Finite Group Databasebeta

← #415317

Subgroups — #415317

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2C2×#2583C212yesmaximal
3C2×#1996C212yesmaximal
4#13923C211yesmaximal, characteristic
5#11344C214yesmaximal
6#11344C214yesmaximal
7C4×#342C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2857—22no—
11#2659—22no—
12#2583—28no—
13#2583—28no—
14#2583—24no—
15#2502—22no—
16#2434—22no—
17C2×#398—22no—
18C2×#395—22no—
19#2315C2214yes—
20C2×#389—24no—
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1996—28no—
27C2×#342C2218yes—
28C2×#342C2214yes—
29#1441—22no—
30#1436—22no—
31C22⋊#95C2212yes—
32C22⋊#95C2212yes—
33C23×Q8D412yes—
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C2×C4⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D414yes—
56C22×C4⋊C4D414yes—
57C22×C4⋊C4D412yes—
58C22×C22⋊C4C2312yes—
59C22×C22⋊C4C2314yes—
60C22×C22⋊C4C2312yes—
61C22×C22⋊C4C2314yes—
62C22×C42—22no—
63C22×C42D412yes—
64#398—216no—
65#398—28no—
66#395—216no—
67#395—28no—
68#391—24no—
69#389—28no—
70#389—232no—
71#385—28no—
72#385—28no—
73#385—28no—
74#385—28no—
75#385—28no—
76#385—28no—
77#384—22no—
78#380—28no—
79#380—24no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4D418yes—
84C2×#95—22no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—24no—
90C2×#95D412yes—
91#342—216no—
92#342Q8116yescomplemented
93C25C2411yescharacteristic
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—84no—
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—44no—
110C23×C4—82no—
111C23×C4—82no—
112C23×C4C2×D414yes—
113C23×C4C2×D414yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D414yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D414yes—
123C23×C4#4112yes—
124C4⋊Q8—84no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C2×C4⋊C4—216no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—216no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—416no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—88no—
159C2×C4⋊C4—82no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4C2×D418yes—
173C2×C22⋊C4C2×Q818yes—
174C2×C22⋊C4C2×Q8116yes—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—28no—
178C2×C42—82no—
179C2×C42—84no—
180#95—216no—
181#95—216no—
182#95—216no—
183#95—28no—
184#95—24no—
185#95—48no—
186#95—42no—
187#95—84no—
188C24—42no—
189C24—42no—
190C24C22×D412yes—
191C24C22×D411yescharacteristic
192C24C22×D411yescharacteristic
193C24C22×D412yes—
194C24C22×Q814yes—
195C24C2×#4111yescharacteristic
196C242−1+414yes—
197C2×Q8—48no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—84no—
201C2×Q8—88no—
202C2×Q8—88no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—216no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—48no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—48no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—48no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—82no—
247C22×C4—84no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—84no—
251C22×C4—82no—
252C22×C4#12012yes—
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12014yes—
256C22×C4#12014yes—
257C22×C4#12114yes—
258C22×C4#12112yes—
259C22×C4#12112yes—
260C22×C4#12114yes—
261C22×C4#12712yes—
262C22×C4#12712yes—
263C22×C4C4⋊Q818yes—
264C4⋊C4—432no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—88no—
273C4⋊C4—816no—
274C4⋊C4—88no—
275C4⋊C4—84no—
276C22⋊C4—28no—
277C22⋊C4—216no—
278C22⋊C4—216no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C22⋊C4C4⋊Q8116yescomplemented
285C42—44no—
286C42—82no—
287C42—88no—
288C23—24no—
289C23—22no—
290C23—24no—
291C23—22no—
292C23—41no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—41no—
298C23—48no—
299C23—48no—
300C23C2×#12012yes—
301C23C2×#12011yescharacteristic
302C23C2×#12111yescharacteristic
303C23C2×#12112yes—
304C23C2×#12711yescharacteristic
305C23C2×C4⋊Q812yes—
306C23C2×C4⋊Q814yes—
307C23#53612yes—
308C23#53614yes—
309C23D4×Q818yes—
310C23D4×Q814yes—
311C23#55714yes—
312Q8—88no—
313Q8—88no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—216no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—432no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—88no—
359C2×C4—82no—
360C2×C4#39212yes—
361C2×C4#39214yes—
362C2×C4#39212yes—
363C22—22no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—22no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—24no—
372C22—216no—
373C22—24no—
374C22—44no—
375C22—44no—
376C22—44no—
377C22—44no—
378C22C2×#39211yescharacteristic
379C22#243414yes—
380C22#243412yes—
381C22#254612yes—
382C22#254614yes—
383C22#265811yescharacteristic
384C22#286811yescharacteristic
385C22#302512yes—
386C22#302612yes—
387C4—22no—
388C4—24no—
389C4—22no—
390C4—416no—
391C4—44no—
392C4—48no—
393C4—84no—
394C4—82no—
395C4—84no—
396C4—162no—
397C2—24no—
398C2—21no—
399C2—21no—
400C2—22no—
401C2—24no—
402C2#1519011yescharacteristic
403C2#2344311yescharacteristic
404C2#2344411yescharacteristic
405C2#2585414yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
92. #342Q8semidirect
284. C22⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#25832
C2×#19962
#139232
#113442
C4×#3422