Finite Group Databasebeta

← #303592

Subgroups — #303592

Order 512.

1070 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1070 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
1D4×C4⋊Q8C211yesmaximal, characteristic
2#33068C211yesmaximal, characteristic
3#30228C211yesmaximal, characteristic, complemented
4#29150C211yesmaximal, characteristic, complemented
5#10418C214yesmaximal, complemented
6#9705C211yesmaximal, characteristic, complemented
7#9641C211yesmaximal, characteristic, complemented
8#8447C214yesmaximal
9#8174C211yesmaximal, characteristic
10C2×D4×Q8C2212yes—
11C22×C4⋊Q8—21no—
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857—21no—
15#2551—22no—
16#2546C2212yes—
17#2440—24no—
18#2434—22no—
19D4×C4⋊C4—22no—
20C4×C4⋊Q8C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#2099—24no—
23#2026—22no—
24#1990C2214yes—
25#1960—21no—
26#1948C2212yes—
27#1920—22no—
28#1864—22no—
29#1836—21no—
30#1831C2211yescharacteristic
31#1803C2214yes—
32#1802C2212yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#332C2214yes—
36C2×Q8⋊C8C2214yes—
37C2×D4⋊C8C2212yes—
38#1279C2212yes—
39C23×Q8—22no—
40D4×Q8—28no—
41D4×Q8—44no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—44no—
48C2×C4⋊Q8—44no—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×#125—21no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—24no—
54C2×C4×Q8C2312yes—
55C2×C4×D4—22no—
56C2×C4×D4D412yes—
57C2×C4×D4C2311yescharacteristic
58C22×C4⋊C4—24no—
59C22×C42—21no—
60C2×#106—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—24no—
64C2×#102—22no—
65C2×#97—21no—
66C2×C4×C8C2312yes—
67#398—24no—
68#395—28no—
69#391—22no—
70#391D412yes—
71#389—24no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389—22no—
76#385—28no—
77#384—22no—
78#384D411yescharacteristic
79#384D414yes—
80#384C2312yes—
81#382—21no—
82#380—24no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4C2311yescharacteristic
86C4×C22⋊C4—21no—
87C4×C22⋊C4—22no—
88C43C2311yescharacteristic
89#340—22no—
90#340—24no—
91#340—28no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#332—44no—
97Q8⋊C8—44no—
98D4⋊C8—24no—
99C22×Q8—24no—
100C22×Q8—22no—
101C22×Q8—48no—
102C22×Q8—44no—
103C22×Q8—44no—
104C22×Q8—84no—
105C22×Q8C2×D412yes—
106C22×D4C2×D411yescharacteristic
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—24no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—24no—
115C4⋊Q8—24no—
116C4⋊Q8—416no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119C4⋊Q8—44no—
120C4⋊Q8—44no—
121C4⋊Q8—42no—
122C4⋊Q8—42no—
123C4⋊Q8—82no—
124C4⋊Q8SD1614yes—
125#125—82no—
126C22⋊Q8—416no—
127C22⋊Q8—48no—
128C4×Q8—24no—
129C4×Q8—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—22no—
133C4×D4—44no—
134C4×D4—44no—
135C4×D4#4114yes—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—416no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D412yes—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—24no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—21no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42C2×D412yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D412yes—
164C2×C42C2×D411yescharacteristic
165C2×C42C2411yescharacteristic
166#106—44no—
167C4⋊C8—22no—
168C4⋊C8—24no—
169C4⋊C8—22no—
170#103—84no—
171#102—44no—
172#97—42no—
173C4×C8—22no—
174C4×C8—22no—
175#95—28no—
176#95—22no—
177#95—24no—
178#95—21no—
179#95—42no—
180C24—21no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—24no—
184C2×Q8—48no—
185C2×Q8—48no—
186C2×Q8—44no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—44no—
190C2×Q8—84no—
191C2×Q8—816no—
192C2×Q8—816no—
193C2×Q8C2×SD1614yes—
194C2×D4—21no—
195C2×D4—22no—
196C2×D4#12412yes—
197C2×D4#13714yes—
198C22×C4—28no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—48no—
212C22×C4—42no—
213C22×C4—84no—
214C22×C4#12012yes—
215C22×C4#12012yes—
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C2×C8—44no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—28no—
230C4⋊C4—24no—
231C4⋊C4—28no—
232C4⋊C4—44no—
233C4⋊C4—416no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4#12412yes—
246C4⋊C4#13714yes—
247C22⋊C4—28no—
248C22⋊C4—24no—
249C22⋊C4—48no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—42no—
254C42—42no—
255C42—42no—
256C42—41no—
257C42—41no—
258C42—44no—
259C42C2×SD1612yes—
260C42AGL(1,ℤ/8)12yes—
261C42C2×#4112yes—
262C42H(5,𝔽2)12yes—
263C23—22no—
264C23—24no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23C2×#12011yescharacteristic
269Q8—28no—
270Q8—44no—
271Q8—816no—
272Q8—84no—
273Q8—84no—
274D4—22no—
275D4—24no—
276D4#49114yes—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—416no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—816no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4#45014yes—
305C2×C4C2×#12411yescharacteristic
306C2×C4#53511yescharacteristic
307C2×C4#54612yes—
308C2×C4#54612yes—
309C2×C4C22×SD1611yescharacteristic
310C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
311C2×C4C2×#13712yes—
312C2×C4C2×#13712yes—
313C2×C4#57612yes—
314C8—82no—
315C8—82no—
316C8—82no—
317C22—24no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—44no—
322C22#243511yescharacteristic
323C22C2×#45011yescharacteristic
324C22#300412yes—
325C22#300811yescharacteristic
326C22#302012yes—
327C4—22no—
328C4—24no—
329C4—42no—
330C4—44no—
331C4—44no—
332C4—42no—
333C4—82no—
334C4—84no—
335C4—162no—
336C4C2×#49111yescharacteristic
337C4#313911yescharacteristic
338C4#321812yes—
339C4#329512yes—
340C4#332512yes—
341C2—44no—
342C2#1877712yes—
343C2#1882812yes—
344C2#2586111yescharacteristic
345C2#2587911yescharacteristic
346C2#2588111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
3. #30228C2semidirect
4. #29150C2semidirect
5. #10418C2semidirect
6. #9705C2semidirect
7. #9641C2semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#330682
#302282
#291502
#104182
#97052
#96412
#84472
#81742