Finite Group Databasebeta

← #329608

Subgroups — #329608

Order 512.

716 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32958C212yesmaximal
2#29878C211yesmaximal, characteristic, complemented
3#21847C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10622C211yesmaximal, characteristic, complemented
6#10483C212yesmaximal, complemented
7#10481C212yesmaximal, complemented
8#10279C211yesmaximal, characteristic, complemented
9#10278C211yesmaximal, characteristic, complemented
10#9127C212yesmaximal, complemented
11#8599C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#477C2212yes—
15C2×#476C2212yes—
16C2×#431C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2857C2212yes—
19#2087—22no—
20#2076—22no—
21#1990C2212yes—
22#1984C2211yescharacteristic
23#1984C2211yescharacteristic
24#1954C2211yescharacteristic
25#1954C2211yescharacteristic
26#1869—22no—
27#1865—22no—
28#1862—22no—
29#1854—22no—
30#1834C2211yescharacteristic
31#1803C2212yes—
32#1802C2212yes—
33#1797—22no—
34#1797C2212yes—
35#1788—22no—
36C2×#335C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×#41D411yescharacteristic
44C23×C8—21no—
45C2×C4⋊Q8C2312yes—
46C2×C22⋊Q8—22no—
47C4×#41—42no—
48C4×#41—42no—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×#117—21no—
56C2×#117D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#500—42no—
60#477—42no—
61#477—42no—
62#476—42no—
63#476—42no—
64#431—42no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#106—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—42no—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8D412yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—22no—
86#389—22no—
87#389—22no—
88#384C2312yes—
89#380—22no—
90#380—22no—
91#340—22no—
92#340—24no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#335—42no—
99#333—42no—
100#333—42no—
101C2×#41—22no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41Q1614yes—
105C22×Q8—22no—
106C22×Q8C2×D411yescharacteristic
107C22×D4—21no—
108C22×D4C2×D411yescharacteristic
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—42no—
120C22×C8—82no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C22⋊Q8—84no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—42no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—42no—
134C4×D4—44no—
135C4×D4—42no—
136C4×D4—42no—
137#117—44no—
138#117—42no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4C2×D411yescharacteristic
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—42no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42C2411yescharacteristic
164#107—44no—
165#107—44no—
166#106—44no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—42no—
174C4⋊C8—42no—
175#103—44no—
176#103—44no—
177#102—44no—
178#102—44no—
179C22⋊C8—42no—
180C22⋊C8—44no—
181C22⋊C8—82no—
182C22⋊C8—82no—
183C4×C8—22no—
184C4×C8—22no—
185#95—42no—
186#95—42no—
187C24—21no—
188C24#12111yescharacteristic
189#41—48no—
190#41—48no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8C2×Q1612yes—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4C2×Q1612yes—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—82no—
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C2×Q1612yes—
232C22×C4#13512yes—
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4111yescharacteristic
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—84no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—82no—
266C4⋊C4—84no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—84no—
271C42—21no—
272C42—22no—
273C42—21no—
274C42—22no—
275C42—42no—
276C42—42no—
277C42—41no—
278C42—41no—
279C42H(5,𝔽2)12yes—
280C42H(5,𝔽2)12yes—
281C23—21no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—41no—
289C23—42no—
290C23#46712yes—
291C23C2×#12111yescharacteristic
292Q8—24no—
293Q8—42no—
294Q8—162no—
295Q8—162no—
296D4—24no—
297D4—48no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—48no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#46712yes—
331C2×C4#53412yes—
332C2×C4#53411yescharacteristic
333C2×C4#53511yescharacteristic
334C2×C4#53812yes—
335C2×C4C22×Q1611yescharacteristic
336C2×C4C2×#13511yescharacteristic
337C2×C4#57512yes—
338C2×C4#57712yes—
339C2×C4#57812yes—
340C8—42no—
341C8—42no—
342C8—82no—
343C8—82no—
344C8—81no—
345C8—81no—
346C22—22no—
347C22—24no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—44no—
353C22—81no—
354C22—81no—
355C22—82no—
356C22#261711yescharacteristic
357C22#304612yes—
358C22C2×#46711yescharacteristic
359C22#305411yescharacteristic
360C22#306812yes—
361C4—22no—
362C4—22no—
363C4—41no—
364C4—42no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—162no—
369C4—162no—
370C4#318912yes—
371C4#319112yes—
372C4#320112yes—
373C4#322211yescharacteristic
374C4#322311yescharacteristic
375C2—42no—
376C2—82no—
377C2#1941712yes—
378C2#1945512yes—
379C2#2972611yescharacteristic
380C2#2973111yescharacteristic
381C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
2. #29878C2semidirect
4. #10624C2semidirect
5. #10622C2semidirect
6. #10483C2semidirect
7. #10481C2semidirect
8. #10279C2semidirect
9. #10278C2semidirect
10. #9127C2semidirect
11. #8599C2semidirect

Maximal subgroups

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

subgroupindex
#329582
#298782
#218472
#106242
#106222
#104832
#104812
#102792
#102782
#91272
#85992