Finite Group Databasebeta

← #329624

Subgroups — #329624

Order 512.

820 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32956C211yesmaximal, characteristic
2#32953C211yesmaximal, characteristic
3#29844C211yesmaximal, characteristic, complemented
4#21786C211yesmaximal, characteristic
5#10616C212yesmaximal, complemented
6#10485C212yesmaximal, complemented
7#10481C212yesmaximal, complemented
8#10275C212yesmaximal, complemented
9#9132C211yesmaximal, characteristic, complemented
10#9128C211yesmaximal, characteristic, complemented
11#8599C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#487C2211yescharacteristic, complemented
14C2×#476C2212yes—
15C2×#474C2212yes—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2068—22no—
21#2036—22no—
22#1990C2211yescharacteristic, complemented
23#1956C2211yescharacteristic, complemented
24#1954—22no—
25#1954C2212yes—
26#1954C2212yes—
27#1864—22no—
28#1862—22no—
29#1853—21no—
30#1852—21no—
31#1845C2211yescharacteristic, complemented
32#1802C2212yes—
33#1802C2212yes—
34#1797C2211yescharacteristic
35#1797C2211yescharacteristic
36#1786—21no—
37#1786—21no—
38#1779—22no—
39C2×#335C2212yescomplemented
40C2×#333C2212yescomplemented
41C2×#332C2212yescomplemented
42#1296C2212yescomplemented
43#1296C2212yescomplemented
44#1277C2212yescomplemented
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4—42no—
54C2×C4×D4—42no—
55C2×C4×D4D412yes—
56C2×C4×D4C2312yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—21no—
59C22×C42C2311yescharacteristic
60#487—42no—
61#476—42no—
62#476—42no—
63#474—42no—
64#474—42no—
65#431—42no—
66#431—42no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#106—22no—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#103—22no—
75C2×#103—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8D412yes—
88C2×C4×C8—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#391—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99#380—21no—
100#380—21no—
101#380—21no—
102#380—21no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—21no—
107#336—21no—
108#335—42no—
109#333—42no—
110#332—42no—
111C25—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4Q1614yes—
119C22×D4C2×D412yes—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C22⋊Q8—82no—
140C22⋊Q8—82no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—44no—
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—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C2×D412yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42C2411yescharacteristic
176#107—44no—
177#107—82no—
178#106—44no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186#103—44no—
187#103—82no—
188#102—44no—
189#102—44no—
190#102—44no—
191#102—44no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—82no—
197C22⋊C8—82no—
198C22⋊C8—82no—
199C22⋊C8—82no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—42no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24—22no—
208C24—24no—
209C24—42no—
210C24—42no—
211C24—81no—
212C24—81no—
213C24—81no—
214C24—81no—
215C24#12111yescharacteristic
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—81no—
222C2×Q8—81no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—48no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—48no—
233C2×D4C2×Q1614yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4C2×Q1612yes—
266C22×C4#13512yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—21no—
307C42—22no—
308C42—21no—
309C42—22no—
310C42H(5,𝔽2)12yes—
311C42H(5,𝔽2)12yes—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—24no—
319C23—48no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—84no—
327C23—84no—
328C23—84no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23#46712yes—
333C23C2×#12111yescharacteristic
334Q8—82no—
335Q8—82no—
336Q8—162no—
337D4—24no—
338D4—24no—
339D4—48no—
340D4—42no—
341D4—42no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—88no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4#46712yes—
379C2×C4#53412yes—
380C2×C4#53411yescharacteristic
381C2×C4#53511yescharacteristic
382C2×C4#53812yes—
383C2×C4C22×Q1611yescharacteristic
384C2×C4C2×#13511yescharacteristic
385C2×C4#57512yes—
386C2×C4#57612yes—
387C2×C4#57712yes—
388C8—42no—
389C8—42no—
390C8—82no—
391C8—82no—
392C8—82no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—816no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22#261711yescharacteristic
409C22#304111yescharacteristic
410C22#304211yescharacteristic
411C22C2×#46711yescharacteristic
412C22#305411yescharacteristic
413C22#306811yescharacteristic
414C22#306911yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#318912yes—
425C4#319112yes—
426C4#319812yes—
427C4#322412yes—
428C2—42no—
429C2—84no—
430C2#1938711yescharacteristic
431C2#1939311yescharacteristic
432C2#1942511yescharacteristic
433C2#1943111yescharacteristic
434C2#2972711yescharacteristic
435C2#2973111yescharacteristic
436C2#2975311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
3. #29844C2semidirect
5. #10616C2semidirect
6. #10485C2semidirect
7. #10481C2semidirect
8. #10275C2semidirect
9. #9132C2semidirect
10. #9128C2semidirect
11. #8599C2semidirect
13. C2×#487C22semidirect
22. #1990C22semidirect
23. #1956C22semidirect
31. #1845C22semidirect
39. C2×#335C22semidirect
40. C2×#333C22semidirect
41. C2×#332C22semidirect
42. #1296C22semidirect
43. #1296C22semidirect
44. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#329562
#329532
#298442
#217862
#106162
#104852
#104812
#102752
#91322
#91282
#85992