Finite Group Databasebeta

← #303596

Subgroups — #303596

Order 512.

1080 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1080 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#33066C211yesmaximal, characteristic
3#30228C211yesmaximal, characteristic, complemented
4#28725C211yesmaximal, characteristic, complemented
5#10419C214yesmaximal, complemented
6#9705C211yesmaximal, characteristic, complemented
7#9641C211yesmaximal, characteristic, complemented
8#8447C214yesmaximal
9#8174C211yesmaximal, characteristic
10C2×D4×Q8C2212yes—
11C22×C4⋊Q8—21no—
12C2×#491—21no—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2857—21no—
16#2551—22no—
17#2546C2212yes—
18#2440—24no—
19#2434—22no—
20D4×C4⋊C4—22no—
21C4×C4⋊Q8C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#2099—24no—
24#2074—22no—
25#1992—21no—
26#1990C2214yes—
27#1947C2212yes—
28#1918—22no—
29#1862—22no—
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—
60#491—44no—
61C2×#107—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103—24no—
65C2×#102—22no—
66C2×#97—21no—
67C2×C4×C8C2312yes—
68#398—24no—
69#395—28no—
70#391—22no—
71#391D412yes—
72#389—24no—
73#389—24no—
74#389—22no—
75#389—22no—
76#389—22no—
77#385—28no—
78#384—22no—
79#384D411yescharacteristic
80#384D414yes—
81#384C2312yes—
82#382—21no—
83#380—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D412yes—
86C4×C4⋊C4C2311yescharacteristic
87C4×C22⋊C4—21no—
88C4×C22⋊C4—22no—
89C43C2311yescharacteristic
90#340—22no—
91#340—24no—
92#340—28no—
93#340—22no—
94#336—21no—
95#336—22no—
96#336—22no—
97#332—44no—
98Q8⋊C8—44no—
99D4⋊C8—24no—
100C22×Q8—24no—
101C22×Q8—22no—
102C22×Q8—48no—
103C22×Q8—44no—
104C22×Q8—44no—
105C22×Q8—84no—
106C22×Q8C2×D412yes—
107C22×D4C2×D411yescharacteristic
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—24no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—24no—
116C4⋊Q8—24no—
117C4⋊Q8—416no—
118C4⋊Q8—44no—
119C4⋊Q8—44no—
120C4⋊Q8—44no—
121C4⋊Q8—44no—
122C4⋊Q8—42no—
123C4⋊Q8—42no—
124C4⋊Q8—82no—
125C4⋊Q8D814yes—
126#125—42no—
127#125—42no—
128C22⋊Q8—416no—
129C22⋊Q8—48no—
130C4×Q8—24no—
131C4×Q8—24no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—22no—
135C4×D4—44no—
136C4×D4—44no—
137C4×D4#4114yes—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—416no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—24no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42C2×D412yes—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D412yes—
166C2×C42C2×D411yescharacteristic
167C2×C42C2411yescharacteristic
168#107—44no—
169C4⋊C8—22no—
170C4⋊C8—24no—
171C4⋊C8—22no—
172#103—48no—
173#102—44no—
174#97—42no—
175C4×C8—22no—
176C4×C8—22no—
177#95—28no—
178#95—22no—
179#95—24no—
180#95—21no—
181#95—42no—
182C24—21no—
183C2×Q8—22no—
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—48no—
187C2×Q8—48no—
188C2×Q8—44no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—44no—
192C2×Q8—84no—
193C2×Q8—816no—
194C2×Q8—816no—
195C2×Q8C2×D814yes—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4#12412yes—
199C2×D4#13714yes—
200C22×C4—28no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—48no—
214C22×C4—42no—
215C22×C4—84no—
216C22×C4#12012yes—
217C22×C4#12012yes—
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—28no—
234C4⋊C4—44no—
235C4⋊C4—416no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—48no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4#12412yes—
249C4⋊C4#13714yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—48no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42—41no—
260C42—41no—
261C42—44no—
262C42C2×D812yes—
263C42AGL(1,ℤ/8)12yes—
264C42C2×#4112yes—
265C42H(5,𝔽2)12yes—
266C23—22no—
267C23—24no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23C2×#12011yescharacteristic
272Q8—28no—
273Q8—44no—
274Q8—816no—
275Q8—84no—
276Q8—84no—
277D4—22no—
278D4—24no—
279D4#49114yescomplemented
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—416no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—816no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#44714yes—
308C2×C4C2×#12411yescharacteristic
309C2×C4#53511yescharacteristic
310C2×C4#54612yes—
311C2×C4#54612yes—
312C2×C4C22×D811yescharacteristic
313C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
314C2×C4C2×#13712yes—
315C2×C4C2×#13712yes—
316C2×C4#57712yes—
317C8—82no—
318C8—82no—
319C8—82no—
320C22—24no—
321C22—42no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22#243511yescharacteristic
326C22C2×#44711yescharacteristic
327C22#300412yes—
328C22#300911yescharacteristic
329C22#302012yes—
330C4—22no—
331C4—24no—
332C4—42no—
333C4—44no—
334C4—44no—
335C4—42no—
336C4—82no—
337C4—84no—
338C4—162no—
339C4C2×#49111yescharacteristic
340C4#313911yescharacteristic
341C4#321812yes—
342C4#329412yes—
343C4#332412yes—
344C2—44no—
345C2#1877612yes—
346C2#1882912yes—
347C2#2586111yescharacteristic
348C2#2587911yescharacteristic
349C2#2588511yescharacteristic
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. #28725C2semidirect
5. #10419C2semidirect
6. #9705C2semidirect
7. #9641C2semidirect
279. D4#491semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#330662
#302282
#287252
#104192
#97052
#96412
#84472
#81742