Finite Group Databasebeta

← #201490

Subgroups — #201490

Order 512.

1661 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3#21844C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic
5#9477C218yesmaximal, complemented
6#7941C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10#2920C2212yes—
11#2848—22no—
12#2633C414yescomplemented
13#2547C412yescomplemented
14#2542C412yescomplemented
15#2536—22no—
16#2502—21no—
17#2482C2214yes—
18#2435—21no—
19#2326—22no—
20#2283C2211yescharacteristic
21#2275—24no—
22#2274—21no—
23#2273C2214yes—
24C42×D4—24no—
25C4×#117C2212yes—
26C2×C4×C22⋊C4—22no—
27#1994—24no—
28#1986—24no—
29#1826C2218yes—
30#1466—22no—
31#1462C2214yes—
32#1460—22no—
33#1284C2218yes—
34C23×D4D411yescharacteristic
35C24×C4D412yes—
36#533—22no—
37#533—22no—
38#533—22no—
39#533—22no—
40C2×C4⋊Q8C2×C411yescharacteristic, complemented
41C2×C4⋊Q8C2×C411yescharacteristic, complemented
42C2×#125C2×C412yescomplemented
43C2×#124—22no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—24no—
47#518—216no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×D4—28no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D412yes—
59C2×#117—22no—
60C2×#117C2312yes—
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4—24no—
63C22×C22⋊C4C2312yes—
64C22×C42—24no—
65C22×C42C2311yescharacteristic
66#432—22no—
67#432—22no—
68#431—22no—
69#431—22no—
70#423—22no—
71#407C2314yes—
72C4×M16—22no—
73#399C2×C418yescomplemented
74#398—24no—
75#397C2×C418yescomplemented
76#394—24no—
77#391—22no—
78#390—22no—
79#388—28no—
80#385—28no—
81#385—24no—
82#384—22no—
83#383C2×C412yescomplemented
84#382—21no—
85#381C2312yes—
86#380—24no—
87#380—21no—
88#379—24no—
89C4×C4⋊C4—28no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C412yescomplemented
92C4×C4⋊C4D414yes—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—216no—
99C4×C22⋊C4C2314yes—
100#376—21no—
101#376—22no—
102C2×#95—24no—
103C43D414yes—
104#339—216no—
105#323—28no—
106C25C2×D412yes—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×D4—28no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C22⋊C412yes—
113C22×D4C22⋊C412yes—
114C23×C4—28no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—84no—
120C23×C4C22⋊C414yes—
121C23×C4C22⋊C414yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C2×M16—24no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131#125—24no—
132#125—24no—
133#124—48no—
134C22⋊Q8—41no—
135C22⋊Q8—42no—
136C22⋊Q8—41no—
137C22⋊Q8—42no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—88no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×D4—264no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154#117—216no—
155#117—24no—
156#117—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—232no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4C22×C414yes—
180C2×C22⋊C4C2×D414yes—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42—28no—
186C2×C42—88no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D418yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4112yes—
195C2×C42#4112yes—
196C4⋊C8—42no—
197C4⋊C8—42no—
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—
211C24—22no—
212C24—28no—
213C24—44no—
214C24—82no—
215C24C2×C22⋊C414yes—
216C24C22×D411yescharacteristic
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—41no—
221C2×Q8—44no—
222C2×D4—216no—
223C2×D4—216no—
224C2×D4—28no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4#10418yes—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—22no—
232C22×C4—232no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—88no—
250C22×C4—88no—
251C22×C4—84no—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12012yes—
259C22×C4#12011yescharacteristic
260C22×C4#12312yes—
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4112yes—
267M16—48no—
268C2×C8—44no—
269C4⋊C4—216no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—88no—
282C22⋊C4—216no—
283C22⋊C4—232no—
284C22⋊C4—28no—
285C22⋊C4—28no—
286C22⋊C4—44no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—416no—
290C22⋊C4—88no—
291C42—216no—
292C42—48no—
293C42—48no—
294C42—44no—
295C42—44no—
296C42—88no—
297C42C4×D412yes—
298C42C4×D412yes—
299C42C4×D412yes—
300C42C4×D412yes—
301C42#12112yes—
302C42#12112yes—
303C42#12112yes—
304C42#12112yes—
305C23—28no—
306C23—28no—
307C23—22no—
308C23—22no—
309C23—24no—
310C23—24no—
311C23—41no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—82no—
317C23—82no—
318C23—816no—
319C23C2×#10418yes—
320C23C22×C22⋊C411yescharacteristic
321C23C2×#12011yescharacteristic
322C23C2×#12311yescharacteristic
323Q8—82no—
324Q8—84no—
325Q8—82no—
326D4—216no—
327D4—216no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—216no—
336C2×C4—24no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—832no—
360C2×C4#38612yes—
361C2×C4#38611yescharacteristic
362C2×C4#38611yescharacteristic
363C2×C4#38611yescharacteristic
364C2×C4#38612yes—
365C2×C4#38611yescharacteristic
366C2×C4C2×#10414yes—
367C2×C4C2×C4×D412yes—
368C2×C4C2×#12112yes—
369C8—84no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—28no—
378C22—44no—
379C22—48no—
380C22—416no—
381C22—44no—
382C22—44no—
383C22—84no—
384C22—88no—
385C22C2×#38611yescharacteristic
386C22C22×#10412yes—
387C22#302812yes—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—416no—
395C4—44no—
396C4—84no—
397C4—82no—
398C4—82no—
399C4#189614yes—
400C4#191112yes—
401C2—22no—
402C2—22no—
403C2—41no—
404C2—48no—
405C2#2096211yescharacteristic
406C2C2×#189611yescharacteristic
407C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
3. #21844C2semidirect
5. #9477C2semidirect
6. #7941C2semidirect
12. #2633C4semidirect
13. #2547C4semidirect
14. #2542C4semidirect
40. C2×C4⋊Q8C2×C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
42. C2×#125C2×C4semidirect
73. #399C2×C4semidirect
75. #397C2×C4semidirect
83. #383C2×C4semidirect
91. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#352402
C4×#5182
#218442
#173642
#94772
#79412