Finite Group Databasebeta

← #238536

Subgroups — #238536

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2#21759C212yesmaximal
3C22×#328C212yesmaximal
4#13908C211yesmaximal, characteristic
5#10407C214yesmaximal, complemented
6#10406C214yesmaximal, complemented
7#7907C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C22×C22⋊C8C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2583—24no—
13#2502—22no—
14#2434—22no—
15#2315—21no—
16#2315—21no—
17C2×#380—21no—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1892—24no—
22#1872C2218yes—
23#1824—22no—
24#1787—22no—
25C2×#328—28no—
26C2×#328C2214yes—
27C2×#328C2214yes—
28#1444C2212yes—
29#1442C2212yes—
30#1436—22no—
31C22⋊#95—21no—
32#1280C2214yes—
33C23×Q8D412yes—
34C24×C4—21no—
35C24×C4C2311yescharacteristic
36C23×C8—22no—
37C2×C4⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8D414yes—
45C2×C22⋊Q8C2314yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4D411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42D412yes—
55C2×#103—28no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8C2314yes—
60#398—28no—
61#395—28no—
62#391—24no—
63#389—24no—
64#389—24no—
65#385—24no—
66#385—24no—
67#385—24no—
68#385—24no—
69#384—22no—
70#380—24no—
71#380—24no—
72#380—22no—
73#380—22no—
74#380—44no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—22no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95D414yes—
84C2×#95C2312yes—
85C2×#95C2311yescharacteristic
86C2×#95C2311yescharacteristic
87#340—28no—
88#328—28no—
89#328—28no—
90C25C2×D411yescharacteristic
91C22×Q8—22no—
92C22×Q8—24no—
93C22×Q8—24no—
94C22×Q8—84no—
95C22×Q8C2×D414yes—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—44no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—82no—
106C23×C4—82no—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4#4114yes—
114C23×C4#4112yes—
115C23×C4C2411yescharacteristic
116C22×C8—24no—
117C22×C8—44no—
118C22×C8—84no—
119C4⋊Q8—84no—
120C22⋊Q8—28no—
121C22⋊Q8—28no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—24no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—82no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4C2×D414yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4SD1614yes—
159C2×C22⋊C4Q1614yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—82no—
163#103—88no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C22⋊C8—44no—
167#95—28no—
168#95—28no—
169#95—24no—
170#95—24no—
171#95—21no—
172#95—21no—
173#95—44no—
174#95—41no—
175#95—41no—
176#95—44no—
177#95—82no—
178#95C2×D412yes—
179#95C2×D412yes—
180C24—42no—
181C24—42no—
182C24#12012yes—
183C24#12112yes—
184C24#12412yes—
185C24C2×SD1612yes—
186C24C2×Q1612yes—
187C24#13712yes—
188C24#13712yes—
189C24C22×D411yescharacteristic
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—48no—
193C2×Q8—48no—
194C2×Q8—44no—
195C2×Q8—88no—
196C2×Q8—88no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—48no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—48no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—82no—
233C22×C4—84no—
234C22×C4—82no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—88no—
239C22×C4—88no—
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12114yes—
243C22×C4#12114yes—
244C22×C4#12112yes—
245C22×C4#12314yes—
246C22×C4#12314yes—
247C22×C4#12412yes—
248C22×C4C2×SD1614yes—
249C22×C4C2×Q1614yes—
250C22×C4C22×D412yes—
251C22×C4C22×D411yescharacteristic
252C22×C4C2×#4112yes—
253C22×C4C2×#4111yescharacteristic
254C2×C8—48no—
255C2×C8—44no—
256C2×C8—88no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—48no—
264C4⋊C4—84no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—88no—
269C22⋊C4—28no—
270C22⋊C4—28no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—84no—
275C42—44no—
276C42—82no—
277C23—22no—
278C23—24no—
279C23—22no—
280C23—24no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—44no—
285C23—44no—
286C23—42no—
287C23—44no—
288C23—44no—
289C23#39414yes—
290C23#44814yes—
291C23#45114yes—
292C23#46114yes—
293C23#46214yes—
294C23#48714yes—
295C23#49114yes—
296C23C2×#12011yescharacteristic
297C23C2×#12111yescharacteristic
298C23C2×#12411yescharacteristic
299C23C22×SD1611yescharacteristic
300C23C22×Q1611yescharacteristic
301C23C2×#13711yescharacteristic
302C23C2×#13711yescharacteristic
303Q8—48no—
304Q8—84no—
305Q8—88no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—816no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—88no—
344C2×C4—88no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#39412yes—
353C2×C4#39412yes—
354C2×C4C2×#12112yes—
355C2×C4C2×#12312yes—
356C8—84no—
357C22—21no—
358C22—24no—
359C22—21no—
360C22—22no—
361C22—22no—
362C22—28no—
363C22—22no—
364C22—22no—
365C22—28no—
366C22—21no—
367C22—24no—
368C22—21no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—44no—
373C22#204318yes—
374C22C2×#39411yescharacteristic
375C22C2×#44811yescharacteristic
376C22C2×#45111yescharacteristic
377C22#301812yes—
378C22#302612yes—
379C22C2×#46111yescharacteristic
380C22C2×#46211yescharacteristic
381C22C2×#48711yescharacteristic
382C22C2×#49111yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—44no—
388C4—44no—
389C4—44no—
390C4—82no—
391C4—84no—
392C4—84no—
393C4—162no—
394C2—21no—
395C2—21no—
396C2—21no—
397C2—21no—
398C2—28no—
399C2#1838212yes—
400C2#1838312yes—
401C2#2349811yescharacteristic
402C2C2×#204311yescharacteristic
403C2#2351511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10407C2semidirect
6. #10406C2semidirect

Maximal subgroups

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

subgroupindex
#350112
#217592
C22×#3282
#139082
#104072
#104062
#79072