Finite Group Databasebeta

← #375376

Subgroups — #375376

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2C2×#1996C214yesmaximal
3#13778C211yesmaximal, characteristic
4#13778C211yesmaximal, characteristic
5#11353C214yesmaximal
6#10854C214yesmaximal
7C22×C22⋊Q8C2212yes—
8#2659—24no—
9#2659—24no—
10C2×#398C414yescomplemented
11C2×#395C414yescomplemented
12C2×#380C2212yes—
13C2×#380C2212yes—
14C2×#380C2212yes—
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2212yes—
17C2×C4×C22⋊C4C2212yes—
18C22×#95C2211yescharacteristic
19#1996—216no—
20C2×#342C22116yes—
21#1436—24no—
22#1436—24no—
23C22⋊#95C2212yes—
24C22⋊#95C2212yes—
25C23×Q8C2×C412yes—
26C24×C4C2311yescharacteristic
27C24×C4C2311yescharacteristic
28C24×C4C2311yescharacteristic
29C2×C22⋊Q8—24no—
30C2×C22⋊Q8—24no—
31C2×C22⋊Q8—28no—
32C2×C22⋊Q8—28no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C414yes—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C22⋊C4C2314yes—
42C22×C22⋊C4C2314yes—
43C22×C22⋊C4C2314yes—
44C22×C42—22no—
45C22×C42—22no—
46#398—216no—
47#395—216no—
48#380—24no—
49#380—24no—
50#380—24no—
51#380—24no—
52#380—24no—
53#380—24no—
54#380D418yes—
55#380Q818yescomplemented
56C4×C22⋊C4—28no—
57C4×C22⋊C4—28no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95C2×C414yes—
67C2×#95C2×C414yes—
68C2×#95C2×C414yes—
69C2×#95D414yes—
70C2×#95D414yes—
71#342—232no—
72C25C2411yescharacteristic
73C22×Q8—24no—
74C22×Q8—28no—
75C22×Q8—82no—
76C22×Q8—82no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4—81no—
86C23×C4C22×C414yes—
87C23×C4C22×C414yes—
88C23×C4C22×C412yes—
89C23×C4C22×C412yes—
90C23×C4C22×C412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D414yes—
98C23×C4C2×Q814yes—
99C23×C4#4114yes—
100C23×C4#4114yes—
101C22⋊Q8—48no—
102C22⋊Q8—48no—
103C22⋊Q8—48no—
104C22⋊Q8—48no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—216no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4—82no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4—82no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4C2×D418yes—
134C2×C22⋊C4C2×Q818yes—
135C2×C42—44no—
136C2×C42—44no—
137C2×C42—82no—
138C2×C42—82no—
139C2×C42—82no—
140C2×C42—82no—
141#95—28no—
142#95—28no—
143#95—28no—
144#95—28no—
145#95—44no—
146#95—44no—
147#95—82no—
148#95—82no—
149#95—82no—
150#95—82no—
151#95C22⋊C4116yes—
152C24—41no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24C23×C411yescharacteristic
157C24C22×D411yescharacteristic
158C24C22×D411yescharacteristic
159C24C22×D412yes—
160C24C22×Q812yes—
161C24C2×#4112yes—
162C24C2×#4112yes—
163C242−1+414yes—
164C2×Q8—48no—
165C2×Q8—84no—
166C2×Q8—84no—
167C2×Q8—816no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—88no—
216C22×C4C2×C22⋊C414yes—
217C22×C4C2×C22⋊C414yes—
218C22×C4C2×C22⋊C414yes—
219C22×C4C4×D418yes—
220C22×C4C4×Q818yes—
221C22×C4#12114yes—
222C22×C4#12114yes—
223C22×C4C22⋊Q814yes—
224C22×C4C22⋊Q814yes—
225C22×C4#12414yes—
226C22×C4#12414yes—
227C22×C4#12714yes—
228C22×C4C4⋊Q814yes—
229C4⋊C4—48no—
230C4⋊C4—416no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—88no—
236C4⋊C4—88no—
237C22⋊C4—216no—
238C22⋊C4—216no—
239C22⋊C4—48no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C42—88no—
245C42—88no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—44no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—44no—
268C23C22×C22⋊C411yescharacteristic
269C23C2×C4×D412yes—
270C23C2×C4×Q812yes—
271C23#51912yes—
272C23C2×#12112yes—
273C23C2×C22⋊Q812yes—
274C23C2×#12411yescharacteristic
275C23C2×#12411yescharacteristic
276C23C2×#12711yescharacteristic
277C23C2×C4⋊Q811yescharacteristic
278C23#53612yes—
279C23#53612yes—
280C23#54714yes—
281C23D4×Q814yes—
282C23#55614yes—
283C23#55714yes—
284Q8—816no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—88no—
327C2×C4#39018yes—
328C2×C4#39118yes—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—21no—
337C22—22no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—22no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22C2×#39011yescharacteristic
356C22C2×#39111yescharacteristic
357C22#231011yescharacteristic
358C22#236514yes—
359C22#254212yes—
360C22#254612yes—
361C22#254712yes—
362C22#256812yes—
363C22#288311yescharacteristic
364C22#288311yescharacteristic
365C22#288311yescharacteristic
366C22#288411yescharacteristic
367C4—28no—
368C4—28no—
369C4—44no—
370C4—88no—
371C4—88no—
372C4—84no—
373C4—84no—
374C2—22no—
375C2—22no—
376C2—22no—
377C2—24no—
378C2—22no—
379C2#1415511yescharacteristic
380C2#2121311yescharacteristic
381C2#2121411yescharacteristic
382C2#2133311yescharacteristic
383C2#2133411yescharacteristic
384C2#2182711yescharacteristic
385C2#2182811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×#398C4semidirect
11. C2×#395C4semidirect
55. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C2×#19962
#137782
#113532
#108542