Finite Group Databasebeta

← #6528491

Subgroups — #6528491

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#29105C214yesmaximal
4#27961C214yesmaximal
5#27815C218yesmaximal, complemented
6#27729C218yesmaximal, complemented
7#21834C212yesmaximal
8#20734C212yesmaximal
9#20533C211yesmaximal, characteristic
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2212yes—
14#2857C2214yes—
15#2659—22no—
16#2567—24no—
17#2567C2218yes—
18#2552C2214yes—
19#2551C2214yes—
20#2550—24no—
21C2×#395—22no—
22#2316—21no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389—21no—
26#2285C2212yes—
27C2×#384C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2100C22116yes—
30#2072C2218yes—
31#2024C2218yes—
32#1990C2218yes—
33#1990C2214yes—
34#1990C2214yes—
35#1987C2218yes—
36#1933C22116yes—
37#1834C2218yes—
38#1824C2214yes—
39#1824C2218yes—
40#1803C2214yes—
41#1803C2214yes—
42#1787C2214yes—
43C8×C4⋊C4C2218yes—
44C4×#103C2218yes—
45C4×C22⋊C8C2214yes—
46C23×Q8—21no—
47C23×Q8C2311yescharacteristic
48#544—48no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—28no—
52C2×C4⋊Q8C2314yes—
53C2×C4⋊Q8C2314yes—
54C2×#125—24no—
55C2×C22⋊Q8—24no—
56C2×C4×Q8—22no—
57C2×C4×Q8—28no—
58C2×C4×Q8C2314yes—
59C2×#117—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#431—28no—
72C2×C4⋊C8C2314yes—
73C2×#103C2318yes—
74C2×#103C2318yes—
75C2×C22⋊C8C2314yes—
76C2×C4×C8C2314yes—
77C2×C4×C8C2318yes—
78#398—28no—
79#395—216no—
80#395—28no—
81#391—24no—
82#391—24no—
83#391C2318yes—
84#389—28no—
85#389—28no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389C2318yes—
90#384—24no—
91#384C2318yes—
92#384C2312yes—
93#384C2312yes—
94#384C2314yes—
95#382—22no—
96#380—24no—
97#380—22no—
98#380C2314yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C4×C4⋊C4C2314yes—
102C4×C4⋊C4C2314yes—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340C2318yes—
107#340C2318yes—
108#340C23116yes—
109#340C23116yes—
110#336C2318yes—
111C22×Q8—24no—
112C22×Q8—21no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—42no—
117C22×Q8—44no—
118C22×Q8—82no—
119C22×Q8—82no—
120C22×Q8C2412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8C2418yes—
130C4⋊Q8—28no—
131C4⋊Q8—28no—
132C4⋊Q8—48no—
133C4⋊Q8—416no—
134C4⋊Q8—48no—
135#125—416no—
136C22⋊Q8—416no—
137C4×Q8—28no—
138C4×Q8—28no—
139C4×Q8—48no—
140#117—48no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4SD1618yes—
162C2×C4⋊C4Q1618yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2412yes—
166C2×C4⋊C4C2414yes—
167C2×C4⋊C4C2414yes—
168C2×C4⋊C4C2414yes—
169C2×C22⋊C4—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—44no—
173C2×C42—44no—
174C2×C42C2×D414yes—
175C2×C42C2×D412yes—
176C2×C42#4118yes—
177C2×C42C2414yes—
178C2×C42C2412yes—
179C4⋊C8—28no—
180#103—216no—
181#103—48no—
182C22⋊C8—28no—
183C4×C8—28no—
184C4×C8—48no—
185#95—28no—
186#95—28no—
187#95—24no—
188#95—24no—
189#95—28no—
190#95C2×D418yes—
191C24C22×D411yescharacteristic
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—416no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—48no—
200C2×Q8—48no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×Q8—816no—
206C2×Q82−1+414yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4#12414yes—
230C22×C4#12414yes—
231C22×C4C2×SD1618yes—
232C22×C4C2×SD1614yes—
233C22×C4C2×Q1618yes—
234C22×C4C2×Q1614yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D412yes—
237C22×C4C22×D414yes—
238C22×C4C22×D414yes—
239C22×C4C22×D412yes—
240C22×C4C2×#4114yes—
241C22×C4C2511yescharacteristic
242C2×C8—28no—
243C2×C8—28no—
244C2×C8—28no—
245C4⋊C4—24no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—416no—
257C4⋊C4—84no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—88no—
261C4⋊C4—88no—
262C4⋊C4—84no—
263C4⋊C4C2×SD16116yes—
264C4⋊C4C2×Q16116yes—
265C22⋊C4—48no—
266C42—28no—
267C42—44no—
268C42—48no—
269C42—48no—
270C42#12418yes—
271C42C2×#4114yes—
272C42C2×#4114yes—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23C2×#12412yes—
278C23C22×SD1612yes—
279C23C22×Q1612yes—
280C23C23×D411yescharacteristic
281Q8—28no—
282Q8—48no—
283Q8—44no—
284Q8—816no—
285Q8—168no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—28no—
295C2×C4—24no—
296C2×C4—44no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—84no—
315C2×C4—88no—
316C2×C4#487116yes—
317C2×C4C2×#12412yes—
318C2×C4C2×#12414yes—
319C2×C4C2×#12414yes—
320C2×C4C22×SD1618yes—
321C2×C4C22×SD1614yes—
322C2×C4C22×Q1618yes—
323C2×C4C22×Q1614yes—
324C2×C4C22×#4112yes—
325C2×C4C2×2−1+412yes—
326C2×C4#58512yes—
327C8—48no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22C2×#48714yes—
335C22C22×#12411yescharacteristic
336C22#348511yescharacteristic
337C22#348611yescharacteristic
338C22C23×SD1611yescharacteristic
339C22C23×Q1611yescharacteristic
340C22#358511yescharacteristic
341C22#358511yescharacteristic
342C4—28no—
343C4—28no—
344C4—22no—
345C4—24no—
346C4—44no—
347C4—84no—
348C4—84no—
349C4—82no—
350C4—84no—
351C4C2×#48718yes—
352C4#349514yes—
353C2—24no—
354C2#3564411yescharacteristic
355C2C22×#48711yescharacteristic
356C2#6061911yescharacteristic
357C2#6169711yescharacteristic
358C2#6169811yescharacteristic
359C2#6182011yescharacteristic
360C2#6182111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #27815C2semidirect
6. #27729C2semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25672
#291052
#279612
#278152
#277292
#218342
#207342
#205332