Finite Group Databasebeta

← #189866

Subgroups — #189866

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#21818C211yesmaximal, characteristic
3#13961C211yesmaximal, characteristic
4#8717C212yesmaximal
5#8716C212yesmaximal
6#8714C214yesmaximal
7#8025C212yesmaximal
8#8013C212yesmaximal
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2212yes—
11#2356—22no—
12#2356—22no—
13#2355C414yescomplemented
14#2355C414yescomplemented
15#2313—21no—
16#2313—21no—
17#2310—21no—
18#2285C2211yescharacteristic
19#2285C2211yescharacteristic
20#2283C2212yes—
21#2283C2212yes—
22C2×#383C2211yescharacteristic
23#2274—21no—
24C2×#376C2211yescharacteristic
25#1984—22no—
26#1984—22no—
27#1817C2212yes—
28#1817C2212yes—
29#1446—21no—
30#1445—21no—
31#1436—21no—
32#1436—21no—
33#1434C2212yes—
34#1434C2212yes—
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C23×Q8D411yescharacteristic
40C2×C4×Q8—24no—
41C2×C4×Q8—24no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8D412yes—
45C2×C4×Q8D412yes—
46C2×#117—21no—
47C2×#117—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55#431—24no—
56#431—24no—
57C2×C4⋊C8—22no—
58C2×C22⋊C8C2314yes—
59C2×C4×C8—22no—
60#391—24no—
61#391—24no—
62#387—24no—
63#387—24no—
64#387—24no—
65#387—24no—
66#384C2×C412yes—
67#384C2×C412yes—
68#383—22no—
69#383—22no—
70#383—24no—
71#382—21no—
72#382—21no—
73#382C2×C414yes—
74#382C2×C414yes—
75#381—22no—
76#381—22no—
77#380—21no—
78#380—21no—
79#380C2312yes—
80#380C2312yes—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4C2×C414yes—
86C4×C4⋊C4C2×C414yes—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4C2×C412yes—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91#376—22no—
92#376—22no—
93#376—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95C2312yes—
102C2×#95C2312yes—
103#336—24no—
104#336—22no—
105#336—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—42no—
110C22×Q8C22⋊C412yes—
111C22×Q8C22⋊C412yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C4×Q8—28no—
123C4×Q8—28no—
124C4×Q8—24no—
125C4×Q8—24no—
126C4×Q8—24no—
127C4×Q8—24no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—44no—
159C2×C42—42no—
160C2×C42C22×C412yes—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42#4114yes—
169C2×C42#4114yes—
170C4⋊C8—44no—
171C22⋊C8—24no—
172C22⋊C8—24no—
173C4×C8—44no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—22no—
180#95—24no—
181#95—24no—
182#95—22no—
183#95—22no—
184#95—24no—
185#95—24no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—42no—
190#95—42no—
191#95—42no—
192#95—44no—
193C24C22×D411yescharacteristic
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—48no—
201C2×Q8—48no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4#12312yes—
244C22×C4#12312yes—
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4#13514yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4112yes—
251C22×C4C2×#4112yes—
252C2×C8—48no—
253C2×C8—44no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C42—22no—
269C42—22no—
270C42—24no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—44no—
276C42—48no—
277C42—44no—
278C42—84no—
279C42—84no—
280C42#11718yes—
281C42#12314yes—
282C42#12314yes—
283C23—22no—
284C23—22no—
285C23C22×C22⋊C411yescharacteristic
286C23C2×#12311yescharacteristic
287C23C2×#12311yescharacteristic
288C23C2×#13512yes—
289C23#57512yes—
290Q8—48no—
291Q8—48no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4#38114yes—
345C2×C4#38114yes—
346C2×C4#41612yes—
347C2×C4#41612yes—
348C2×C4C2×#11712yes—
349C2×C4C2×#12312yes—
350C8—84no—
351C22—22no—
352C22—22no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22C2×#38111yescharacteristic
360C22#288412yes—
361C22C2×#41611yescharacteristic
362C22#289611yescharacteristic
363C22#290112yes—
364C22#309112yes—
365C22#309112yes—
366C4—22no—
367C4—22no—
368C4—24no—
369C4—22no—
370C4—22no—
371C4—24no—
372C4—42no—
373C4—42no—
374C4—41no—
375C4—41no—
376C4—84no—
377C4—84no—
378C4—84no—
379C4—162no—
380C2—22no—
381C2—22no—
382C2#1761612yes—
383C2#1762412yes—
384C2#2069111yescharacteristic
385C2#2069911yescharacteristic
386C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2355C4semidirect
14. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#218182
#139612
#87172
#87162
#87142
#80252
#80132