Finite Group Databasebeta

← #318408

Subgroups — #318408

Order 512.

690 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28280C211yesmaximal, characteristic
2#28252C211yesmaximal, characteristic
3#22264C211yesmaximal, characteristic
4#22246C211yesmaximal, characteristic, complemented
5C2×#1691C212yesmaximal, complemented
6#10481C212yesmaximal, complemented
7#10476C212yesmaximal, complemented
8#8810C211yesmaximal, characteristic, complemented
9#8654C212yesmaximal, complemented
10#8590C211yesmaximal, characteristic, complemented
11#8589C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#476C2212yes—
14C2×#460—21no—
15C2×#461C2212yes—
16C2×#436C2212yes—
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C4×#127C2211yescharacteristic
20#2065—22no—
21#2055—22no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1982—21no—
25#1954C2212yes—
26#1948—21no—
27#1920—22no—
28#1919—22no—
29#1870—22no—
30#1870—22no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1834C2211yescharacteristic
34#1831C2211yescharacteristic
35#1803—21no—
36#1802C2212yes—
37#1802C2212yes—
38#1744C2211yescharacteristic
39#1691—28no—
40C2×#335C2212yes—
41C2×#335C2212yes—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46C22×SD16—22no—
47C22×M16—22no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127D411yescharacteristic
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C2×C4×D4C2312yes—
56#498—22no—
57#498D414yescomplemented
58#476—24no—
59#476—42no—
60#461—24no—
61#461—42no—
62#460—44no—
63#436—24no—
64#436—24no—
65C2×#106—22no—
66C2×#106—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#103—22no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83#391—21no—
84#391—21no—
85#390—21no—
86#390—21no—
87#390D412yes—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#384D411yescharacteristic
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C43C2311yescharacteristic
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#336—21no—
106#336—21no—
107#336D412yes—
108#336D412yes—
109#335—24no—
110#335—42no—
111#333—24no—
112#332—24no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4C2×D412yes—
118C23×C4—22no—
119C2×SD16—44no—
120C2×SD16—44no—
121C2×SD16—82no—
122C2×M16—44no—
123C2×M16—82no—
124C2×M16—82no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C2×D412yes—
129C22×C8C2×D412yes—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8C2×D412yes—
137C4⋊Q8C2×D412yes—
138#127—22no—
139#127—22no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—82no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—82no—
148C4×D4—82no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C2×D412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42C2411yescharacteristic
176#106—28no—
177#106—44no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—42no—
185C4⋊C8—42no—
186C4⋊C8—81no—
187C4⋊C8—81no—
188#103—44no—
189#102—44no—
190#102—44no—
191#102—82no—
192C22⋊C8—44no—
193C22⋊C8—82no—
194C22⋊C8—82no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8C2×D412yes—
198C4×C8#4112yes—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203C24—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—81no—
213C2×Q8—81no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4—81no—
225C2×D4—81no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4111yescharacteristic
244SD16—88no—
245M16—88no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8#12114yes—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—21no—
288C42—22no—
289C42—22no—
290C42—21no—
291C42—41no—
292C42—41no—
293C42—82no—
294C42—82no—
295C42AGL(1,ℤ/8)12yes—
296C42#13712yes—
297C42C22×D411yescharacteristic
298C42C2×#4111yescharacteristic
299C42H(5,𝔽2)11yescharacteristic
300C422−1+411yescharacteristic
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23C2×#12111yescharacteristic
309Q8—42no—
310Q8—42no—
311Q8—82no—
312Q8—82no—
313Q8—81no—
314Q8—81no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—84no—
320D4—84no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4#46812yes—
353C2×C4#46812yes—
354C2×C4C2×#12111yescharacteristic
355C2×C4#53711yescharacteristic
356C2×C4D4211yescharacteristic
357C2×C4#54711yescharacteristic
358C2×C4#54711yescharacteristic
359C2×C4#55511yescharacteristic
360C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
361C2×C4C2×#13711yescharacteristic
362C2×C4#57611yescharacteristic
363C2×C4#57712yes—
364C2×C4#57712yes—
365C2×C4#57811yescharacteristic
366C8—24no—
367C8—42no—
368C8—82no—
369C8—82no—
370C8—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22#253711yescharacteristic
379C22#304211yescharacteristic
380C22#304211yescharacteristic
381C22C2×#46811yescharacteristic
382C22#306111yescharacteristic
383C22#306811yescharacteristic
384C22#306911yescharacteristic
385C4—21no—
386C4—21no—
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—82no—
392C4—82no—
393C4—82no—
394C4—161no—
395C4—161no—
396C4#304312yes—
397C4#326012yes—
398C4#327411yescharacteristic
399C4#327611yescharacteristic
400C4#335311yescharacteristic
401C4#335711yescharacteristic
402C2—162no—
403C2#1942611yescharacteristic
404C2#1942711yescharacteristic
405C2#1951811yescharacteristic
406C2#1951911yescharacteristic
407C2#2823011yescharacteristic
408C2#2824211yescharacteristic
409C2#2826011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22246C2semidirect
5. C2×#1691C2semidirect
6. #10481C2semidirect
7. #10476C2semidirect
8. #8810C2semidirect
9. #8654C2semidirect
10. #8590C2semidirect
11. #8589C2semidirect
57. #498D4semidirect

Maximal subgroups

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

subgroupindex
#282802
#282522
#222642
#222462
C2×#16912
#104812
#104762
#88102
#86542
#85902
#85892