Finite Group Databasebeta

← #504341

Subgroups — #504341

Order 512.

480 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28831C211yesmaximal, characteristic, complemented
2#23179C211yesmaximal, characteristic
3#13534C211yesmaximal, characteristic, complemented
4#13533C211yesmaximal, characteristic, complemented
5#13467C211yesmaximal, characteristic, complemented
6#13467C211yesmaximal, characteristic, complemented
7#13255C211yesmaximal, characteristic
8#13211C211yesmaximal, characteristic
9#13184C211yesmaximal, characteristic, complemented
10#13163C211yesmaximal, characteristic, complemented
11#12950C211yesmaximal, characteristic, complemented
12#12950C211yesmaximal, characteristic, complemented
13#12824C211yesmaximal, characteristic, complemented
14#12823C211yesmaximal, characteristic, complemented
15C4×#365C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17C2×#443C2211yescharacteristic
18C2×#443C2211yescharacteristic
19#2935—21no—
20#2570—21no—
21C2×#365C2211yescharacteristic
22C2×#365C2211yescharacteristic
23C2×#365C2211yescharacteristic
24C2×#365C2211yescharacteristic
25C2×#364C2211yescharacteristic
26C2×#364C2211yescharacteristic
27#2150C2211yescharacteristic
28C2×#360C2211yescharacteristic
29C2×#360C2211yescharacteristic
30C2×#360C2211yescharacteristic
31C2×#360C2211yescharacteristic
32C2×#360C2211yescharacteristic
33C2×#360C2211yescharacteristic
34C2×#360C2211yescharacteristic
35C2×#360C2211yescharacteristic
36C4×M32C2211yescharacteristic
37#2085—21no—
38#2083—21no—
39#1980—21no—
40#1974C2211yescharacteristic
41#1974C2211yescharacteristic
42#1949—21no—
43#1941C2211yescharacteristic
44#1941C2211yescharacteristic
45#1876—21no—
46#1876—21no—
47#1846C2211yescharacteristic
48#1846C2211yescharacteristic
49#1844C2211yescharacteristic
50C4×#108C2211yescharacteristic
51C4×#106C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1383C2211yescharacteristic
54#1383C2211yescharacteristic
55#1383C2211yescharacteristic
56#1383C2211yescharacteristic
57#1376C2211yescharacteristic
58#1376C2211yescharacteristic
59C2×#135—21no—
60C2×#135C2311yescharacteristic
61C2×#135C2311yescharacteristic
62C2×#131—21no—
63C2×#131—21no—
64#532—21no—
65C2×M32C2311yescharacteristic
66C2×M32C2311yescharacteristic
67C2×M32C2311yescharacteristic
68C2×M32C2311yescharacteristic
69#495—21no—
70#495Q811yescharacteristic
71#495Q811yescharacteristic
72#494Q811yescharacteristic
73#493Q811yescharacteristic
74#492—21no—
75#443—22no—
76#443—22no—
77#443—41no—
78#443—41no—
79#436—41no—
80#435—41no—
81#435—41no—
82C8×D4—41no—
83C2×#108C2311yescharacteristic
84C2×#108C2311yescharacteristic
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106C2311yescharacteristic
89C2×#106C2311yescharacteristic
90C2×#106C2311yescharacteristic
91C2×#106C2311yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×#104—21no—
94C2×#104—21no—
95C2×#104—21no—
96C2×#104—21no—
97#408—21no—
98#408—21no—
99C4×M16—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#398—41no—
105#389—21no—
106#389—21no—
107#387—41no—
108#387—41no—
109#384—21no—
110#382—21no—
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#365—22no—
115#365—22no—
116#365—41no—
117#365—41no—
118#364—41no—
119#364—41no—
120C4⋊C16—21no—
121C4⋊C16—21no—
122#360—22no—
123#360—22no—
124#360—41no—
125#360—41no—
126#360—41no—
127#360—41no—
128#360—41no—
129#360—41no—
130#346—21no—
131C4×C16—21no—
132#341—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C2×#41—21no—
139C2×#41—21no—
140#135—21no—
141#135—21no—
142#135—21no—
143#135—21no—
144#135—21no—
145#135—21no—
146#135—21no—
147#135—21no—
148#135—81no—
149#135—81no—
150C2×Q16—21no—
151C2×Q16C2×Q811yescharacteristic
152C2×Q16C2×Q811yescharacteristic
153C2×SD16—21no—
154C2×SD16—21no—
155C2×SD16—21no—
156C2×SD16—21no—
157C2×D8—21no—
158C2×D8C2×Q811yescharacteristic
159C2×D8C2×Q811yescharacteristic
160#131—42no—
161#131—42no—
162#131—81no—
163#131—81no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—41no—
167C2×M16—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174C4⋊Q8—21no—
