Finite Group Databasebeta

← #511690

Subgroups — #511690

Order 512.

469 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33257C211yesmaximal, characteristic, complemented
2#33240C211yesmaximal, characteristic
3#13529C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic
5#13483C211yesmaximal, characteristic, complemented
6#13482C211yesmaximal, characteristic
7#13477C211yesmaximal, characteristic, complemented
8#13476C211yesmaximal, characteristic
9#13475C211yesmaximal, characteristic, complemented
10#13474C211yesmaximal, characteristic
11#13453C211yesmaximal, characteristic, complemented
12#13453C211yesmaximal, characteristic, complemented
13#13408C211yesmaximal, characteristic
14#13408C211yesmaximal, characteristic
15#12879C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#488—21no—
19#2864—21no—
20#2861—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1968—21no—
40#1968—21no—
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1941—21no—
46#1941—21no—
47#1846C2211yescharacteristic
48#1846C2211yescharacteristic
49#1835—21no—
50C42×C8C2211yescharacteristic
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—21no—
63C2×#124—21no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#499—41no—
69#499—41no—
70#495—41no—
71#495—41no—
72#488—44no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#391—21no—
93#391—21no—
94#390—21no—
95#390—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#388—41no—
101#388—41no—
102#388—41no—
103#388—41no—
104#387—41no—
105#387—41no—
106#387—41no—
107#387—41no—
108#382—21no—
109#382—21no—
110C43D411yescharacteristic
111#367—41no—
112#367—41no—
113#366—41no—
114#366—41no—
115C4⋊C16—41no—
116C4⋊C16—41no—
117#358—41no—
118#358—41no—
119#358—41no—
120#358—41no—
121#357—41no—
122#357—41no—
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—42no—
151#125—42no—
152#124—42no—
153#124—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42D811yescharacteristic
173C2×C42D811yescharacteristic
174C2×C42D811yescharacteristic
175C2×C42D811yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—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#106—42no—
204#106—42no—
205#106—42no—
206#103—42no—
207#103—42no—
208#102—42no—
209#102—42no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—41no—
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#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C22×D411yescharacteristic
258Q16—41no—
259Q16—41no—
260Q16—41no—
261Q16—41no—
262SD16—84no—
263D8—41no—
264D8—41no—
265D8—41no—
266D8—41no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8AGL(1,ℤ/8)12yes—
275C2×C8#13712yes—
276C2×C8H(5,𝔽2)11yescharacteristic
277C2×C8H(5,𝔽2)11yescharacteristic
278C2×C8H(5,𝔽2)11yescharacteristic
279C2×C82−1+411yescharacteristic
280C2×C82−1+411yescharacteristic
281C2×C82−1+411yescharacteristic
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—22no—
313C42—41no—
314C42—41no—
315C42—41no—
316C16—41no—
317C16—41no—
318C16—41no—
319C16—41no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23C22×D811yescharacteristic
328Q8—81no—
329Q8—81no—
330Q8—81no—
331Q8—81no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
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—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
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#50812yes—
375C2×C4#53511yescharacteristic
376C2×C4#53611yescharacteristic
377C2×C4#53711yescharacteristic
378C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
379C2×C4C2×#13711yescharacteristic
380C2×C4#57511yescharacteristic
381C8—22no—
382C8—22no—
383C8—41no—
384C8#56211yescharacteristic
385C8#56211yescharacteristic
386C8#56311yescharacteristic
387C8#56311yescharacteristic
388C22—161no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22#320511yescharacteristic
396C22#322611yescharacteristic
397C22#324511yescharacteristic
398C22C2×#50811yescharacteristic
399C22#341411yescharacteristic
400C22#341411yescharacteristic
401C22#341411yescharacteristic
402C4—22no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4#282011yescharacteristic
416C4#321511yescharacteristic
417C4#323411yescharacteristic
418C4#325411yescharacteristic
419C2—321no—
420C2#3281811yescharacteristic
421C2#3377811yescharacteristic
422C2#3378511yescharacteristic
423C2#3379911yescharacteristic
424C2#3380611yescharacteristic
425C2#3381611yescharacteristic
426C2#3382211yescharacteristic
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
1. #33257C2semidirect
3. #13529C2semidirect
5. #13483C2semidirect
7. #13477C2semidirect
9. #13475C2semidirect
11. #13453C2semidirect
12. #13453C2semidirect
15. #12879C2semidirect

Maximal subgroups

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

subgroupindex
#332572
#332402
#135292
#135282
#134832
#134822
#134772
#134762
#134752
#134742
#134532
#134082
#128792