Finite Group Databasebeta

← #498231

Subgroups — #498231

Order 512.

525 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33649C211yesmaximal, characteristic
2#23030C211yesmaximal, characteristic
3#22607C211yesmaximal, characteristic
4#13174C211yesmaximal, characteristic, complemented
5#13154C211yesmaximal, characteristic, complemented
6#13141C211yesmaximal, characteristic, complemented
7#13119C211yesmaximal, characteristic, complemented
8#13118C211yesmaximal, characteristic, complemented
9#13066C211yesmaximal, characteristic
10#12957C211yesmaximal, characteristic, complemented
11#12930C211yesmaximal, characteristic
12#12677C211yesmaximal, characteristic, complemented
13#12648C211yesmaximal, characteristic, complemented
14#12616C211yesmaximal, characteristic
15#12567C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#441C2211yescharacteristic
19#2924—21no—
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2362—21no—
23#2237C412yescomplemented
24#2237C412yescomplemented
25#2236C412yescomplemented
26#2235C412yescomplemented
27#2159C2211yescharacteristic
28#2159C2211yescharacteristic
29C2×#365C2211yescharacteristic
30C2×#365C2211yescharacteristic
31C2×#362C2211yescharacteristic
32C2×#362C2211yescharacteristic
33#2141C2211yescharacteristic
34#2141C2211yescharacteristic
35#2112C2211yescharacteristic
36C2×#347C2211yescharacteristic
37C2×#347C2211yescharacteristic
38#1971C2211yescharacteristic
39#1946C2211yescharacteristic
40#1946C2211yescharacteristic
41#1943C2211yescharacteristic
42#1928—21no—
43#1922—21no—
44#1922—21no—
45#1873—21no—
46#1863—21no—
47#1826—21no—
48#1810C2211yescharacteristic
49#1776C2211yescharacteristic
50#1773C2211yescharacteristic
51#1773C2211yescharacteristic
52#1752C2211yescharacteristic
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1386C2211yescharacteristic
58#1386C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×M32—21no—
70C2×M32—22no—
71C2×M32—41no—
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74C2×M32C2311yescharacteristic
75C2×M32C2311yescharacteristic
76C22×C16—21no—
77#467—21no—
78#467—21no—
79#467C2×C412yes—
80#467C2×C412yes—
81#441—22no—
82#441—41no—
83#433—41no—
84#433—41no—
85#432—41no—
86#431—41no—
87C2×#108C2311yescharacteristic
88C2×#108C2311yescharacteristic
89#427—21no—
90#427—21no—
91#427—22no—
92#427—41no—
93#427C2×C411yescharacteristic
94#427C2×C411yescharacteristic
95#427C2311yescharacteristic
96#427C2311yescharacteristic
97C2×#107—21no—
98C2×#107C2×C411yescharacteristic
99C2×#107C2×C411yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#106—21no—
102C2×#106C2311yescharacteristic
103C2×#106C2311yescharacteristic
104#424—21no—
105#424—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109#407—21no—
110C2×C22⋊C8D411yescharacteristic
111#405—21no—
112#405—21no—
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115#391—21no—
116#389—21no—
117#388—41no—
118#388—41no—
119#387—41no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367C2×C412yes—
124#367C2×C412yes—
125#366C2×C412yes—
126#366C2×C412yes—
127#365—22no—
128#365—41no—
129#362—22no—
130#362—41no—
131#358C2×C412yes—
132#358C2×C412yes—
133#358C2×C412yes—
134#358C2×C412yes—
135C22⋊C16C2×C412yes—
136C22⋊C16C2×C412yes—
137#347—22no—
138#347—41no—
139#341—21no—
140#340—21no—
141#340—21no—
142#339—21no—
143#339—21no—
144#339—21no—
145#339—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148C22×Q8—21no—
149C23×C4C2×D411yescharacteristic
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16C22×C412yes—
156C2×M16—21no—
157C2×M16—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8#4111yescharacteristic
168C22×C8#4111yescharacteristic
169C22×C8C2411yescharacteristic
