Finite Group Databasebeta

← #508701

Subgroups — #508701

Order 512.

499 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33240C211yesmaximal, characteristic
2C4×#499C211yesmaximal, characteristic, complemented
3#13529C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic
5#13453C211yesmaximal, characteristic, complemented
6#13453C211yesmaximal, characteristic, complemented
7#13408C211yesmaximal, characteristic
8#13408C211yesmaximal, characteristic
9#13366C211yesmaximal, characteristic, complemented
10#13365C211yesmaximal, characteristic
11#13133C211yesmaximal, characteristic, complemented
12#13132C211yesmaximal, characteristic, complemented
13#13131C211yesmaximal, characteristic
14#13130C211yesmaximal, characteristic
15#12894C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×#488—21no—
20#2861—21no—
21C4×#125—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34#1991—21no—
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1968—21no—
40#1968—21no—
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1941—21no—
44#1846C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#106—21no—
48C42×C8C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1374C2211yescharacteristic
54#1374C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×#125—21no—
63C2×#125—21no—
64C2×#124—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#499—22no—
70#499—22no—
71#499—41no—
72#499—41no—
73#495—41no—
74#495—41no—
75#488—44no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×C4×C8D411yescharacteristic
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#391—21no—
97#390—21no—
98#390—21no—
99#389—21no—
100#389—21no—
101#388—41no—
102#388—41no—
103#388—41no—
104#388—41no—
105#387—41no—
106#387—41no—
107#382—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—41no—
111C4×C4⋊C4—41no—
112C43D411yescharacteristic
113#367—41no—
114#367—41no—
115#366—41no—
116#366—41no—
117#358—22no—
118#358—22no—
119#358—41no—
120#358—41no—
121#357—22no—
122#357—22no—
123#357—41no—
124#357—41no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129C22×Q8—21no—
130C22×D4—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×SD16—42no—
138C2×SD16—42no—
139C2×SD16—42no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2411yescharacteristic
150#125—24no—
151#125—42no—
152#125—42no—
153#125—42no—
154#124—42no—
155#124—42no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42D811yescharacteristic
177C2×C42D811yescharacteristic
178C2×C42D811yescharacteristic
179C2×C42D811yescharacteristic
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#4112yes—
206#107#4112yes—
207#106—24no—
208#106—42no—
209#106—42no—
210#103—42no—
211#103—42no—
212#102—42no—
213#102—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233C24—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C22×D411yescharacteristic
260Q16—41no—
261Q16—41no—
262Q16—41no—
263Q16—41no—
264SD16—84no—
265D8—41no—
266D8—41no—
267D8—41no—
268D8—41no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8AGL(1,ℤ/8)12yes—
277C2×C8#13712yes—
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C2×C82−1+411yescharacteristic
283C2×C82−1+411yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4—44no—
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—41no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—22no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—82no—
319C42—82no—
320C16—41no—
321C16—41no—
322C16—41no—
323C16—41no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23C22×D811yescharacteristic
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
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—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
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—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4#50812yes—
379C2×C4#52511yescharacteristic
380C2×C4#53511yescharacteristic
381C2×C4#53611yescharacteristic
382C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C2×#13711yescharacteristic
384C2×C4#57511yescharacteristic
385C8—22no—
386C8—22no—
387C8—41no—
388C8#54211yescharacteristic
389C8#54211yescharacteristic
390C8#54211yescharacteristic
391C8#54211yescharacteristic
392C22—161no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22#310011yescharacteristic
400C22#320511yescharacteristic
401C22#322611yescharacteristic
402C22C2×#50811yescharacteristic
403C22#341411yescharacteristic
404C22#341411yescharacteristic
405C22#341411yescharacteristic
406C4—22no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—82no—
413C4—82no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4#263211yescharacteristic
420C4#311211yescharacteristic
421C4#321511yescharacteristic
422C4#323511yescharacteristic
423C2—321no—
424C2#3005911yescharacteristic
425C2#3366211yescharacteristic
426C2#3367111yescharacteristic
427C2#3377811yescharacteristic
428C2#3378511yescharacteristic
429C2#3379811yescharacteristic
430C2#3380611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#499C2semidirect
3. #13529C2semidirect
5. #13453C2semidirect
6. #13453C2semidirect
9. #13366C2semidirect
11. #13133C2semidirect
12. #13132C2semidirect
15. #12894C2semidirect

Maximal subgroups

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

subgroupindex
#332402
C4×#4992
#135292
#135282
#134532
#134082
#133662
#133652
#131332
#131322
#131312
#131302
#128942