Finite Group Databasebeta

← #510720

Subgroups — #510720

Order 512.

456 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27995C211yesmaximal, characteristic, complemented
2#26482C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic
7#13474C211yesmaximal, characteristic
8#13453C211yesmaximal, characteristic, complemented
9#13277C211yesmaximal, characteristic, complemented
10#13165C211yesmaximal, characteristic, complemented
11#13159C211yesmaximal, characteristic, complemented
12#13147C211yesmaximal, characteristic
13#13130C211yesmaximal, characteristic
14#12935C211yesmaximal, characteristic
15#12811C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2507—21no—
19#2454—21no—
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#2100—21no—
33#2087—21no—
34#2065—21no—
35#2025—21no—
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1921—21no—
45#1864—21no—
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×#125—21no—
61C2×C4×D4—21no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66#499—22no—
67#499—41no—
68C4×D8—22no—
69C4×D8—41no—
70C2×#107—21no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#106—21no—
79C2×C4⋊C8D411yescharacteristic
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#399—41no—
87#398—41no—
88#398—41no—
89#396—41no—
90#390—21no—
91#390—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—41no—
99#387—41no—
100#385—41no—
101#384—21no—
102#380—41no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D411yescharacteristic
105#367—41no—
106#366—22no—
107#366—41no—
108#366—41no—
109C4⋊C16—41no—
110C4⋊C16—41no—
111C4⋊C16—41no—
112#357—41no—
113#357—41no—
114#357—41no—
115#357Q814yes—
116C4×C16—41no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#336D411yescharacteristic
125#336D411yescharacteristic
126C22×D4—21no—
127C23×C4—41no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8C2×Q812yes—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2411yescharacteristic
138#125—42no—
139#125—81no—
140C4×D4—42no—
141C4×D4—81no—
142C2×C4⋊C4—21no—
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—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×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4D811yescharacteristic
160C2×C4⋊C4D811yescharacteristic
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C42—41no—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C16—21no—
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—
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—21no—
192#107—21no—
193#107—21no—
194#107—42no—
195#107C2×Q812yes—
196#107#4112yes—
197#106—42no—
198C4⋊C8—41no—
199#102—42no—
200#102—42no—
201#102—81no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95D811yescharacteristic
216#95D811yescharacteristic
217C24—41no—
218C2×D4—22no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—161no—
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C22×D411yescharacteristic
241D8—22no—
242D8—22no—
243D8—41no—
244D8—41no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8C22×Q811yescharacteristic
255C2×C8C2×#4111yescharacteristic
256C2×C8H(5,𝔽2)11yescharacteristic
257C2×C8H(5,𝔽2)11yescharacteristic
258C2×C82−1+411yescharacteristic
259C2×C82−1+411yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
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—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—161no—
280C4⋊C4—161no—
281C4⋊C4—161no—
282C4⋊C4—161no—
283C4⋊C4—161no—
284C4⋊C4—161no—
285C4⋊C4—161no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4—161no—
289C22⋊C4—161no—
290C22⋊C4—161no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—161no—
298C16—22no—
299C16—41no—
300C16—41no—
301C16—41no—
302C23—42no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23C22×D811yescharacteristic
310D4—42no—
311D4—42no—
312D4—81no—
313D4—81no—
314C2×C4—21no—
315C2×C4—21no—
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—42no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—81no—
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—164no—
352C2×C4#50812yes—
353C2×C4#51012yes—
354C2×C4#53711yescharacteristic
355C2×C4#54611yescharacteristic
356C2×C4D4×Q811yescharacteristic
357C2×C4#57511yescharacteristic
358C2×C4#57711yescharacteristic
359C2×C4#57811yescharacteristic
360C8—41no—
361C8—41no—
362C8—41no—
363C8#54111yescharacteristic
364C8#54411yescharacteristic
365C8#55411yescharacteristic
366C8#56211yescharacteristic
367C22—82no—
368C22—82no—
369C22—82no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22#324511yescharacteristic
375C22#329411yescharacteristic
376C22Q8×D811yescharacteristic
377C22C2×#50811yescharacteristic
378C22C2×#51011yescharacteristic
379C22#341511yescharacteristic
380C22#341511yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—82no—
388C4—82no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4#270411yescharacteristic
395C4#325811yescharacteristic
396C4#332711yescharacteristic
397C4#338811yescharacteristic
398C2—162no—
399C2#3186611yescharacteristic
400C2#3381411yescharacteristic
401C2#3382511yescharacteristic
402C2#3386411yescharacteristic
403C2#3390011yescharacteristic
404C2#3394711yescharacteristic
405C2#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
1. #27995C2semidirect
3. #13516C2semidirect
4. #13515C2semidirect
5. #13505C2semidirect
8. #13453C2semidirect
9. #13277C2semidirect
10. #13165C2semidirect
11. #13159C2semidirect

Maximal subgroups

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

subgroupindex
#279952
#264822
#135162
#135152
#135052
#134762
#134742
#134532
#132772
#131652
#131592
#131472
#131302
#129352
#128112