Finite Group Databasebeta

← #275236

Subgroups — #275236

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33108C211yesmaximal, characteristic
2#27961C211yesmaximal, characteristic
3#22599C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5#10658C212yesmaximal, complemented
6#10590C212yesmaximal, complemented
7#9613C212yesmaximal, complemented
8#9566C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8724C211yesmaximal, characteristic
12C2×#483—21no—
13C2×#431—21no—
14C22×#107—21no—
15C22×C4⋊C8C2211yescharacteristic
16#2857C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#2100—22no—
20#1990C2212yes—
21#1981C2211yescharacteristic
22#1981C2211yescharacteristic
23#1948C2211yescharacteristic
24#1948C2211yescharacteristic
25#1947—21no—
26#1947—21no—
27#1947C2212yes—
28#1947C2212yes—
29#1939C2212yes—
30#1927—21no—
31#1927—21no—
32#1916—22no—
33#1872—21no—
34#1834C2212yes—
35#1824—21no—
36#1824—21no—
37#1817—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C8×C4⋊C4C2212yes—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×C8—21no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C22⋊Q8—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#483—44no—
60#431—44no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—44no—
64C2×#106—21no—
65C2×#106—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×#103—21no—
74C2×#103—21no—
75C2×#103—21no—
76C2×#103—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D411yescharacteristic
80C2×C22⋊C8D411yescharacteristic
81C2×C4×C8—22no—
82C2×C4×C8C2312yes—
83#391—21no—
84#389—21no—
85#389—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384C2311yescharacteristic
90#384C2312yes—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4C2311yescharacteristic
98C4×C4⋊C4C2311yescharacteristic
99C2×#95—21no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—21no—
105#340—21no—
106#340—22no—
107#340—22no—
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336D412yes—
113#334—24no—
114#332—24no—
115C22×Q8—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—41no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4#4111yescharacteristic
122C22×C8—24no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—42no—
128C22×C8—42no—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C22⋊Q8—42no—
134C22⋊Q8—42no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4SD1614yes—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C22⋊C4—41no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42C2411yescharacteristic
175#107—44no—
176#107—44no—
177#106—42no—
178#106—42no—
179C4⋊C8—24no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—44no—
186C4⋊C8#4114yes—
187#103—42no—
188#103—42no—
189#103—42no—
190#103—42no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—44no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—41no—
203#95—41no—
204C24#12111yescharacteristic
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C2×SD1612yes—
240C22×C4#13512yes—
241C22×C4#13512yes—
242C22×C4AGL(1,ℤ/8)12yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—48no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—81no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4C2×SD1612yes—
288C4⋊C4C2×SD1612yes—
289C4⋊C4AGL(1,ℤ/8)12yes—
290C22⋊C4—82no—
291C42—21no—
292C42—21no—
293C42—22no—
294C42—22no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—42no—
300C42—42no—
301C42C2×#4112yes—
302C422−1+412yes—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#46312yes—
308C23#46512yes—
309C23#47312yes—
310C23C2×#12111yescharacteristic
311Q8—81no—
312Q8—81no—
313Q8—81no—
314Q8—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—84no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—164no—
355C2×C4#46312yes—
356C2×C4#46512yes—
357C2×C4#52511yescharacteristic
358C2×C4#53611yescharacteristic
359C2×C4#54712yes—
360C2×C4#55212yes—
361C2×C4C22×SD1611yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C2×C4C2×#13511yescharacteristic
364C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
365C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
366C2×C4#57511yescharacteristic
367C2×C4#57812yes—
368C8—42no—
369C8—42no—
370C8—44no—
371C8—82no—
372C8—81no—
373C8—81no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22#167914yes—
381C22#241311yescharacteristic
382C22C2×#46311yescharacteristic
383C22#304411yescharacteristic
384C22C2×#46511yescharacteristic
385C22#305511yescharacteristic
386C22C2×#47311yescharacteristic
387C22#306811yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—81no—
396C4—161no—
397C4—161no—
398C4—162no—
399C4#310311yescharacteristic
400C4#310511yescharacteristic
401C4#322711yescharacteristic
402C4#323211yescharacteristic
403C4#333612yes—
404C4#334312yes—
405C2—24no—
406C2C2×#167911yescharacteristic
407C2#1916611yescharacteristic
408C2#1940311yescharacteristic
409C2#1947611yescharacteristic
410C2#2457711yescharacteristic
411C2#2460311yescharacteristic
412C2#2462211yescharacteristic
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
5. #10658C2semidirect
6. #10590C2semidirect
7. #9613C2semidirect
8. #9566C2semidirect

Maximal subgroups

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

subgroupindex
#331082
#279612
#225992
C2×#19472
#106582
#105902
#96132
#95662
#91312
#88302
#87242