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C4×Q8—42no—
175C4×Q8—81no—
176#117—21no—
177#117—21no—
178#117—21no—
179#117—21no—
180#117—41no—
181#117—41no—
182#117—42no—
183#117—81no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202M32—21no—
203M32—21no—
204M32—21no—
205M32—21no—
206M32—21no—
207M32—21no—
208M32—21no—
209M32—42no—
210M32#4112yes—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—42no—
214C2×C16C22×C412yes—
215C2×C16C22×C412yes—
216#108—21no—
217#108—21no—
218#108—21no—
219#108—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—42no—
224#107C22×C411yescharacteristic
225#107C22×C411yescharacteristic
226#107C22×C411yescharacteristic
227#107C22×C411yescharacteristic
228#107C22×C412yes—
229#106—21no—
230#106—21no—
231#106—21no—
232#106—21no—
233#106—21no—
234#106—21no—
235#106—21no—
236#106#4112yes—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241#103—22no—
242#103—22no—
243#103—42no—
244#103—81no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—41no—
248C22⋊C8—41no—
249#97—21no—
250#97—21no—
251#97—21no—
252#97—21no—
253#97—41no—
254#97—41no—
255C4×C8—41no—
256C4×C8—41no—
257#95—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262C24AGL(1,ℤ/8)11yescharacteristic
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4AGL(1,ℤ/8)11yescharacteristic
284C22×C4AGL(1,ℤ/8)11yescharacteristic
285C22×C4AGL(1,ℤ/8)11yescharacteristic
286C22×C4C22×D411yescharacteristic
287Q16—22no—
288Q16—22no—
289Q16—41no—
290Q16—41no—
291M16—41no—
292M16—41no—
293M16—41no—
294M16—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8C4×D411yescharacteristic
314C2×C8C4×D411yescharacteristic
315C2×C8C23×C411yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C82−1+411yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—81no—
344C22⋊C4—82no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—82no—
354C42—82no—
355C42—82no—
356C16—22no—
357C16—22no—
358C16—22no—
359C16—41no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—41no—
364C23#44211yescharacteristic
365C23#44211yescharacteristic
366C23C2×AGL(1,ℤ/8)11yescharacteristic
367Q8—42no—
368Q8—42no—
369Q8—81no—
370Q8—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—161no—
405C2×C4#44211yescharacteristic
406C2×C4#44211yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4#54611yescharacteristic
409C2×C4#54711yescharacteristic
410C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
411C2×C4#57711yescharacteristic
412C2×C4#57811yescharacteristic
413C8—21no—
414C8—21no—
415C8—41no—
416C8—41no—
417C8—81no—
418C8—81no—
419C8C4×#4111yescharacteristic
420C8#52011yescharacteristic
421C8#55111yescharacteristic
422C8#55211yescharacteristic
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—81no—
432C22—81no—
433C22C2×#44211yescharacteristic
434C22#331011yescharacteristic
435C22#334311yescharacteristic
436C4—21no—
437C4—21no—
438C4—81no—
439C4—81no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—161no—
448C4#235911yescharacteristic
449C4#294711yescharacteristic
450C4#330811yescharacteristic
451C4#334511yescharacteristic
452C2—21no—
453C2—21no—
454C2—41no—
455C2—41no—
456C2#2293811yescharacteristic
457C2#3346811yescharacteristic
458C2#3347011yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13174C2semidirect
5. #13154C2semidirect
6. #13141C2semidirect
7. #13119C2semidirect
8. #13118C2semidirect
10. #12957C2semidirect
12. #12677C2semidirect
13. #12648C2semidirect
23. #2237C4semidirect
24. #2237C4semidirect
25. #2236C4semidirect
26. #2235C4semidirect

Maximal subgroups

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

subgroupindex
#336492
#230302
#226072
#131742
#131542
#131412
#131192
#131182
#130662
#129572
#129302
#126772
#126482
#126162
#125672