Finite Group Databasebeta

← #276379

Subgroups — #276379

Order 512.

733 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33195C211yesmaximal, characteristic, complemented
2#28508C211yesmaximal, characteristic, complemented
3#22795C211yesmaximal, characteristic, complemented
4C4×#459C211yesmaximal, characteristic
5#10615C212yesmaximal, complemented
6#10472C212yesmaximal
7#9898C211yesmaximal, characteristic, complemented
8#9614C212yesmaximal, complemented
9#9601C211yesmaximal, characteristic, complemented
10#9563C212yesmaximal, complemented
11#8585C211yesmaximal, characteristic, complemented
12C2×#474—21no—
13C2×#474—21no—
14C2×#474—21no—
15C2×#459C2211yescharacteristic
16C2×C4×D8—21no—
17C2×#431—21no—
18#2868C2211yescharacteristic
19C4×#127C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#2093—22no—
22#1988—21no—
23#1982C2211yescharacteristic
24#1957C2212yes—
25#1954C2212yes—
26#1947—21no—
27#1947C2212yes—
28#1947C2211yescharacteristic
29#1925—21no—
30#1923—22no—
31#1879—21no—
32#1845C2212yes—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1802—21no—
36#1802C2211yescharacteristic
37#1802C2212yes—
38#1797—21no—
39C8×C4⋊C4C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#102—21no—
43C2×#335C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×D8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×#127C2311yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C22×C42—21no—
61#474—44no—
62#474—44no—
63#474—44no—
64#459—22no—
65#459—22no—
66C4×D8—44no—
67#431—44no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—21no—
71C2×#107—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×#102—21no—
83C2×#102—22no—
84C2×#102—21no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—21no—
88C2×#102—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—21no—
93C2×C4×C8C2312yes—
94#390—21no—
95#390—21no—
96#390—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384C2311yescharacteristic
101#384C2312yes—
102C4×C4⋊C4D411yescharacteristic
103C4×C4⋊C4D411yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C43C2311yescharacteristic
109#340—22no—
110#340—21no—
111#340—22no—
112#340—21no—
113#336—22no—
114#336—21no—
115#336D412yes—
116#335—24no—
117#331—24no—
118C22×Q8—41no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—41no—
127C2×D8—24no—
128C2×D8—42no—
129C2×D8—42no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C2×D412yes—
135C22×C8C2×D412yes—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—82no—
141#127—21no—
142#127—21no—
143#127—21no—
144#127#4112yes—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—41no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—82no—
157C4×D4D814yes—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
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—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#107—44no—
189#107—44no—
190#107—42no—
191#107—44no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203C4⋊C8—81no—
204C4⋊C8D814yescomplemented
205C4⋊C8#4112yes—
206#102—24no—
207#102—44no—
208#102—42no—
209#102—44no—
210#102—44no—
211#102—42no—
212#102—42no—
213C22⋊C8—44no—
214C22⋊C8—82no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—44no—
218C4×C8—42no—
219#95—21no—
220#95—21no—
221#95—21no—
222C24—21no—
223C24—41no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×D4—21no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—81no—
247C2×D4—81no—
248C2×D4C2×D812yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276D8—48no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8C2×D814yes—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4C2×D812yes—
313C4⋊C4#13712yes—
314C22⋊C4—24no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—22no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—42no—
330C42—42no—
331C42C2×D812yes—
332C42C2×D812yes—
333C42#13512yes—
334C42AGL(1,ℤ/8)12yes—
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C422−1+411yescharacteristic
338C422−1+411yescharacteristic
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23C2×#12111yescharacteristic
354Q8—164no—
355D4—22no—
356D4—22no—
357D4—42no—
358D4—42no—
359D4—41no—
360D4—42no—
361D4—42no—
362D4—41no—
363D4—82no—
364D4—82no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—48no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—84no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—164no—
400C2×C4#45912yes—
401C2×C4#45912yes—
402C2×C4#46612yes—
403C2×C4#46612yes—
404C2×C4#47212yes—
405C2×C4#52511yescharacteristic
406C2×C4#53611yescharacteristic
407C2×C4#54711yescharacteristic
408C2×C4#54811yescharacteristic
409C2×C4#55211yescharacteristic
410C2×C4#55611yescharacteristic
411C2×C4C22×D811yescharacteristic
412C2×C4C22×D811yescharacteristic
413C2×C4C2×#13511yescharacteristic
414C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C2×C4#57511yescharacteristic
417C2×C4#57711yescharacteristic
418C2×C4#57811yescharacteristic
419C8—24no—
420C8—42no—
421C8—42no—
422C8—82no—
423C8—82no—
424C8—81no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22#241511yescharacteristic
440C22C2×#45911yescharacteristic
441C22#304611yescharacteristic
442C22C2×#46611yescharacteristic
443C22#305611yescharacteristic
444C22C2×#47211yescharacteristic
445C22#306911yescharacteristic
446C4—21no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—161no—
455C4—162no—
456C4#166714yes—
457C4#310011yescharacteristic
458C4#310711yescharacteristic
459C4#322611yescharacteristic
460C4#323311yescharacteristic
461C4#333411yescharacteristic
462C4#335411yescharacteristic
463C4#335811yescharacteristic
464C4#336711yescharacteristic
465C2—82no—
466C2—162no—
467C2C2×#166711yescharacteristic
468C2#1916311yescharacteristic
469C2#1941611yescharacteristic
470C2#1950311yescharacteristic
471C2#2467611yescharacteristic
472C2#2470011yescharacteristic
473C2#2472011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #33195C2semidirect
2. #28508C2semidirect
3. #22795C2semidirect
5. #10615C2semidirect
7. #9898C2semidirect
8. #9614C2semidirect
9. #9601C2semidirect
10. #9563C2semidirect
11. #8585C2semidirect
204. C4⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#331952
#285082
#227952
C4×#4592
#106152
#104722
#98982
#96142
#96012
#95632
#85852