Finite Group Databasebeta

← #237796

Subgroups — #237796

Order 512.

960 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35120C211yesmaximal, characteristic
2#20741C212yesmaximal
3#13850C211yesmaximal, characteristic
4#10347C212yesmaximal
5#10345C212yesmaximal
6#10343C212yesmaximal
7#10341C212yesmaximal
8#8021C212yesmaximal
9#7910C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12#2855—22no—
13#2659C2212yes—
14#2646C2212yes—
15#2612—24no—
16#2606—22no—
17#2605—22no—
18#2329—22no—
19#2283C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#1824C2214yes—
23#1805—24no—
24#1787C2214yes—
25#1763—22no—
26#1763—22no—
27#1446C2211yescharacteristic
28#1446C2211yescharacteristic
29#1445C2211yescharacteristic
30#1445C2211yescharacteristic
31#1441—22no—
32C4×#95—22no—
33#1294C2214yes—
34#1294C2214yes—
35#1280C2212yes—
36#1280C2212yes—
37C23×Q8C2311yescharacteristic
38C2×C22⋊Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—24no—
43C2×C4×Q8Q814yescomplemented
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2311yescharacteristic
47C22×C42—21no—
48C22×C42—21no—
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51#432—24no—
52#432—24no—
53C2×C4⋊C8—22no—
54C2×#103—24no—
55C2×#103—24no—
56C2×C22⋊C8C2314yes—
57C2×#97—22no—
58#400—24no—
59#398—24no—
60#398—24no—
61#395—24no—
62#391—24no—
63#391—22no—
64#391—22no—
65#389—24no—
66#389—22no—
67#389—22no—
68#387—24no—
69#387—24no—
70#387—24no—
71#387—24no—
72#385—24no—
73#383—22no—
74#382D414yes—
75#381—22no—
76#380—22no—
77#380C2314yes—
78C4×C4⋊C4—24no—
79C4×C4⋊C4D412yes—
80C4×C4⋊C4D412yes—
81C4×C22⋊C4—22no—
82#376—22no—
83#376—22no—
84#376—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95C2311yescharacteristic
91C2×#95C2311yescharacteristic
92C2×#95C2311yescharacteristic
93C2×#95C2311yescharacteristic
94#340—24no—
95#340—24no—
96C22×Q8—22no—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—44no—
101C22×Q8C2×Q812yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—24no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—44no—
117C4×Q8—44no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×Q812yes—
136C2×C22⋊C4—24no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—24no—
140C2×C42—24no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—42no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42—42no—
147C2×C42—44no—
148C2×C42—42no—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C2×C42C2×Q812yes—
152C2×C42#4114yes—
153C2×C42#4112yes—
154C4⋊C8—44no—
155#103—48no—
156#103—48no—
157C22⋊C8—24no—
158C22⋊C8—24no—
159#97—44no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—24no—
171#95—22no—
172#95—22no—
173#95—44no—
174#95—44no—
175#95—44no—
176#95—44no—
177C24C22×D411yescharacteristic
178C2×Q8—28no—
179C2×Q8—24no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—48no—
185C2×Q8—42no—
186C2×Q8—82no—
187C2×Q8—82no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12012yes—
227C22×C4C22⋊Q812yes—
228C22×C4C22⋊Q812yes—
229C22×C4#12412yes—
230C22×C4#12412yes—
231C22×C4#13714yes—
232C22×C4C22×D412yes—
233C22×C4C22×Q811yescharacteristic
234C22×C4C2×#4112yes—
235C22×C4C2×#4111yescharacteristic
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C4⋊C4—24no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—48no—
244C4⋊C4—48no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C42—44no—
254C42—44no—
255C42—44no—
256C42—44no—
257C42—82no—
258C42—82no—
259C42—84no—
260C42C22⋊Q814yes—
261C42C22⋊Q814yes—
262C42#12414yes—
263C42#12414yes—
264C23—21no—
265C23—21no—
266C23—22no—
267C23C2×#12011yescharacteristic
268C23C2×C22⋊Q811yescharacteristic
269C23C2×#12411yescharacteristic
270C23C2×#13712yes—
271C23#57512yes—
272Q8—28no—
273Q8—48no—
274Q8—44no—
275Q8—164no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4#39312yes—
326C2×C4#39312yes—
327C2×C4#39314yes—
328C2×C4C2×C22⋊Q812yes—
329C2×C4C2×#12412yes—
330C8—84no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22C2×#39311yescharacteristic
340C22#300712yes—
341C22#301811yescharacteristic
342C22#301811yescharacteristic
343C22#302512yes—
344C22#307812yes—
345C22#313911yescharacteristic
346C22#314011yescharacteristic
347C4—22no—
348C4—22no—
349C4—24no—
350C4—22no—
351C4—22no—
352C4—44no—
353C4—44no—
354C4—82no—
355C4—82no—
356C4—84no—
357C4—84no—
358C4—82no—
359C4—82no—
360C2—21no—
361C2—22no—
362C2—21no—
363C2#1817712yes—
364C2#1818512yes—
365C2#2345511yescharacteristic
366C2#2346111yescharacteristic
367C2#2346711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
43. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351202
#207412
#138502
#103472
#103452
#103432
#103412
#80212
#79102