Finite Group Databasebeta

← #362714

Subgroups — #362714

Order 512.

3118 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic
2C2×#1996C214yesmaximal, complemented
3#14338C211yesmaximal, characteristic
4#13775C211yesmaximal, characteristic
5#11381C214yesmaximal, complemented
6#10725C214yesmaximal
7C2×#543—216no—
8C2×#543—24no—
9C2×#543—24no—
10C22×C22⋊Q8C414yescomplemented
11C22×C22⋊Q8C414yescomplemented
12C22×C22⋊Q8C2212yes—
13C22×C22⋊Q8C2212yes—
14C23×C22⋊C4C2212yes—
15C23×C22⋊C4C2211yescharacteristic
16C2×#380C2212yes—
17C2×#379C2212yes—
18C2×#379C2212yes—
19C2×C4×C22⋊C4C2212yes—
20#1996—216no—
21C2×#342C22116yes—
22#1439—24no—
23#1437—24no—
24C22⋊#95C2212yes—
25C22⋊#95C2212yes—
26C26C2311yescharacteristic
27C23×Q8C2×C412yes—
28C23×Q8C2×C412yes—
29C24×C4C2×C412yes—
30C24×C4C2×C412yes—
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33#543—232no—
34#543—232no—
35#543—232no—
36#543—232no—
37C2×C22⋊Q8—216no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—28no—
43C2×C22⋊Q8—216no—
44C2×C22⋊Q8—216no—
45C2×C22⋊Q8—216no—
46C2×C22⋊Q8—216no—
47C2×C22⋊Q8—216no—
48C2×C22⋊Q8—216no—
49C2×C22⋊Q8D4116yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4—28no—
58C22×C22⋊C4—216no—
59C22×C22⋊C4—24no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4C2×C414yes—
62C22×C22⋊C4C2×C414yes—
63C22×C22⋊C4D414yes—
64C22×C22⋊C4D414yes—
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2314yes—
68C22×C42—22no—
69#380—28no—
70#379—24no—
71#379—28no—
72#379Q818yescomplemented
73C4×C22⋊C4—28no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—24no—
77C2×#95—24no—
78#342—232no—
79C25—24no—
80C25—24no—
81C25—82no—
82C25—82no—
83C25C22×C412yes—
84C25C2×D412yes—
85C25C2×D412yes—
86C25C2×Q814yes—
87C25#4114yes—
88C25C2411yescharacteristic
89C22×Q8—28no—
90C22×Q8—24no—
91C22×Q8—24no—
92C22×Q8—24no—
93C22×Q8—84no—
94C22×Q8—84no—
95C22×Q8C22⋊C418yescomplemented
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—28no—
100C23×C4—28no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4—84no—
106C23×C4—84no—
107C23×C4C22×C412yes—
108C23×C4C22×C414yes—
109C23×C4C22×C414yes—
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D418yes—
116C22⋊Q8—2128no—
117C22⋊Q8—2128no—
118C22⋊Q8—232no—
119C22⋊Q8—232no—
120C22⋊Q8—216no—
121C22⋊Q8—216no—
122C22⋊Q8—216no—
123C22⋊Q8—216no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—84no—
135C2×C4⋊C4C22⋊C4116yescomplemented
136C2×C4⋊C4C22⋊C418yescomplemented
137C2×C22⋊C4—232no—
138C2×C22⋊C4—232no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—216no—
158C2×C22⋊C4—48no—
159C2×C22⋊C4—84no—
160C2×C22⋊C4—84no—
161C2×C22⋊C4—84no—
162C2×C22⋊C4—84no—
163C2×C22⋊C4C22⋊C4116yes—
164C2×C22⋊C4C2×D4116yes—
165C2×C22⋊C4C2×Q818yes—
166C2×C42—44no—
167C2×C42—84no—
168#95—44no—
169#95—44no—
170#95—84no—
171#95—84no—
172C24—24no—
173C24—24no—
174C24—24no—
175C24—24no—
176C24—24no—
177C24—24no—
178C24—216no—
179C24—216no—
180C24—44no—
181C24—44no—
182C24—42no—
183C24—42no—
184C24—44no—
185C24—44no—
186C24—44no—
187C24—44no—
188C24—88no—
189C24—88no—
190C24—88no—
191C24—88no—
192C24—816no—
193C24C2×C22⋊C414yes—
194C24C4×Q818yes—
195C24C22⋊Q818yes—
196C24#12414yes—
197C24C4⋊Q814yes—
