Finite Group Databasebeta

← #511204

Subgroups — #511204

Order 512.

468 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28534C211yesmaximal, characteristic
2#27995C211yesmaximal, characteristic, complemented
3#13538C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic
5#13507C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic
7#13453C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13178C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic
11#13130C211yesmaximal, characteristic
12#13104C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic
14#12935C211yesmaximal, characteristic
15#12905C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#478—21no—
18C2×C4×D8C2211yescharacteristic
19#2566—21no—
20#2507—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#2100—21no—
34#2094—21no—
35#2092—21no—
36#2087—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1921—21no—
44#1846C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
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×D8C2311yescharacteristic
60C2×#125—21no—
61C2×C4×D4—21no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66#499—41no—
67#499—41no—
68#478—44no—
69C4×D8—41no—
70C4×D8—41no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#398—41no—
90#398—41no—
91#396—41no—
92#396—41no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—41no—
100#387—41no—
101#385—41no—
102#385—41no—
103#384—21no—
104#384—21no—
105#382—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108#367—41no—
109#367—41no—
110#366—41no—
111#366—41no—
112#366—41no—
113#366—41no—
114C4⋊C16—22no—
115#357—22no—
116#357—41no—
117#357—41no—
118#357Q814yes—
119C4×C16—22no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#336D411yescharacteristic
126#336D411yescharacteristic
127C22×D4—21no—
128C23×C4—41no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8C2×Q812yes—
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2411yescharacteristic
139#125—42no—
140#125—42no—
141C4×D4—42no—
142C4×D4—42no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
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×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4D811yescharacteristic
161C2×C4⋊C4D811yescharacteristic
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16C2×Q812yes—
182C2×C16#4112yes—
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—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—42no—
197#107—42no—
198#107C2×Q812yes—
199#106—42no—
200#106—42no—
201C4⋊C8—22no—
202#102—42no—
203#102—42no—
204#102—42no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95D811yescharacteristic
219#95D811yescharacteristic
220C24—41no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—82no—
235C22×C4—161no—
236C22×C4—161no—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4C22×D411yescharacteristic
244D8—22no—
245D8—22no—
246D8—41no—
247D8—41no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8AGL(1,ℤ/8)12yes—
257C2×C8C22×Q811yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8H(5,𝔽2)11yescharacteristic
260C2×C8H(5,𝔽2)11yescharacteristic
261C2×C82−1+411yescharacteristic
262C2×C82−1+411yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
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—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—161no—
283C4⋊C4—161no—
284C4⋊C4—161no—
285C4⋊C4—161no—
286C4⋊C4—161no—
287C4⋊C4—161no—
288C4⋊C4—161no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C22⋊C4—161no—
293C22⋊C4—161no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—161no—
301C16—22no—
302C16—22no—
303C16—41no—
304C16—41no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23C22×D811yescharacteristic
313D4—42no—
314D4—42no—
315D4—81no—
316D4—81no—
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—42no—
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—164no—
355C2×C4#50812yes—
356C2×C4#51012yes—
357C2×C4#53511yescharacteristic
358C2×C4#54711yescharacteristic
359C2×C4D4×Q811yescharacteristic
360C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
361C2×C4#57811yescharacteristic
362C2×C4#57811yescharacteristic
363C8—22no—
364C8—41no—
365C8—41no—
366C8#53911yescharacteristic
367C8#54411yescharacteristic
368C8#55411yescharacteristic
369C8#56311yescharacteristic
370C22—82no—
371C22—82no—
372C22—82no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22#320511yescharacteristic
378C22#333411yescharacteristic
379C22Q8×D811yescharacteristic
380C22C2×#50811yescharacteristic
381C22C2×#51011yescharacteristic
382C22#341511yescharacteristic
383C22#341511yescharacteristic
384C4—22no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—82no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4—161no—
395C4—161no—
396C4—161no—
397C4#272011yescharacteristic
398C4#322311yescharacteristic
399C4#334311yescharacteristic
400C4#338611yescharacteristic
401C2—162no—
402C2#3234211yescharacteristic
403C2#3377611yescharacteristic
404C2#3378611yescharacteristic
405C2#3390611yescharacteristic
406C2#3391311yescharacteristic
407C2#3394711yescharacteristic
408C2#3395211yescharacteristic
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
2. #27995C2semidirect
3. #13538C2semidirect
5. #13507C2semidirect
7. #13453C2semidirect
8. #13453C2semidirect
9. #13178C2semidirect
12. #13104C2semidirect
15. #12905C2semidirect

Maximal subgroups

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

subgroupindex
#285342
#279952
#135382
#135272
#135072
#134742
#134532
#131782
#131472
#131302
#131042
#129352
#129052