Finite Group Databasebeta

← #279367

Subgroups — #279367

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33197C211yesmaximal, characteristic, complemented
2#28508C211yesmaximal, characteristic, complemented
3#22696C211yesmaximal, characteristic, complemented
4#21503C211yesmaximal, characteristic
5#10651C212yesmaximal, complemented
6#10599C212yesmaximal, complemented
7#9905C211yesmaximal, characteristic, complemented
8#9604C211yesmaximal, characteristic, complemented
9#9221C212yesmaximal
10#8644C212yesmaximal, complemented
11#8588C211yesmaximal, characteristic, complemented
12C2×#476—21no—
13C2×#474—22no—
14C2×#459C2211yescharacteristic
15C2×#442—21no—
16C2×C8×D4C2212yes—
17#2868C2211yescharacteristic
18C4×#127C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#2093—21no—
21#2093—21no—
22#1987—21no—
23#1982C2211yescharacteristic
24#1957C2212yes—
25#1948—21no—
26#1948C2212yes—
27#1948C2211yescharacteristic
28#1939C2212yes—
29#1926—21no—
30#1879—21no—
31#1850—22no—
32#1845C2212yes—
33#1842C2211yescharacteristic
34#1831C2211yescharacteristic
35#1802—21no—
36#1802C2212yes—
37#1802C2211yescharacteristic
38#1797—21no—
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1762—21no—
42C2×#336—21no—
43C2×#335C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×D8—21no—
50C23×C8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×#127C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C22×C42—21no—
62#476—44no—
63#474—48no—
64#459—22no—
65#459Q814yescomplemented
66#442—44no—
67C8×D4—24no—
68C8×D4—42no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#106—21no—
72C2×#106—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8D412yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×#102—21no—
82C2×#102—22no—
83C2×#102—21no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C22⋊C8—22no—
89C2×#97—21no—
90C2×C4×C8C2312yes—
91#390—21no—
92#390—21no—
93#390—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384C2311yescharacteristic
98#384C2312yes—
99C4×C4⋊C4D411yescharacteristic
100C4×C4⋊C4D411yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C43C2311yescharacteristic
106#340—22no—
107#340—21no—
108#340—22no—
109#340—21no—
110#336—22no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336D411yescharacteristic
115#336D411yescharacteristic
116#335—42no—
117#331—42no—
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—22no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—44no—
135C22×C8—44no—
136C22×C8—82no—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C4⋊Q8—44no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—82no—
143#127—21no—
144#127—21no—
145#127—21no—
146#127C2×Q812yes—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—41no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—82no—
156C4×D4D814yes—
157C4×D4C2×Q812yes—
158C4×D4#4112yes—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—82no—
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#106—42no—
191#106—44no—
192C4⋊C8—24no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—82no—
203C4⋊C8C2×Q812yes—
204#102—24no—
205#102—42no—
206#102—44no—
207#102—42no—
208#102—44no—
209#102—44no—
210#102—42no—
211C22⋊C8—44no—
212#97—42no—
213C4×C8—22no—
214C4×C8—22no—
215#95—21no—
216#95—21no—
217#95—21no—
218C24—21no—
219C24—41no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4C22⋊Q812yes—
241C2×D4C2×D812yes—
242C2×D4#13712yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4111yescharacteristic
268D8—48no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—88no—
287C2×C8AGL(1,ℤ/8)14yes—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—81no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4C22⋊Q812yes—
305C4⋊C4C2×D812yes—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—22no—
314C42—21no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—41no—
319C42—42no—
320C42—81no—
321C42—81no—
322C42C2×D812yes—
323C42#13512yes—
324C42C22×Q811yescharacteristic
325C42C2×#4111yescharacteristic
326C42H(5,𝔽2)11yescharacteristic
327C422−1+411yescharacteristic
328C23—22no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23C2×#12111yescharacteristic
343Q8—164no—
344D4—22no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—44no—
349D4—42no—
350D4—82no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—82no—
355D4#47414yescomplemented
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—164no—
387C2×C4#46612yes—
388C2×C4#46612yes—
389C2×C4#47212yes—
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4#53711yescharacteristic
392C2×C4#54611yescharacteristic
393C2×C4D4×Q811yescharacteristic
394C2×C4#55211yescharacteristic
395C2×C4#55411yescharacteristic
396C2×C4C22×D811yescharacteristic
397C2×C4C2×#13511yescharacteristic
398C2×C4C2×AGL(1,ℤ/8)12yes—
399C2×C4C2×#13711yescharacteristic
400C2×C4#57511yescharacteristic
401C2×C4#57712yes—
402C8—24no—
403C8—42no—
404C8—42no—
405C8—81no—
406C8—82no—
407C8—82no—
408C22—24no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22#241711yescharacteristic
423C22#303811yescharacteristic
424C22#304211yescharacteristic
425C22C2×#46611yescharacteristic
426C22#305611yescharacteristic
427C22C2×#47211yescharacteristic
428C22#306911yescharacteristic
429C4—22no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—161no—
439C4—162no—
440C4C2×#47411yescharacteristic
441C4#307611yescharacteristic
442C4#324511yescharacteristic
443C4#324811yescharacteristic
444C4#330612yes—
445C4#338412yes—
446C2—44no—
447C2—162no—
448C2#1926711yescharacteristic
449C2#1931311yescharacteristic
450C2#1939211yescharacteristic
451C2#1946511yescharacteristic
452C2#2482911yescharacteristic
453C2#2483511yescharacteristic
454C2#2485511yescharacteristic
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. #33197C2semidirect
2. #28508C2semidirect
3. #22696C2semidirect
5. #10651C2semidirect
6. #10599C2semidirect
7. #9905C2semidirect
8. #9604C2semidirect
10. #8644C2semidirect
11. #8588C2semidirect
65. #459Q8semidirect
355. D4#474semidirect

Maximal subgroups

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

subgroupindex
#331972
#285082
#226962
#215032
#106512
#105992
#99052
#96042
#92212
#86442
#85882