Finite Group Databasebeta

← #274117

Subgroups — #274117

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33107C211yesmaximal, characteristic
2#25903C211yesmaximal, characteristic
3#22625C211yesmaximal, characteristic, complemented
4C2×#1981C211yesmaximal, characteristic, complemented
5#10602C212yesmaximal, complemented
6#10540C212yesmaximal, complemented
7#9738C212yesmaximal, complemented
8#9565C212yesmaximal, complemented
9#9133C211yesmaximal, characteristic
10#8829C211yesmaximal, characteristic, complemented
11#8340C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#486C2212yes—
14C2×#481—21no—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103—21no—
17C22×#103—21no—
18#2856C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#2023—22no—
21#1990C2212yes—
22#1982C2211yescharacteristic
23#1982C2211yescharacteristic
24#1981—21no—
25#1981—21no—
26#1981C2212yes—
27#1981C2212yes—
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1939C2212yes—
31#1926—21no—
32#1926—21no—
33#1916—22no—
34#1874—21no—
35#1874—21no—
36#1874—21no—
37#1797—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C8×C4⋊C4C2212yes—
41C4×C22⋊C8—21no—
42C4×C22⋊C8C2211yescharacteristic
43C2×#335C2212yes—
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#481—44no—
70C2×#106—21no—
71C2×#106—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—42no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—44no—
84C2×#103—44no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8—22no—
96C2×C4×C8C2312yes—
97#390—21no—
98#390—21no—
99#390—21no—
100#384C2311yescharacteristic
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4D411yescharacteristic
104C4×C4⋊C4D411yescharacteristic
105C4×C4⋊C4C2311yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C2×#95—21no—
108#340—22no—
109#340—22no—
110#340—21no—
111#340—21no—
112#336—22no—
113#336—22no—
114#336—21no—
115#336—21no—
116#336D412yes—
117#335—42no—
118#332—24no—
119#332—42no—
120#331—42no—
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—41no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—41no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C4⋊Q8—24no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154C4⋊Q8—42no—
155C4⋊Q8#4114yes—
156#127—22no—
157#127—22no—
158#121—42no—
159#121—42no—
160#121—42no—
161#121—42no—
162#121—82no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4D814yes—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—82no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#106—42no—
197#106—42no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—42no—
205C4⋊C8—82no—
206#103—44no—
207#103—44no—
208#103—44no—
209#103—44no—
210#102—44no—
211#102—44no—
212#102—42no—
213#102—42no—
214#102—42no—
215#102—42no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—82no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227C24—41no—
228C24#12111yescharacteristic
229C2×Q8—24no—
230C2×Q8—44no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—82no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—82no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—81no—
254C2×D4—81no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4C2×D812yes—
286C22×C4#13512yes—
287C22×C4#13512yes—
288C22×C4#13712yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—84no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—81no—
329C4⋊C4C2×D812yes—
330C4⋊C4C2×D812yes—
331C4⋊C4AGL(1,ℤ/8)12yes—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C42—21no—
336C42—21no—
337C42—22no—
338C42—22no—
339C42—42no—
340C42—42no—
341C42—81no—
342C42—81no—
343C42C2×#4112yes—
344C42H(5,𝔽2)12yes—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23#46412yes—
357C23#46612yes—
358C23#47112yes—
359C23C2×#12111yescharacteristic
360Q8—44no—
361Q8—44no—
362Q8—88no—
363Q8—82no—
364Q8—82no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—82no—
370D4—82no—
371D4—162no—
372D4—164no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—88no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4#46412yes—
411C2×C4#46612yes—
412C2×C4#52511yescharacteristic
413C2×C4#53511yescharacteristic
414C2×C4#54612yes—
415C2×C4#55112yes—
416C2×C4C22×D811yescharacteristic
417C2×C4C2×#13511yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
420C2×C4C2×#13711yescharacteristic
421C2×C4#57511yescharacteristic
422C2×C4#57712yes—
423C8—42no—
424C8—42no—
425C8—81no—
426C8—81no—
427C8—82no—
428C8—82no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22#168214yes—
443C22#241111yescharacteristic
444C22C2×#46411yescharacteristic
445C22#304211yescharacteristic
446C22C2×#46611yescharacteristic
447C22#305511yescharacteristic
448C22C2×#47111yescharacteristic
449C22#306911yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—41no—
457C4—81no—
458C4—84no—
459C4—161no—
460C4#310011yescharacteristic
461C4#310511yescharacteristic
462C4#320511yescharacteristic
463C4#321111yescharacteristic
464C4#330112yes—
465C4#331612yes—
466C2—24no—
467C2—321no—
468C2C2×#168211yescharacteristic
469C2#1917711yescharacteristic
470C2#1939211yescharacteristic
471C2#1946211yescharacteristic
472C2#2448311yescharacteristic
473C2#2450011yescharacteristic
474C2#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. #22625C2semidirect
4. C2×#1981C2semidirect
5. #10602C2semidirect
6. #10540C2semidirect
7. #9738C2semidirect
8. #9565C2semidirect
10. #8829C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#331072
#259032
#226252
C2×#19812
#106022
#105402
#97382
#95652
#91332
#88292
#83402