Finite Group Databasebeta

← #278060

Subgroups — #278060

Order 512.

701 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24701C212yesmaximal
3#21487C211yesmaximal, characteristic
4C2×#1695C211yesmaximal, characteristic, complemented
5C2×#1564C211yesmaximal, characteristic, complemented
6#10601C212yesmaximal, complemented
7#10481C212yesmaximal, complemented
8#9739C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9128C212yesmaximal, complemented
11#8652C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#476C2212yes—
14C2×#436C2211yescharacteristic
15C2×#435—21no—
16C2×#431C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2077—22no—
21#2034—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1954C2211yescharacteristic
25#1948C2212yes—
26#1942—21no—
27#1942—21no—
28#1941—22no—
29#1939C2211yescharacteristic
30#1934—22no—
31#1923—21no—
32#1869—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1770C2211yescharacteristic
39#1753—21no—
40#1695—24no—
41#1564—24no—
42C2×#335C2212yes—
43C2×#334C2211yescharacteristic
44C2×#334C2211yescharacteristic
45C2×#333C2212yes—
46C2×#333C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×M16—21no—
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C2×C4⋊Q8C2312yes—
53C2×C22⋊Q8—22no—
54C2×#121D411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#500—22no—
59#500—22no—
60#476—24no—
61#476—42no—
62#436—22no—
63#436—22no—
64#435—44no—
65#431—24no—
66#431—42no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#106—22no—
72C2×#106—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#102—22no—
79C2×#102—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×#97—21no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8D411yescharacteristic
90C2×C4×C8D411yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#391—22no—
94#390D411yescharacteristic
95#389—22no—
96#389—22no—
97#389D411yescharacteristic
98#388—21no—
99#386—21no—
100#384C2312yes—
101#380—22no—
102#380—22no—
103#380D411yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105C4×C22⋊C4—21no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#335—24no—
116#334—22no—
117#334D414yescomplemented
118#333—24no—
119#333—42no—
120C22×Q8—22no—
121C22×D4—21no—
122C22×D4C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—41no—
139C22×C8SD1614yes—
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C22⋊Q8—84no—
149#121—42no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—42no—
155C4×D4—42no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—42no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42C2411yescharacteristic
181#107—42no—
182#107—42no—
183#107—42no—
184#107—42no—
185#106—44no—
186#106—44no—
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—42no—
195C4⋊C8C2×D414yes—
196#102—44no—
197#102—44no—
198C22⋊C8—44no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202C22⋊C8—42no—
203#97—42no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—44no—
209C4×C8C2×D412yes—
210C4×C8#4112yes—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—42no—
215#95—42no—
216C24—21no—
217C24#12111yescharacteristic
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4C2×SD1612yes—
256C22×C4#13712yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4111yescharacteristic
260M16—84no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—28no—
266C2×C8—21no—
267C2×C8—24no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8#12112yes—
283C2×C8#12112yes—
284C2×C8C2×SD1612yes—
285C2×C8C2×SD1612yes—
286C2×C8#13512yes—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—84no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—41no—
316C42—41no—
317C42C22×D411yescharacteristic
318C42C2×#4111yescharacteristic
319C42H(5,𝔽2)11yescharacteristic
320C422−1+411yescharacteristic
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23#46112yes—
335C23C2×#12111yescharacteristic
336Q8—162no—
337Q8—162no—
338D4—42no—
339D4—42no—
340D4—84no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#46112yes—
378C2×C4#47212yes—
379C2×C4C2×#12111yescharacteristic
380C2×C4#53711yescharacteristic
381C2×C4D4211yescharacteristic
382C2×C4#54711yescharacteristic
383C2×C4#54811yescharacteristic
384C2×C4#55311yescharacteristic
385C2×C4C22×SD1611yescharacteristic
386C2×C4C2×#13511yescharacteristic
387C2×C4C2×#13711yescharacteristic
388C2×C4#57512yes—
389C2×C4#57511yescharacteristic
390C2×C4#57812yes—
391C8—22no—
392C8—22no—
393C8—44no—
394C8—41no—
395C8—81no—
396C8—82no—
397C8—81no—
398C8#46514yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#241611yescharacteristic
414C22C2×#46111yescharacteristic
415C22#303511yescharacteristic
416C22#305411yescharacteristic
417C22#305911yescharacteristic
418C22C2×#47211yescharacteristic
419C22#306711yescharacteristic
420C4—22no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—162no—
429C4—162no—
430C4C2×#46511yescharacteristic
431C4#305611yescharacteristic
432C4#324611yescharacteristic
433C4#324811yescharacteristic
434C4#326912yes—
435C4#334912yes—
436C2—42no—
437C2—161no—
438C2#1913711yescharacteristic
439C2#1916011yescharacteristic
440C2#1919311yescharacteristic
441C2#1922611yescharacteristic
442C2#2475111yescharacteristic
443C2#2478511yescharacteristic
444C2#2480311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
1. #33209C2semidirect
4. C2×#1695C2semidirect
5. C2×#1564C2semidirect
6. #10601C2semidirect
7. #10481C2semidirect
8. #9739C2semidirect
9. #9621C2semidirect
10. #9128C2semidirect
11. #8652C2semidirect
117. #334D4semidirect

Maximal subgroups

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

subgroupindex
#332092
#247012
#214872
C2×#16952
C2×#15642
#106012
#104812
#97392
#96212
#91282
#86522