175#127—21no—
176#124—21no—
177#124—21no—
178#121—41no—
179#121—41no—
180C4×D4—41no—
181C4×D4—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195M32—21no—
196M32—21no—
197M32—21no—
198M32—21no—
199M32—21no—
200M32—21no—
201M32—21no—
202M32#4112yes—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207#108—21no—
208#108—21no—
209#108—21no—
210#108#4112yes—
211#107—81no—
212#107—81no—
213#106—21no—
214#106—21no—
215#106—21no—
216#106—21no—
217#106—21no—
218#106—21no—
219#106—21no—
220#106—21no—
221#106—81no—
222C4⋊C8SD1611yescharacteristic
223C4⋊C8SD1611yescharacteristic
224C4⋊C8SD1611yescharacteristic
225C4⋊C8SD1611yescharacteristic
226#104—42no—
227#104—42no—
228#104—81no—
229#104—81no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234#97—41no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—41no—
240C4×C8C2×Q811yescharacteristic
241C4×C8C2×Q811yescharacteristic
242C4×C8#4111yescharacteristic
243C4×C8#4111yescharacteristic
244#95—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4AGL(1,ℤ/8)11yescharacteristic
277C22×C4AGL(1,ℤ/8)11yescharacteristic
278C22×C4C22×D411yescharacteristic
279Q16—21no—
280Q16—21no—
281Q16—21no—
282Q16—21no—
283Q16—81no—
284SD16—41no—
285SD16—41no—
286SD16—41no—
287SD16—41no—
288SD16—81no—
289SD16—81no—
290D8—21no—
291D8—21no—
292D8—21no—
293D8—21no—
294D8—81no—
295M16—22no—
296M16—41no—
297M16—41no—
298M16—41no—
299M16—82no—
300M16—82no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—82no—
310C2×C8—82no—
311C2×C8C22⋊Q811yescharacteristic
312C2×C8C22⋊Q811yescharacteristic
313C2×C8C2×SD1611yescharacteristic
314C2×C8C2×SD1611yescharacteristic
315C2×C8C2×SD1611yescharacteristic
316C2×C8C2×SD1611yescharacteristic
317C2×C8C22×Q811yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8H(5,𝔽2)11yescharacteristic
322C2×C8H(5,𝔽2)11yescharacteristic
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—161no—
337C4⋊C4—161no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—161no—
347C42C22⋊Q811yescharacteristic
348C42C22⋊Q811yescharacteristic
349C42C2×SD1611yescharacteristic
350C42C2×SD1611yescharacteristic
351C16—22no—
352C16—41no—
353C16—41no—
354C16—41no—
355C23—41no—
356C23—41no—
357C23C2×AGL(1,ℤ/8)11yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—41no—
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—41no—
366D4—82no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—161no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#47611yescharacteristic
399C2×C4#47611yescharacteristic
400C2×C4#47611yescharacteristic
401C2×C4#47611yescharacteristic
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4#54611yescharacteristic
404C2×C4#55011yescharacteristic
405C2×C4C22×SD1611yescharacteristic
406C2×C4#57611yescharacteristic
407C2×C4#57711yescharacteristic
408C8—21no—
409C8—21no—
410C8—21no—
411C8—21no—
412C8—21no—
413C8—21no—
414C8—42no—
415C8—81no—
416C8#52911yescharacteristic
417C8#54011yescharacteristic
418C8#55411yescharacteristic
419C8#55411yescharacteristic
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C2×#47611yescharacteristic
429C22#333211yescharacteristic
430C22#339911yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—42no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4#245411yescharacteristic
444C4#308111yescharacteristic
445C4#329511yescharacteristic
446C4#339211yescharacteristic
447C2—21no—
448C2—21no—
449C2—161no—
450C2—161no—
451C2#2653011yescharacteristic
452C2#3363711yescharacteristic
453C2#3363811yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radical#3147C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28831C2semidirect
3. #13534C2semidirect
4. #13533C2semidirect
5. #13467C2semidirect
6. #13467C2semidirect
9. #13184C2semidirect
10. #13163C2semidirect
11. #12950C2semidirect
12. #12950C2semidirect
13. #12824C2semidirect
14. #12823C2semidirect
15. C4×#365C2semidirect

Maximal subgroups

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

subgroupindex
#288312
#231792
#135342
#135332
#134672
#132552
#132112
#131842
#131632
#129502
#128242
#128232
C4×#3652