Finite Group Databasebeta

← #329787

Subgroups — #329787

Order 512.

678 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic
2#33108C212yesmaximal
3C2×#1939C211yesmaximal, characteristic
4#10651C212yesmaximal, complemented
5#10602C214yesmaximal, complemented
6#10545C212yesmaximal, complemented
7#9129C212yesmaximal
8#8596C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#483—22no—
11C22×C4⋊C8—21no—
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2857C2212yes—
15C2×C4×C4⋊C4C2211yescharacteristic
16#1990C2212yes—
17#1981C2214yes—
18#1948C2214yes—
19#1942—21no—
20#1942—21no—
21#1939—21no—
22#1939—21no—
23#1939C2212yes—
24#1939C2212yes—
25#1927—24no—
26#1872—22no—
27#1845C2211yescharacteristic
28#1808—21no—
29#1808—21no—
30#1797C2212yes—
31C2×#336—21no—
32C2×#335C2212yes—
33C2×#332C2214yes—
34#1296C2214yes—
35#1277C2212yes—
36C23×C8—21no—
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C2×C4⋊Q8C2312yes—
40C2×C22⋊Q8—22no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45#500—42no—
46#483—48no—
47C2×#107—21no—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107—21no—
51C2×#106—24no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—42no—
56C2×C4⋊C8—44no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2314yes—
59C2×#103—24no—
60C2×#103—24no—
61C2×C22⋊C8—22no—
62C2×C4×C8—42no—
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#391—22no—
68#389—22no—
69#389—22no—
70#384C2312yes—
71#380—22no—
72#380—22no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4D412yes—
76C4×C4⋊C4C2312yes—
77C2×#95—21no—
78#340—22no—
79#340—22no—
80#340—24no—
81#336—22no—
82#336—22no—
83#336—22no—
84#335—42no—
85#332—44no—
86C22×Q8—22no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—24no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8SD1614yes—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C22⋊Q8—44no—
110C22⋊Q8—44no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4D814yes—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C22⋊C4—42no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42C2411yescharacteristic
135#107—42no—
136#107—42no—
137#107—42no—
138#107—42no—
139#106—48no—
140C4⋊C8—24no—
141C4⋊C8—24no—
142C4⋊C8—24no—
143C4⋊C8—42no—
144C4⋊C8—44no—
145C4⋊C8—44no—
146#103—48no—
147#103—48no—
148C22⋊C8—44no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—41no—
152C4×C8—41no—
153#95—22no—
154#95—22no—
155#95—42no—
156#95—42no—
157C24#12111yescharacteristic
158C2×Q8—44no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—44no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4C2×D812yes—
185C22×C4C2×SD1612yes—
186C22×C4#13512yes—
187C22×C4#13712yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—44no—
201C2×C8—44no—
202C2×C8—42no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—42no—
206C2×C8—44no—
207C2×C8—42no—
208C2×C8—84no—
209C2×C8C2×SD1612yes—
210C2×C8C2×SD1612yes—
211C2×C8#13712yes—
212C4⋊C4—22no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—84no—
225C4⋊C4C2×D814yes—
226C22⋊C4—84no—
227C42—21no—
228C42—21no—
229C42—22no—
230C42—22no—
231C42—42no—
232C42—42no—
233C42—44no—
234C42H(5,𝔽2)12yes—
235C422−1+412yes—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23#46112yes—
241C23#46612yes—
242C23#47312yes—
243C23C2×#12111yescharacteristic
244Q8—84no—
245Q8—84no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—84no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#46112yes—
277C2×C4#46612yes—
278C2×C4#53411yescharacteristic
279C2×C4#53611yescharacteristic
280C2×C4#53712yes—
281C2×C4#53912yes—
282C2×C4C22×D811yescharacteristic
283C2×C4C22×SD1611yescharacteristic
284C2×C4C2×#13511yescharacteristic
285C2×C4C2×#13711yescharacteristic
286C2×C4C2×#13711yescharacteristic
287C2×C4#57511yescharacteristic
288C2×C4#57512yes—
289C8—22no—
290C8—22no—
291C8—22no—
292C8—41no—
293C8—82no—
294C8—84no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22#167314yes—
302C22#261811yescharacteristic
303C22C2×#46111yescharacteristic
304C22#303711yescharacteristic
305C22C2×#46611yescharacteristic
306C22#305411yescharacteristic
307C22C2×#47311yescharacteristic
308C22#306711yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—41no—
313C4—44no—
314C4—82no—
315C4—162no—
316C4—162no—
317C4#318711yescharacteristic
318C4#319511yescharacteristic
319C4#322911yescharacteristic
320C4#323611yescharacteristic
321C4#324512yes—
322C4#325012yes—
323C2—24no—
324C2C2×#167311yescharacteristic
325C2#1913711yescharacteristic
326C2#1926111yescharacteristic
327C2#1929011yescharacteristic
328C2#2975611yescharacteristic
329C2#2977011yescharacteristic
330C2#2978911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10651C2semidirect
5. #10602C2semidirect
6. #10545C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#331082
C2×#19392
#106512
#106022
#105452
#91292
#85962