Finite Group Databasebeta

← #278182

Subgroups — #278182

Order 512.

728 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24843C212yesmaximal
3C8×#121C211yesmaximal, characteristic
4C2×#1675C211yesmaximal, characteristic, complemented
5C2×#1567C211yesmaximal, characteristic, complemented
6#10593C212yesmaximal, complemented
7#10481C212yesmaximal, complemented
8#9730C211yesmaximal, characteristic, complemented
9#9614C211yesmaximal, characteristic, complemented
10#9135C212yesmaximal, complemented
11#8644C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#476C2212yes—
14C2×C8×D4—21no—
15C2×C8×D4C2211yescharacteristic
16C2×#423C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2088—22no—
21#2067—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1954C2211yescharacteristic
25#1947C2212yes—
26#1942—21no—
27#1942—21no—
28#1939C2211yescharacteristic
29#1923—21no—
30#1895—22no—
31#1869—21no—
32#1845C2211yescharacteristic
33#1809—22no—
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40#1675—24no—
41#1567—24no—
42C2×#335C2211yescharacteristic
43C2×#335C2212yes—
44C2×#335C2211yescharacteristic
45C2×#333C2212yes—
46C2×#333C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×M16—22no—
50C23×C8—21no—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C22⋊Q8—22no—
55C2×#121D411yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#500—22no—
60#500—22no—
61#476—24no—
62#476—42no—
63C8×D4—22no—
64C8×D4—22no—
65C8×D4—44no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—22no—
71C2×#106—22no—
72#423—24no—
73#423—42no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#102—22no—
81C2×#102—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—21no—
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#335—22no—
117#335D414yescomplemented
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—44no—
128C2×M16—44no—
129C2×M16—82no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—41no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C22⋊Q8—84no—
152#121—22no—
153#121Q1614yes—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—42no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185#107—42no—
186#107—42no—
187#107—42no—
188#107—42no—
189#107—82no—
190#106—44no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—44no—
199C4⋊C8—42no—
200C4⋊C8C2×D414yes—
201#102—44no—
202#102—82no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—42no—
212C4×C8C2×D412yes—
213C4×C8#4112yes—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—42no—
218#95—42no—
219C24—21no—
220C24#12111yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×D4—24no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4C2×Q1612yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4C2×Q1612yes—
263C22×C4#13512yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4111yescharacteristic
267M16—88no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—24no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—44no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—84no—
290C2×C8#12112yes—
291C2×C8#12112yes—
292C2×C8AGL(1,ℤ/8)12yes—
293C2×C8#13712yes—
294C2×C8#13712yes—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—82no—
310C4⋊C4—84no—
311C4⋊C4C2×Q1612yes—
312C22⋊C4—24no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—84no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—41no—
326C42—41no—
327C42C22×D411yescharacteristic
328C42C2×#4111yescharacteristic
329C42H(5,𝔽2)11yescharacteristic
330C422−1+411yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23#46712yes—
345C23C2×#12111yescharacteristic
346Q8—162no—
347Q8—162no—
348D4—22no—
349D4—22no—
350D4—48no—
351D4—41no—
352D4—41no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
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#46712yes—
392C2×C4#47212yes—
393C2×C4C2×#12111yescharacteristic
394C2×C4#53711yescharacteristic
395C2×C4D4211yescharacteristic
396C2×C4#54711yescharacteristic
397C2×C4#54811yescharacteristic
398C2×C4#55311yescharacteristic
399C2×C4C22×Q1611yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4#57511yescharacteristic
405C2×C4#57611yescharacteristic
406C2×C4#57811yescharacteristic
407C8—22no—
408C8—22no—
409C8—44no—
410C8—41no—
411C8—81no—
412C8—82no—
413C8—81no—
414C8#46814yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#241611yescharacteristic
430C22#303611yescharacteristic
431C22#303911yescharacteristic
432C22C2×#46711yescharacteristic
433C22#306111yescharacteristic
434C22C2×#47211yescharacteristic
435C22#306711yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—22no—
440C4—42no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—162no—
445C4—162no—
446C4C2×#46811yescharacteristic
447C4#305611yescharacteristic
448C4#325211yescharacteristic
449C4#325411yescharacteristic
450C4#328011yescharacteristic
451C4D4×Q1611yescharacteristic
452C4#333611yescharacteristic
453C4#333811yescharacteristic
454C2—42no—
455C2—82no—
456C2#1925611yescharacteristic
457C2#1927211yescharacteristic
458C2#1933211yescharacteristic
459C2#1936111yescharacteristic
460C2#2475911yescharacteristic
461C2#2478111yescharacteristic
462C2#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×#1675C2semidirect
5. C2×#1567C2semidirect
6. #10593C2semidirect
7. #10481C2semidirect
8. #9730C2semidirect
9. #9614C2semidirect
10. #9135C2semidirect
11. #8644C2semidirect
117. #335D4semidirect

Maximal subgroups

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

subgroupindex
#332092
#248432
C8×#1212
C2×#16752
C2×#15672
#105932
#104812
#97302
#96142
#91352
#86442