Finite Group Databasebeta

← #497782

Subgroups — #497782

Order 512.

528 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2159C211yesmaximal, characteristic
2#22613C211yesmaximal, characteristic
3#22612C211yesmaximal, characteristic
4#13168C211yesmaximal, characteristic, complemented
5#13165C211yesmaximal, characteristic, complemented
6#13112C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13109C211yesmaximal, characteristic, complemented
10#12868C211yesmaximal, characteristic
11#12866C211yesmaximal, characteristic
12#12836C211yesmaximal, characteristic
13#12674C211yesmaximal, characteristic, complemented
14C4×#367C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19#2346—21no—
20#2345—21no—
21#2159C418yescomplemented
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×#346C2211yescharacteristic
30C2×C4×C16C2211yescharacteristic
31#1942C2211yescharacteristic
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1921—21no—
36#1921—21no—
37#1920—21no—
38#1920—21no—
39#1825—21no—
40#1824—21no—
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C8×C22⋊C4C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#117—21no—
62C22×C4⋊C4—21no—
63C2×M32—22no—
64C2×M32—22no—
65C2×M32C2×C414yes—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#427—22no—
71#427C2×C414yes—
72C2×#107—21no—
73C2×#107—22no—
74C2×#107C2×C414yes—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#106—21no—
84C2×C22⋊C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#388—41no—
92#388—41no—
93#387—41no—
94#387—41no—
95#382—41no—
96#380—21no—
97#380—21no—
98#380—41no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4—41no—
104C4×C22⋊C4D411yescharacteristic
105#376—41no—
106#367—22no—
107#367C2×C414yes—
108#367C2×C414yes—
109#366—22no—
110#366C2×C414yes—
111#366C2×C414yes—
112C22⋊C16—22no—
113#346—22no—
114C4×C16—22no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#336D411yescharacteristic
126#336D411yescharacteristic
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8C22×C412yes—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8#4112yes—
138C22×C8C2411yescharacteristic
139#117—24no—
140#117—42no—
141C2×C4⋊C4—24no—
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—21no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
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⋊C4D811yescharacteristic
160C2×C22⋊C4D811yescharacteristic
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168M32—28no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16C22×C412yes—
178C2×C16C22×C412yes—
179C2×C16#4112yes—
180C2×C16#4112yes—
181#107—24no—
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#107C22×C412yes—
195#107C22×C412yes—
196#107C22×C412yes—
197#107C22×C412yes—
198#106—24no—
199C22⋊C8—22no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95D811yescharacteristic
215#95D811yescharacteristic
216C24C2×D811yescharacteristic
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—44no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—161no—
231C22×C4—161no—
232C22×C4C4×D412yes—
233C22×C4C2×D811yescharacteristic
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4#13512yes—
239C22×C4C22×D411yescharacteristic
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8C4×D412yes—
253C2×C8C23×C411yescharacteristic
254C2×C8C2×#4111yescharacteristic
255C2×C8C2×#4111yescharacteristic
256C2×C8C2×#4111yescharacteristic
257C2×C82−1+411yescharacteristic
258C2×C82−1+411yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—44no—
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—161no—
277C4⋊C4—161no—
278C4⋊C4—161no—
279C4⋊C4—161no—
280C22⋊C4—22no—
281C22⋊C4—41no—
282C22⋊C4—44no—
283C22⋊C4—82no—
284C22⋊C4—161no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—44no—
290C42—82no—
291C42—82no—
292C42—161no—
293C42—161no—
294C16—22no—
295C16—41no—
296C16—41no—
297C16#11714yes—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23C4×D812yes—
305C23C22×D811yescharacteristic
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
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—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
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—164no—
348C2×C4C4×D812yes—
349C2×C4#51012yes—
350C2×C4C2×C4×D411yescharacteristic
351C2×C4#54711yescharacteristic
352C2×C4#54711yescharacteristic
353C2×C4C2×#13511yescharacteristic
354C2×C4#57811yescharacteristic
355C2×C4#57811yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C8C2×#11711yescharacteristic
360C8#51911yescharacteristic
361C8#55211yescharacteristic
362C8#55211yescharacteristic
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C2×C4×D811yescharacteristic
371C22#333411yescharacteristic
372C22#333411yescharacteristic
373C22C2×#51011yescharacteristic
374C22#341411yescharacteristic
375C22#341511yescharacteristic
376C22#341611yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—82no—
383C4—82no—
384C4—82no—
385C4—82no—
386C4—161no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4#234411yescharacteristic
391C4#295011yescharacteristic
392C4#333611yescharacteristic
393C4#333611yescharacteristic
394C2—42no—
395C2#2261111yescharacteristic
396C2C4×#51011yescharacteristic
397C2#3345411yescharacteristic
398C2#3391111yescharacteristic
399C2#3391411yescharacteristic
400C2#3391811yescharacteristic
401C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13168C2semidirect
5. #13165C2semidirect
6. #13112C2semidirect
7. #13111C2semidirect
8. #13110C2semidirect
9. #13109C2semidirect
13. #12674C2semidirect
14. C4×#367C2semidirect
21. #2159C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21592
#226132
#226122
#131682
#131652
#131122
#131112
#131102
#131092
#128682
#128662
#128362
#126742
C4×#3672
#126122