidirect
4. #17365C2semidirect
5. #9476C2semidirect
6. #7941C2semidirect
12. #2632C4semidirect
14. #2567C4semidirect
15. #2548C4semidirect
222. C2×Q8#104semidirect

Maximal subgroups

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

subgroupindex
#352492
C4×#5192
#218152
#173652
#94762
#79412