198C24C23×C411yescharacteristic
199C24C22×D411yescharacteristic
200C24C22×D411yescharacteristic
201C24C22×D414yes—
202C24C22×Q812yes—
203C24C2×#4112yes—
204C242−1+414yes—
205C2×Q8—216no—
206C2×Q8—216no—
207C2×Q8—88no—
208C2×Q8—88no—
209C2×Q8—88no—
210C2×Q8—88no—
211C2×Q8—816no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—216no—
218C22×C4—216no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—48no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—88no—
240C22×C4—88no—
241C22×C4—88no—
242C22×C4—88no—
243C22×C4—816no—
244C22×C4—816no—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C414yes—
247C22×C4C2×C22⋊C418yes—
248C22×C4C2×C22⋊C418yes—
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C414yes—
251C22×C4#12014yes—
252C22×C4#12014yes—
253C22×C4#12014yes—
254C22×C4#12014yes—
255C4⋊C4—216no—
256C4⋊C4—232no—
257C4⋊C4—232no—
258C4⋊C4—216no—
259C4⋊C4—88no—
260C22⋊C4—232no—
261C22⋊C4—216no—
262C22⋊C4—216no—
263C22⋊C4—232no—
264C22⋊C4—216no—
265C22⋊C4—48no—
266C22⋊C4—416no—
267C22⋊C4—816no—
268C22⋊C4—816no—
269C42—88no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—24no—
277C23—216no—
278C23—28no—
279C23—28no—
280C23—24no—
281C23—216no—
282C23—28no—
283C23—28no—
284C23—44no—
285C23—44no—
286C23—416no—
287C23—44no—
288C23—44no—
289C23—48no—
290C23—48no—
291C23—48no—
292C23—48no—
293C23—48no—
294C23—48no—
295C23—48no—
296C23—416no—
297C23—88no—
298C23—88no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23—832no—
304C23—832no—
305C23#39118yes—
306C23C22×C22⋊C412yes—
307C23C22×C22⋊C411yescharacteristic
308C23C2×C4×Q812yes—
309C23#51912yes—
310C23C2×#12012yes—
311C23C2×#12012yes—
312C23C2×C22⋊Q812yes—
313C23C2×#12411yescharacteristic
314C23C2×C4⋊Q811yescharacteristic
315C23#53612yes—
316C23#53612yes—
317C23#54818yes—
318C23D4×Q818yes—
319Q8—816no—
320Q8—816no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—416no—
340C2×C4—48no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—816no—
344C2×C4—816no—
345C2×C4—84no—
346C2×C4—88no—
347C2×C4—84no—
348C2×C4—88no—
349C2×C4—84no—
350C2×C4—88no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4#37918yes—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—22no—
359C22—24no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—216no—
368C22—28no—
369C22—48no—
370C22—48no—
371C22—432no—
372C22—48no—
373C22—44no—
374C22—44no—
375C22—816no—
376C22—88no—
377C22—88no—
378C22C2×#37911yescharacteristic
379C22C2×#39111yescharacteristic
380C22#231011yescharacteristic
381C22Q8×C22⋊C414yes—
382C22#240914yes—
383C22#243412yes—
384C22#243612yes—
385C22#288312yes—
386C22#288312yes—
387C4—28no—
388C4—48no—
389C4—44no—
390C4—48no—
391C4—88no—
392C4—164no—
393C2—24no—
394C2—22no—
395C2—24no—
396C2—22no—
397C2—48no—
398C2#1392211yescharacteristic
399C2#2058511yescharacteristic
400C2#2058611yescharacteristic
401C2#2133312yes—
402C2#2182712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1996C2semidirect
5. #11381C2semidirect
10. C22×C22⋊Q8C4semidirect
11. C22×C22⋊Q8C4semidirect
72. #379Q8semidirect
95. C22×Q8C22⋊C4semidirect
135. C2×C4⋊C4C22⋊C4semidirect
136. C2×C4⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#19962
#143382
#137752
#113812
#107252