Finite Group Databasebeta

← #511712

Subgroups — #511712

Order 512.

440 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33319C211yesmaximal, characteristic, complemented
2#33272C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13537C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic
6#13527C211yesmaximal, characteristic
7#13507C211yesmaximal, characteristic, complemented
8#13506C211yesmaximal, characteristic, complemented
9#13482C211yesmaximal, characteristic
10#13480C211yesmaximal, characteristic
11#13453C211yesmaximal, characteristic, complemented
12#13452C211yesmaximal, characteristic, complemented
13#13365C211yesmaximal, characteristic
14#13363C211yesmaximal, characteristic
15#12895C211yesmaximal, characteristic, complemented
16C2×#501—21no—
17C2×#500C2211yescharacteristic
18C2×#499C2211yescharacteristic
19#2660—21no—
20#2644—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29#1970C2211yescharacteristic
30#1970C2211yescharacteristic
31#1970C2211yescharacteristic
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1936—21no—
36#1936—21no—
37#1936—21no—
38#1847—21no—
39#1847—21no—
40#1846C2211yescharacteristic
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1726C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1374C2211yescharacteristic
54#1374C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×C4⋊Q8—21no—
61C2×#125—21no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66#501—44no—
67#500—41no—
68#500—41no—
69#499—41no—
70#499—41no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×#97D411yescharacteristic
88C2×#97D411yescharacteristic
89C2×#97D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#398—41no—
94#398—41no—
95#397—41no—
96#397—41no—
97#397—41no—
98#396—41no—
99#395—41no—
100#395—41no—
101#390—21no—
102#390—21no—
103#390—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#384—21no—
108#384—21no—
109#382—21no—
110#382—21no—
111#376D411yescharacteristic
112#367—41no—
113#367—41no—
114#367—41no—
115#366—41no—
116#357—41no—
117#357—41no—
118#357—41no—
119#357—41no—
120#340—21no—
121#340—21no—
122#340—21no—
123C22×Q8—41no—
124C22×D4—21no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8—21no—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2411yescharacteristic
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137#125—42no—
138#125—42no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177#107—21no—
178#107—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—42no—
192#107—42no—
193#107—42no—
194#106—42no—
195#106—42no—
196#106—42no—
197#102—42no—
198#102—42no—
199#102—42no—
200#97—22no—
201#97—22no—
202#97—22no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95D811yescharacteristic
217#95D811yescharacteristic
218#95D811yescharacteristic
219#95D811yescharacteristic
220C24—41no—
221C2×Q8—82no—
222C2×Q8—161no—
223C2×Q8—161no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4C22×D411yescharacteristic
244D8—41no—
245D8—41no—
246D8—41no—
247D8—41no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8AGL(1,ℤ/8)12yes—
255C2×C8#13712yes—
256C2×C8#13712yes—
257C2×C8H(5,𝔽2)11yescharacteristic
258C2×C8H(5,𝔽2)11yescharacteristic
259C2×C82−1+411yescharacteristic
260C2×C82−1+411yescharacteristic
261C2×C82−1+411yescharacteristic
262C2×C82−1+411yescharacteristic
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—161no—
281C4⋊C4—161no—
282C4⋊C4—161no—
283C4⋊C4—161no—
284C4⋊C4—161no—
285C4⋊C4—161no—
286C4⋊C4—161no—
287C22⋊C4—161no—
288C22⋊C4—161no—
289C22⋊C4—161no—
290C22⋊C4—161no—
291C22⋊C4—161no—
292C22⋊C4—161no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C16—41no—
300C16—41no—
301C16—41no—
302C16—41no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23C22×D811yescharacteristic
311Q8—164no—
312D4—81no—
313D4—81no—
314D4—81no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4#53511yescharacteristic
356C2×C4#53611yescharacteristic
357C2×C4#53611yescharacteristic
358C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
359C2×C4C2×#13711yescharacteristic
360C2×C4C2×#13711yescharacteristic
361C8—22no—
362C8—22no—
363C8—22no—
364C8#56311yescharacteristic
365C8#56311yescharacteristic
366C8#56311yescharacteristic
367C8#56311yescharacteristic
368C22—161no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22#320511yescharacteristic
376C22#322611yescharacteristic
377C22#322611yescharacteristic
378C22#341511yescharacteristic
379C22#341611yescharacteristic
380C22#341611yescharacteristic
381C22#341611yescharacteristic
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4—161no—
395C4#282111yescharacteristic
396C4#321811yescharacteristic
397C4#323811yescharacteristic
398C4#323911yescharacteristic
399C2—321no—
400C2#3283411yescharacteristic
401C2#3378711yescharacteristic
402C2#3379011yescharacteristic
403C2#3380711yescharacteristic
404C2#3380711yescharacteristic
405C2#3381011yescharacteristic
406C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33319C2semidirect
3. #13539C2semidirect
4. #13537C2semidirect
7. #13507C2semidirect
8. #13506C2semidirect
11. #13453C2semidirect
12. #13452C2semidirect
15. #12895C2semidirect

Maximal subgroups

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

subgroupindex
#333192
#332722
#135392
#135372
#135282
#135272
#135072
#135062
#134822
#134802
#134532
#134522
#133652
#133632
#128952