Finite Group Databasebeta

← #278217

Subgroups — #278217

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24759C212yesmaximal
3#21487C211yesmaximal, characteristic
4C2×#1693C211yesmaximal, characteristic, complemented
5C2×#1566C211yesmaximal, characteristic, complemented
6#10590C212yesmaximal, complemented
7#10477C212yesmaximal, complemented
8#9740C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9139C212yesmaximal, complemented
11#8652C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#468—22no—
14C2×#461C2212yes—
15C2×#436C2211yescharacteristic
16C2×#435—21no—
17C2×#423C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2857C2212yes—
20C4×#121C2211yescharacteristic
21#2068—22no—
22#2055—22no—
23#1990C2212yes—
24#1981C2212yes—
25#1954C2211yescharacteristic
26#1947C2212yes—
27#1942—21no—
28#1942—21no—
29#1939C2211yescharacteristic
30#1923—21no—
31#1894—22no—
32#1869—21no—
33#1845C2211yescharacteristic
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1770C2211yescharacteristic
39#1753—21no—
40#1693—24no—
41#1566—24no—
42C2×#334C2211yescharacteristic
43C2×#334C2212yes—
44C2×#334C2211yescharacteristic
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×SD16—22no—
50C22×M16—22no—
51C22×M16—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8C2312yes—
55C2×C22⋊Q8—22no—
56C2×#121D411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2311yescharacteristic
59C22×C42C2311yescharacteristic
60#500—22no—
61#500—22no—
62#468—48no—
63#461—24no—
64#461—42no—
65#436—22no—
66#436—22no—
67#435—44no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73#423—24no—
74#423—42no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×#97—21no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—22no—
97#390D411yescharacteristic
98#389—22no—
99#389—22no—
100#389D411yescharacteristic
101#388—21no—
102#386—21no—
103#384C2312yes—
104#380—22no—
105#380—22no—
106#380D411yescharacteristic
107C4×C4⋊C4C2311yescharacteristic
108C4×C22⋊C4—21no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#336—21no—
114#336—21no—
115#336—21no—
116#334—24no—
117#334—22no—
118#334D414yescomplemented
119#332—24no—
120#332—42no—
121C22×Q8—22no—
122C22×D4—21no—
123C22×D4C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C2×SD16—44no—
129C2×SD16—44no—
130C2×SD16—82no—
131C2×M16—44no—
132C2×M16—44no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—42no—
136C2×M16—82no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—41no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C22⋊Q8—44no—
156C22⋊Q8—44no—
157#121—24no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—42no—
163C4×D4—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—42no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2411yescharacteristic
189#107—44no—
190#107—42no—
191#107—42no—
192#107—42no—
193#107—42no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—44no—
202C4⋊C8—42no—
203C4⋊C8C2×D414yes—
204#103—44no—
205#103—44no—
206#103—82no—
207#102—44no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212#97—42no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8C2×D412yes—
218C4×C8#4112yes—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—42no—
223#95—42no—
224C24—21no—
225C24#12111yescharacteristic
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4AGL(1,ℤ/8)14yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270SD16—88no—
271M16—88no—
272M16—84no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—24no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—44no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—44no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8#12112yes—
295C2×C8#12112yes—
296C2×C8AGL(1,ℤ/8)12yes—
297C2×C8#13712yes—
298C2×C8#13712yes—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—82no—
313C4⋊C4—84no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—84no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—41no—
328C42—41no—
329C42C22×D411yescharacteristic
330C42C2×#4111yescharacteristic
331C42H(5,𝔽2)11yescharacteristic
332C422−1+411yescharacteristic
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23#46912yes—
347C23C2×#12111yescharacteristic
348Q8—82no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352D4—42no—
353D4—42no—
354D4—84no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#46912yes—
392C2×C4#47312yes—
393C2×C4C2×#12111yescharacteristic
394C2×C4#53711yescharacteristic
395C2×C4D4211yescharacteristic
396C2×C4#54711yescharacteristic
397C2×C4#54811yescharacteristic
398C2×C4#55311yescharacteristic
399C2×C4C2×AGL(1,ℤ/8)12yes—
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C2×C4#57511yescharacteristic
404C2×C4#57712yes—
405C8—22no—
406C8—22no—
407C8—44no—
408C8—41no—
409C8—81no—
410C8—82no—
411C8—81no—
412C8#46814yescomplemented
413C22—22no—
414C22—22no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22#241611yescharacteristic
428C22#303611yescharacteristic
429C22#303911yescharacteristic
430C22C2×#46911yescharacteristic
431C22#306011yescharacteristic
432C22C2×#47311yescharacteristic
433C22#306711yescharacteristic
434C4—22no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—162no—
443C4—162no—
444C4C2×#46811yescharacteristic
445C4#305611yescharacteristic
446C4#325211yescharacteristic
447C4#325411yescharacteristic
448C4#326512yes—
449C4#334512yes—
450C2—42no—
451C2—161no—
452C2#1928011yescharacteristic
453C2#1929611yescharacteristic
454C2#1933811yescharacteristic
455C2#1936011yescharacteristic
456C2#2475911yescharacteristic
457C2#2478211yescharacteristic
458C2#2480411yescharacteristic
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×#1693C2semidirect
5. C2×#1566C2semidirect
6. #10590C2semidirect
7. #10477C2semidirect
8. #9740C2semidirect
9. #9620C2semidirect
10. #9139C2semidirect
11. #8652C2semidirect
118. #334D4semidirect
412. C8#468semidirect

Maximal subgroups

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

subgroupindex
#332092
#247592
#214872
C2×#16932
C2×#15662
#105902
#104772
#97402
#96202
#91392
#86522