Finite Group Databasebeta

← #303615

Subgroups — #303615

Order 512.

1176 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 182 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
1Q8×#127C211yesmaximal, characteristic
2#33069C211yesmaximal, characteristic, complemented
3#30230C211yesmaximal, characteristic, complemented
4#29149C211yesmaximal, characteristic
5#10421C214yesmaximal, complemented
6#9703C211yesmaximal, characteristic, complemented
7#9642C211yesmaximal, characteristic
8#8447C214yesmaximal, complemented
9#8175C211yesmaximal, characteristic, complemented
10C2×D4×Q8—22no—
11C2×D4×Q8C2212yes—
12#2868—21no—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2542—22no—
16#2542—24no—
17#2542—22no—
18#2542C2212yes—
19C4×#127—21no—
20C4×#127C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#2100—24no—
23#2025—22no—
24#1990C2214yes—
25#1961—21no—
26#1947C2212yes—
27#1921—22no—
28#1863—22no—
29#1836—21no—
30#1831C2211yescharacteristic
31#1803C2212yes—
32#1802C2214yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#332C2214yes—
36C2×Q8⋊C8C2212yes—
37C2×D4⋊C8C2214yes—
38#1279C2212yes—
39C23×Q8—22no—
40D4×Q8—28no—
41D4×Q8—44no—
42D4×Q8—48no—
43D4×Q8—48no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2312yes—
48C2×#127D411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—28no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—24no—
54C2×C4×Q8D412yes—
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2312yes—
60C2×#107—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#102—24no—
65C2×#97—21no—
66C2×C4×C8C2312yes—
67#399—28no—
68#399—24no—
69#390—24no—
70#390—22no—
71#390—22no—
72#390D412yes—
73#389—22no—
74#389—22no—
75#384—24no—
76#384—22no—
77#384—21no—
78#384—21no—
79#384D411yescharacteristic
80#384D414yes—
81#384C2312yes—
82#382—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4C2311yescharacteristic
87C43—21no—
88C43C2311yescharacteristic
89#340—24no—
90#340—22no—
91#340—28no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#332—44no—
97Q8⋊C8—24no—
98D4⋊C8—44no—
99C22×Q8—24no—
100C22×Q8—22no—
101C22×Q8—44no—
102C22×Q8—48no—
103C22×Q8—44no—
104C22×Q8—48no—
105C22×Q8C2×D411yescharacteristic
106C22×D4—22no—
107C22×D4C2×D412yes—
108C23×C4—24no—
109C23×C4—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C4⋊Q8—24no—
114C4⋊Q8—24no—
115C4⋊Q8—48no—
116C4⋊Q8—416no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119#127—22no—
120#127Q1614yes—
121#125—82no—
122C22⋊Q8—416no—
123C22⋊Q8—432no—
124C22⋊Q8—44no—
125C22⋊Q8—48no—
126C22⋊Q8—44no—
127C22⋊Q8—48no—
128C4×Q8—24no—
129C4×Q8—24no—
130C4×Q8—22no—
131C4×Q8#4114yes—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—48no—
135C4×D4—416no—
136C4×D4—44no—
137C4×D4—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D414yes—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D412yes—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—22no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—21no—
162C2×C42C2×D412yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D411yescharacteristic
166C2×C42C2411yescharacteristic
167#107—44no—
168C4⋊C8—22no—
169C4⋊C8—24no—
170C4⋊C8—22no—
171#103—44no—
172#102—84no—
173#97—42no—
174C4×C8—22no—
175C4×C8—22no—
176#95—22no—
177#95—21no—
178#95—42no—
179C24—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—48no—
183C2×Q8—416no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—88no—
189C2×Q8—816no—
190C2×Q8—88no—
191C2×Q8#12412yes—
192C2×Q8AGL(1,ℤ/8)14yes—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—24no—
196C2×D4—28no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4C2×Q1614yes—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—48no—
206C22×C4—416no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4#12012yes—
213C22×C4#12012yes—
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D411yescharacteristic
217C2×C8—44no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C4⋊C4—28no—
226C4⋊C4—21no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—28no—
231C4⋊C4—48no—
232C4⋊C4—44no—
233C4⋊C4—48no—
234C4⋊C4—48no—
235C4⋊C4—416no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—84no—
242C4⋊C4#12412yes—
243C4⋊C4AGL(1,ℤ/8)14yes—
244C22⋊C4—48no—
245C22⋊C4—48no—
246C22⋊C4—416no—
247C22⋊C4—44no—
248C42—28no—
249C42—24no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—42no—
254C42—44no—
255C42—42no—
256C42C2×Q1612yes—
257C42#13712yes—
258C42C2×#4112yes—
259C42H(5,𝔽2)12yes—
260C23—24no—
261C23—42no—
262C23—44no—
263C23—42no—
264C23—44no—
265C23C2×#12011yescharacteristic
266Q8—22no—
267Q8—24no—
268Q8—816no—
269Q8—82no—
270Q8—82no—
271Q8—84no—
272Q8#49014yes—
273D4—28no—
274D4—416no—
275D4—44no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—816no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#45114yes—
301C2×C4C2×#12411yescharacteristic
302C2×C4#53511yescharacteristic
303C2×C4#54612yes—
304C2×C4#54612yes—
305C2×C4C22×Q1611yescharacteristic
306C2×C4C2×AGL(1,ℤ/8)12yes—
307C2×C4C2×AGL(1,ℤ/8)12yes—
308C2×C4C2×#13711yescharacteristic
309C2×C4#57712yes—
310C8—82no—
311C8—82no—
312C8—82no—
313C22—44no—
314C22—48no—
315C22—82no—
316C22—82no—
317C22—84no—
318C22#243511yescharacteristic
319C22C2×#45111yescharacteristic
320C22#300312yes—
321C22#301011yescharacteristic
322C22#302012yes—
323C4—24no—
324C4—22no—
325C4—24no—
326C4—42no—
327C4—42no—
328C4—44no—
329C4—84no—
330C4—82no—
331C4—162no—
332C4C2×#49011yescharacteristic
333C4#313811yescharacteristic
334C4#321712yes—
335C4#329912yes—
336C4#332912yes—
337C2—84no—
338C2#1875912yes—
339C2#1882612yes—
340C2#2586011yescharacteristic
341C2#2587911yescharacteristic
342C2#2588811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33069C2semidirect
3. #30230C2semidirect
5. #10421C2semidirect
6. #9703C2semidirect
8. #8447C2semidirect
9. #8175C2semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#330692
#302302
#291492
#104212
#97032
#96422
#84472
#81752