Finite Group Databasebeta

← #104147

Subgroups — #104147

Order 512.

648 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 100 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#29715C211yesmaximal, characteristic
2#27961C211yesmaximal, characteristic
3#15199C211yesmaximal, characteristic
4#9909C211yesmaximal, characteristic
5#8725C211yesmaximal, characteristic
6#7575C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C2×#451—21no—
9#2857C2211yescharacteristic
10C2×#391—21no—
11C2×#391C2211yescharacteristic
12C2×#384—21no—
13C2×#384C2211yescharacteristic
14#2100—22no—
15#2072—21no—
16#2056—22no—
17#2016—21no—
18#1990—21no—
19#1872—21no—
20#1834—21no—
21#1824—22no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24C4×C22⋊C8C2211yescharacteristic
25#1444—22no—
26#1444—22no—
27C4×#95C2211yescharacteristic
28#1296—22no—
29#1294—22no—
30#1272—22no—
31C23×Q8D411yescharacteristic
32C22×Q16—22no—
33C2×C4⋊Q8D411yescharacteristic
34C2×C22⋊Q8—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—24no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D411yescharacteristic
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#451—42no—
43#451—42no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D412yes—
46C2×#103—22no—
47C2×#103—22no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C4×C8—21no—
52#391—21no—
53#391—21no—
54#391—21no—
55#391—21no—
56#391—21no—
57#391—44no—
58#389—21no—
59#389—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384—44no—
65#384D411yescharacteristic
66#380—21no—
67#380—21no—
68C2×#95—21no—
69C2×#95—24no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D412yes—
73#340—22no—
74#340—24no—
75#340—24no—
76#336—21no—
77#336—21no—
78#336D412yes—
79C22×Q8—21no—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8SD1612yes—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×Q16—44no—
96C2×Q16—82no—
97C2×Q16—82no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—41no—
101C4⋊Q8—42no—
102C22⋊Q8—41no—
103C22⋊Q8—41no—
104C22⋊Q8—41no—
105C22⋊Q8—41no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—84no—
127C2×C4⋊C4—84no—
128C2×C22⋊C4—41no—
129C2×C42—24no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138C4⋊C8—42no—
139C4⋊C8—42no—
140#103—44no—
141#103—82no—
142C22⋊C8—42no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—42no—
146C4×C8—42no—
147#95—24no—
148#95—22no—
149#95—48no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95Q1614yescomplemented
157C24#12011yescharacteristic
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—82no—
171C2×Q8—82no—
172C2×Q8—81no—
173C2×Q8—81no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4#12011yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12711yescharacteristic
210C22×C4C2×SD1611yescharacteristic
211C22×C4C2×Q1612yes—
212C22×C4#13511yescharacteristic
213C22×C4AGL(1,ℤ/8)11yescharacteristic
214C22×C4#13711yescharacteristic
215C22×C4#13712yes—
216Q16—88no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—82no—
235C4⋊C4—81no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—816no—
240C4⋊C4—82no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C42—22no—
246C42—22no—
247C42—41no—
248C42—41no—
249C42—44no—
250C42—44no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23#39211yescharacteristic
256C23#44811yescharacteristic
257C23#44911yescharacteristic
258C23#45112yes—
259C23#45512yes—
260Q8—44no—
261Q8—82no—
262Q8—82no—
263Q8—82no—
264Q8—82no—
265Q8—82no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4#45011yescharacteristic
312C2×C4#45211yescharacteristic
313C2×C4#45312yes—
314C2×C4#45612yes—
315C2×C4#45812yes—
316C2×C4#46212yes—
317C2×C4#46511yescharacteristic
318C2×C4#46811yescharacteristic
319C2×C4#49411yescharacteristic
320C2×C4#49711yescharacteristic
321C8—82no—
322C8—82no—
323C8—161no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22#164412yes—
333C22#164512yes—
334C22#200011yescharacteristic
335C22#200211yescharacteristic
336C22#200411yescharacteristic
337C22#200511yescharacteristic
338C22#200811yescharacteristic
339C22#201611yescharacteristic
340C22#202211yescharacteristic
341C4—22no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—22no—
346C4—21no—
347C4—48no—
348C4—41no—
349C4—82no—
350C4—161no—
351C4—161no—
352C4—162no—
353C4—161no—
354C4#161312yes—
355C4#161812yes—
356C2—22no—
357C2—22no—
358C2#992011yescharacteristic
359C2#995911yescharacteristic
360C2#1006511yescharacteristic
361C2#1009611yescharacteristic
362C2#1024511yescharacteristic
363C2#1029311yescharacteristic
364C2#1033511yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
156. #95Q16semidirect

Maximal subgroups

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

subgroupindex
#297152
#279612
#151992
#99092
#87252
#75752
#70282