Finite Group Databasebeta

← #329604

Subgroups — #329604

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32955C212yesmaximal
2#29808C211yesmaximal, characteristic, complemented
3#21847C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10622C211yesmaximal, characteristic, complemented
6#10487C212yesmaximal, complemented
7#10485C212yesmaximal, complemented
8#10276C211yesmaximal, characteristic, complemented
9#10274C211yesmaximal, characteristic, complemented
10#9131C212yesmaximal, complemented
11#8599C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#494C2211yescharacteristic
14C2×#475C2212yes—
15C2×#474C2212yes—
16C2×#431C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2857C2212yes—
19#2056—22no—
20#2001—22no—
21#1984C2211yescharacteristic
22#1984C2211yescharacteristic
23#1956C2212yes—
24#1954C2211yescharacteristic
25#1954C2211yescharacteristic
26#1869—22no—
27#1864—22no—
28#1863—22no—
29#1854—22no—
30#1834C2211yescharacteristic
31#1803C2212yes—
32#1802C2212yes—
33#1797—22no—
34#1797C2212yes—
35#1788—22no—
36C2×#335C2212yes—
37C2×#332C2212yes—
38C2×#332C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×#41D411yescharacteristic
44C22×Q16—22no—
45C23×C8—21no—
46C2×C4⋊Q8C2312yes—
47C2×C22⋊Q8—22no—
48C4×#41—42no—
49C4×#41—42no—
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8C2311yescharacteristic
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#494—42no—
61#475—42no—
62#475—42no—
63#474—42no—
64#474—42no—
65#431—42no—
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×#103—22no—
78C2×#103—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8D412yes—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#391—22no—
88#389—22no—
89#389—22no—
90#384C2312yes—
91#380—22no—
92#380—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#336—22no—
97#336—22no—
98#336—22no—
99#335—42no—
100#332—42no—
101#332—42no—
102C2×#41—22no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41Q1614yescomplemented
106C22×Q8—22no—
107C22×Q8C2×D411yescharacteristic
108C22×D4—21no—
109C22×D4C2×D411yescharacteristic
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C2×Q16—44no—
117C2×Q16—82no—
118C2×Q16—82no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—42no—
124C22×C8—82no—
125C22×C8C2×D412yes—
126C22×C8C2×D412yes—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C22⋊Q8—82no—
132C22⋊Q8—82no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—42no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—42no—
139C4×D4—44no—
140C4×D4—42no—
141C4×D4—42no—
142#117—44no—
143#117—42no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—42no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42C2411yescharacteristic
169#107—44no—
170#106—44no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178C4⋊C8—42no—
179#103—44no—
180#103—44no—
181#103—82no—
182#103—82no—
183#102—44no—
184#102—44no—
185C22⋊C8—42no—
186C22⋊C8—44no—
187C22⋊C8—82no—
188C22⋊C8—82no—
189C4×C8—22no—
190C4×C8—22no—
191#95—42no—
192#95—42no—
193C24—21no—
194C24#12111yescharacteristic
195#41—48no—
196#41—48no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8C2×Q1612yes—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4C2×Q1612yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C2×Q1612yes—
240C22×C4#13512yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4111yescharacteristic
244Q16—88no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—84no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—84no—
280C42—21no—
281C42—22no—
282C42—21no—
283C42—22no—
284C42—42no—
285C42—42no—
286C42—41no—
287C42—41no—
288C42H(5,𝔽2)12yes—
289C42H(5,𝔽2)12yes—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—41no—
296C23—42no—
297C23—41no—
298C23—42no—
299C23#46712yes—
300C23C2×#12111yescharacteristic
301Q8—24no—
302Q8—42no—
303Q8—82no—
304Q8—82no—
305Q8—82no—
306Q8—82no—
307D4—24no—
308D4—48no—
309D4—42no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—48no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#46712yes—
342C2×C4#53412yes—
343C2×C4#53411yescharacteristic
344C2×C4#53511yescharacteristic
345C2×C4#53812yes—
346C2×C4C22×Q1611yescharacteristic
347C2×C4C2×#13511yescharacteristic
348C2×C4#57512yes—
349C2×C4#57612yes—
350C2×C4#57712yes—
351C8—42no—
352C8—42no—
353C8—82no—
354C8—82no—
355C8—81no—
356C8—81no—
357C22—22no—
358C22—24no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—44no—
364C22—81no—
365C22—81no—
366C22—82no—
367C22#261711yescharacteristic
368C22#304512yes—
369C22C2×#46711yescharacteristic
370C22#305411yescharacteristic
371C22#306912yes—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—42no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—162no—
380C4—162no—
381C4#318912yes—
382C4#319112yes—
383C4#319812yes—
384C4#322011yescharacteristic
385C4#322111yescharacteristic
386C2—42no—
387C2—82no—
388C2#1941112yes—
389C2#1944912yes—
390C2#2972511yescharacteristic
391C2#2973111yescharacteristic
392C2#2975211yescharacteristic
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. #29808C2semidirect
4. #10624C2semidirect
5. #10622C2semidirect
6. #10487C2semidirect
7. #10485C2semidirect
8. #10276C2semidirect
9. #10274C2semidirect
10. #9131C2semidirect
11. #8599C2semidirect
105. C2×#41Q16semidirect

Maximal subgroups

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

subgroupindex
#329552
#298082
#218472
#106242
#106222
#104872
#104852
#102762
#102742
#91312
#85992