Finite Group Databasebeta

← #381189

Subgroups — #381189

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic
2C2×C4×C4⋊C8C211yesmaximal, characteristic
3#14688C211yesmaximal, characteristic
4#11372C212yesmaximal
5#11358C212yesmaximal
6#11228C212yesmaximal
7#11227C212yesmaximal
8#10880C212yesmaximal
9#10880C212yesmaximal
10C22×C4⋊C8C2212yes—
11C22×C4×C8C2212yes—
12#2864—24no—
13#2864C418yescomplemented
14C2×#389C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#387—21no—
19C2×#387—21no—
20C2×#387—21no—
21C2×#387—21no—
22C2×#383—21no—
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C43C2211yescharacteristic
26C4×C4⋊C8—24no—
27C2×#340C2214yes—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#336C2212yes—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35#1434—21no—
36#1434—21no—
37#1434—21no—
38C23×C8C2314yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2314yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—28no—
56C2×C4⋊C8—24no—
57C2×C4×C8—24no—
58C2×C4×C8—44no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389—22no—
63#389C2×C414yes—
64#389C2×C414yes—
65#389C2×C414yes—
66#389C2×C414yes—
67#387—28no—
68#387—28no—
69#387—28no—
70#387—28no—
71#383—44no—
72#382—22no—
73#382—22no—
74#382C2×C414yes—
75#382C2×C414yes—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C43C2×C414yes—
91C43D414yes—
92#340—28no—
93#340—28no—
94#340—44no—
95#340—44no—
96#336—24no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—24no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—24no—
110C22×C8—24no—
111C22×C8—24no—
112C22×C8—24no—
113C22×C8—24no—
114C22×C8—24no—
115C22×C8#4118yes—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C412yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C412yes—
145C2×C42—24no—
146C2×C42—24no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—24no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42—22no—
153C2×C42—42no—
154C2×C42C22⋊C414yes—
155C2×C42C22⋊C414yes—
156C2×C42C22⋊C414yes—
157C2×C42C22⋊C414yes—
158C2×C42C22×C412yes—
159C2×C42C22×C412yes—
160C2×C42C22×C412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C4⋊C8—216no—
165C4×C8—48no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—44no—
175#95—44no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—44no—
180C24C22×D411yescharacteristic
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4#13514yes—
230C22×C4#13514yes—
231C22×C4AGL(1,ℤ/8)12yes—
232C22×C4#13712yes—
233C22×C4C23×C411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4114yes—
236C22×C4H(5,𝔽2)12yes—
237C22×C42−1+414yes—
238C22×C42−1+412yes—
239C2×C8—28no—
240C2×C8—28no—
241C2×C8—28no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—48no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—48no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C42—24no—
265C42—24no—
266C42—24no—
267C42—24no—
268C42—22no—
269C42—22no—
270C42—24no—
271C42—28no—
272C42—28no—
273C42—28no—
274C42—44no—
275C23C22×C22⋊C411yescharacteristic
276C23#53212yes—
277C23#53612yes—
278C23#53712yes—
279C23C2×#13512yes—
280C23C2×#13512yes—
281C23C2×AGL(1,ℤ/8)11yescharacteristic
282C23C2×#13711yescharacteristic
283C23#57512yes—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—28no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4#41614yes—
334C2×C4#41614yes—
335C2×C4#41714yes—
336C2×C4C4×#4112yes—
337C2×C4#51912yes—
338C2×C4#52012yes—
339C2×C4#52512yes—
340C2×C4#53511yescharacteristic
341C2×C4#53612yes—
342C2×C4#53611yescharacteristic
343C2×C4#53914yes—
344C2×C4#53914yes—
345C2×C4#54114yes—
346C2×C4#54114yes—
347C8—48no—
348C22#229711yescharacteristic
349C22#231011yescharacteristic
350C22#231111yescharacteristic
351C22#266414yes—
352C22C2×#41611yescharacteristic
353C22C2×#41611yescharacteristic
354C22C2×#41711yescharacteristic
355C22#289611yescharacteristic
356C22#316814yes—
357C22#317112yes—
358C22#317212yes—
359C22#322812yes—
360C22#323112yes—
361C22#323314yes—
362C22#324712yes—
363C22#324812yes—
364C22#325014yes—
365C4—24no—
366C4—24no—
367C4—24no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—82no—
372C4—82no—
373C4—82no—
374C4—82no—
375C4—164no—
376C4#233614yes—
377C4#263312yes—
378C4#263412yes—
379C2#1428911yescharacteristic
380C2#2140211yescharacteristic
381C2#2140311yescharacteristic
382C2#2183511yescharacteristic
383C2#2183711yescharacteristic
384C2#2188611yescharacteristic
385C2#2188711yescharacteristic
386C2#3038014yes—
387C2#3038114yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2864C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
C2×C4×C4⋊C82
#146882
#113722
#113582
#112282
#112272
#108802