Finite Group Databasebeta

← #330118

Subgroups — #330118

Order 512.

547 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29941C212yesmaximal
2#29808C211yesmaximal, characteristic
3#21973C211yesmaximal, characteristic
4#10658C212yesmaximal, complemented
5#10600C212yesmaximal, complemented
6#10588C212yesmaximal, complemented
7#10538C212yesmaximal, complemented
8#9131C212yesmaximal
9#8600C211yesmaximal, characteristic
10C2×#494C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2857C2212yes—
14#2283C2211yescharacteristic
15#2101—22no—
16#2085—22no—
17#2077—22no—
18#2076—22no—
19#2056—21no—
20#2056—21no—
21#2001—21no—
22#2001—21no—
23#1988C2212yes—
24#1988C2212yes—
25#1956C2212yes—
26#1948C2212yes—
27#1947C2212yes—
28#1939C2212yes—
29#1939C2212yes—
30#1850—21no—
31#1850—21no—
32#1843—21no—
33#1834C2211yescharacteristic
34#1832—21no—
35#1797C2212yes—
36C2×#335C2211yescharacteristic
37C2×#334C2211yescharacteristic
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1277C2212yes—
43C22×Q16—21no—
44C22×Q16—21no—
45C23×C8—21no—
46C2×C4⋊Q8C2312yes—
47C2×C22⋊Q8—22no—
48C22×C42C2311yescharacteristic
49#494—41no—
50#494—41no—
51#431—41no—
52#431—41no—
53C2×#107—22no—
54C2×#106—22no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—41no—
59C2×C4⋊C8—41no—
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8D411yescharacteristic
69C2×C22⋊C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—22no—
73#389—22no—
74#389—22no—
75#384C2312yes—
76#383—21no—
77#382D412yes—
78#381—21no—
79#380—22no—
80#380—22no—
81#380D411yescharacteristic
82#380D411yescharacteristic
83C4×C4⋊C4C2312yes—
84C4×C22⋊C4—21no—
85#376—21no—
86#340—21no—
87#340—21no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#335—41no—
98#334—41no—
99#332—42no—
100#332—42no—
101C22×Q8—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×Q16—42no—
106C2×Q16—42no—
107C2×Q16—82no—
108C2×Q16—82no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—42no—
113C22×C8—82no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C22⋊Q8—82no—
123C22⋊Q8—82no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—42no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42C2411yescharacteristic
144#107—82no—
145#106—82no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—42no—
155C4⋊C8—82no—
156C4⋊C8—82no—
157#103—82no—
158#103—82no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—82no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—42no—
172#95—42no—
173C24#12111yescharacteristic
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
202Q16—84no—
203Q16—84no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—84no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—84no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—42no—
246C42—42no—
247C42H(5,𝔽2)11yescharacteristic
248C42H(5,𝔽2)11yescharacteristic
249C422−1+411yescharacteristic
250C422−1+411yescharacteristic
251C23—21no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23#47312yes—
256C23C2×#12111yescharacteristic
257Q8—82no—
258Q8—82no—
259Q8—82no—
260Q8—82no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4#53411yescharacteristic
291C2×C4#53611yescharacteristic
292C2×C4#53711yescharacteristic
293C2×C4#53711yescharacteristic
294C2×C4#53911yescharacteristic
295C2×C4#53911yescharacteristic
296C2×C4#57612yes—
297C2×C4#57712yes—
298C2×C4#57712yes—
299C2×C4#57812yes—
300C8—41no—
301C8—41no—
302C8—41no—
303C8—41no—
304C8—82no—
305C8—82no—
306C8—82no—
307C22—22no—
308C22—24no—
309C22—41no—
310C22—41no—
311C22—42no—
312C22#261811yescharacteristic
313C22#304512yes—
314C22#306312yes—
315C22C2×#47311yescharacteristic
316C22#306711yescharacteristic
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—82no—
323C4—82no—
324C4—162no—
325C4—162no—
326C4#319911yescharacteristic
327C4#320011yescharacteristic
328C4#324112yes—
329C4#325812yes—
330C4#326111yescharacteristic
331C4#326311yescharacteristic
332C2—42no—
333C2#1957712yes—
334C2#1958312yes—
335C2#2976511yescharacteristic
336C2#2978611yescharacteristic
337C2#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. #10658C2semidirect
5. #10600C2semidirect
6. #10588C2semidirect
7. #10538C2semidirect

Maximal subgroups

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

subgroupindex
#299412
#298082
#219732
#106582
#106002
#105882
#105382
#91312
#86002