Finite Group Databasebeta

← #274265

Subgroups — #274265

Order 512.

824 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33106C211yesmaximal, characteristic
2#25903C211yesmaximal, characteristic
3#22603C211yesmaximal, characteristic, complemented
4C2×#1981C211yesmaximal, characteristic, complemented
5#10590C212yesmaximal, complemented
6#10540C212yesmaximal, complemented
7#9746C212yesmaximal, complemented
8#9574C212yesmaximal, complemented
9#9133C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic, complemented
11#8340C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#486C2212yes—
14C2×#480—21no—
15C2×#432—21no—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103—21no—
18C22×#103—21no—
19#2856C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#2023—22no—
22#1990C2212yes—
23#1982C2211yescharacteristic
24#1982C2211yescharacteristic
25#1981—21no—
26#1981—21no—
27#1981C2212yes—
28#1981C2212yes—
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1939C2212yes—
32#1925—21no—
33#1925—21no—
34#1917—22no—
35#1874—21no—
36#1874—21no—
37#1874—21no—
38#1817—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1770C2212yes—
42C4×C22⋊C8C2211yescharacteristic
43C2×#334C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C23×C8—21no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2312yes—
57C2×#127C2311yescharacteristic
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#486—24no—
68#486—42no—
69#480—44no—
70#432—44no—
71C2×#107—21no—
72C2×#107—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—44no—
85C2×#103—44no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C22⋊C8D411yescharacteristic
96C2×#97—22no—
97C2×C4×C8C2312yes—
98#390—21no—
99#390—21no—
100#390—21no—
101#384C2311yescharacteristic
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D411yescharacteristic
105C4×C4⋊C4D411yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C4×C4⋊C4C2311yescharacteristic
108C2×#95—21no—
109#340—22no—
110#340—22no—
111#340—21no—
112#340—21no—
113#336—22no—
114#336—22no—
115#336—21no—
116#336—21no—
117#336D412yes—
118#334—42no—
119#332—24no—
120#332—42no—
121#331—42no—
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—41no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—41no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4#4111yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—44no—
154C4⋊Q8—44no—
155C4⋊Q8—42no—
156C4⋊Q8#4114yes—
157#127—22no—
158#127—22no—
159#121—42no—
160#121—42no—
161#121—42no—
162#121—42no—
163#121—82no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42C2411yescharacteristic
198#107—42no—
199#107—42no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—42no—
207C4⋊C8—44no—
208#103—44no—
209#103—44no—
210#103—44no—
211#103—44no—
212#102—44no—
213#102—44no—
214#102—42no—
215#102—42no—
216#102—42no—
217#102—42no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222#97—44no—
223C4×C8—22no—
224C4×C8—22no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229C24—41no—
230C24#12111yescharacteristic
231C2×Q8—24no—
232C2×Q8—44no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—82no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—81no—
256C2×D4—81no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4—82no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#13512yes—
288C22×C4AGL(1,ℤ/8)12yes—
289C22×C4AGL(1,ℤ/8)12yes—
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—84no—
313C4⋊C4—21no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—81no—
333C4⋊C4AGL(1,ℤ/8)12yes—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—22no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—42no—
346C42—42no—
347C42C2×#4112yes—
348C42H(5,𝔽2)12yes—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23#46312yes—
361C23#46912yes—
362C23#47112yes—
363C23C2×#12111yescharacteristic
364Q8—44no—
365Q8—44no—
366Q8—88no—
367Q8—82no—
368Q8—82no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—82no—
374D4—82no—
375D4—162no—
376D4—164no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—44no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—88no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4#46312yes—
415C2×C4#46912yes—
416C2×C4#52511yescharacteristic
417C2×C4#53511yescharacteristic
418C2×C4#54612yes—
419C2×C4#55112yes—
420C2×C4C2×#13511yescharacteristic
421C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
422C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
423C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
424C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57812yes—
427C8—42no—
428C8—42no—
429C8—81no—
430C8—81no—
431C8—82no—
432C8—82no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22#170114yes—
447C22#241111yescharacteristic
448C22C2×#46311yescharacteristic
449C22#304411yescharacteristic
450C22C2×#46911yescharacteristic
451C22#305511yescharacteristic
452C22C2×#47111yescharacteristic
453C22#306911yescharacteristic
454C4—22no—
455C4—22no—
456C4—22no—
457C4—41no—
458C4—42no—
459C4—84no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—161no—
464C4#310911yescharacteristic
465C4#311411yescharacteristic
466C4#321311yescharacteristic
467C4#321711yescharacteristic
468C4#330012yes—
469C4#330812yes—
470C2—24no—
471C2—321no—
472C2C2×#170111yescharacteristic
473C2#1916911yescharacteristic
474C2#1944011yescharacteristic
475C2#1947411yescharacteristic
476C2#2448111yescharacteristic
477C2#2450611yescharacteristic
478C2#2452311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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. #22603C2semidirect
4. C2×#1981C2semidirect
5. #10590C2semidirect
6. #10540C2semidirect
7. #9746C2semidirect
8. #9574C2semidirect
10. #8830C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#331062
#259032
#226032
C2×#19812
#105902
#105402
#97462
#95742
#91332
#88302
#83402