Finite Group Databasebeta

← #504609

Subgroups — #504609

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27994C211yesmaximal, characteristic, complemented
2#23186C211yesmaximal, characteristic
3#13481C211yesmaximal, characteristic, complemented
4#13475C211yesmaximal, characteristic, complemented
5#13408C211yesmaximal, characteristic
6#13408C211yesmaximal, characteristic
7#13147C211yesmaximal, characteristic
8#13130C211yesmaximal, characteristic
9#13112C211yesmaximal, characteristic, complemented
10#13111C211yesmaximal, characteristic, complemented
11#13104C211yesmaximal, characteristic, complemented
12#12938C211yesmaximal, characteristic, complemented
13#12937C211yesmaximal, characteristic
14#12811C211yesmaximal, characteristic
15C4×#358C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2506—21no—
20#2367—21no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#2099—21no—
34#2086—21no—
35#1983—21no—
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1918—21no—
43#1880—21no—
44#1877—21no—
45#1845C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×#125—41no—
63C2×#124—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#495—22no—
71#495—22no—
72C4×Q16—22no—
73C4×Q16—41no—
74C4×D8—22no—
75C4×D8—41no—
76C2×#107—21no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×#103—21no—
85C2×#103—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#400—41no—
92#398—41no—
93#391—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#388—41no—
100#388—41no—
101#387—41no—
102#385—41no—
103#384—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D411yescharacteristic
107C4×C22⋊C4—41no—
108#367—22no—
109#366—22no—
110C4⋊C16—41no—
111#358—22no—
112#358—22no—
113#358—22no—
114#358—41no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—41no—
119C4×C16—41no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C22×D4—21no—
130C23×C4—41no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16#4112yes—
137C2×SD16—24no—
138C2×SD16—81no—
139C2×SD16—81no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8#4112yes—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2411yescharacteristic
150#125—84no—
151#124—24no—
152#124—42no—
153C4×Q8—42no—
154C4×Q8—81no—
155C4×D4—42no—
156C4×D4—81no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4D811yescharacteristic
171C2×C4⋊C4D811yescharacteristic
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—42no—
206#107#4112yes—
207#107#4112yes—
208C4⋊C8—41no—
209#103—42no—
210#103—81no—
211#102—42no—
212#102—81no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95D811yescharacteristic
228#95D811yescharacteristic
229C24—41no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—161no—
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C22×D411yescharacteristic
259Q16—22no—
260Q16—22no—
261Q16—41no—
262Q16—41no—
263SD16—84no—
264D8—22no—
265D8—22no—
266D8—41no—
267D8—41no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C2×C82−1+411yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—161no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—82no—
316C42—82no—
317C42—161no—
318C16—41no—
319C16—41no—
320C16—41no—
321C16—41no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23C22×D811yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—42no—
335D4—42no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—164no—
376C2×C4#50812yes—
377C2×C4#50912yes—
378C2×C4#52511yescharacteristic
379C2×C4#54611yescharacteristic
380C2×C4#54811yescharacteristic
381C2×C4#57511yescharacteristic
382C2×C4#57711yescharacteristic
383C2×C4#57711yescharacteristic
384C8—41no—
385C8—41no—
386C8—41no—
387C8#52911yescharacteristic
388C8#53911yescharacteristic
389C8#55211yescharacteristic
390C8#55311yescharacteristic
391C22—82no—
392C22—82no—
393C22—82no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22#310011yescharacteristic
399C22#329411yescharacteristic
400C22#335811yescharacteristic
401C22C2×#50811yescharacteristic
402C22C2×#50911yescharacteristic
403C22#341611yescharacteristic
404C22#341711yescharacteristic
405C4—22no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—82no—
412C4—82no—
413C4—82no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4#246011yescharacteristic
419C4#312311yescharacteristic
420C4#331711yescharacteristic
421C4#337111yescharacteristic
422C2—162no—
423C2#2675211yescharacteristic
424C2#3366611yescharacteristic
425C2#3368011yescharacteristic
426C2#3386311yescharacteristic
427C2#3387511yescharacteristic
428C2#3393111yescharacteristic
429C2#3394011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ 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. #27994C2semidirect
3. #13481C2semidirect
4. #13475C2semidirect
9. #13112C2semidirect
10. #13111C2semidirect
11. #13104C2semidirect
12. #12938C2semidirect
15. C4×#358C2semidirect

Maximal subgroups

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

subgroupindex
#279942
#231862
#134812
#134752
#134082
#131472
#131302
#131122
#131112
#131042
#129382
#129372
#128112
C4×#3582