Finite Group Databasebeta

← #507754

Subgroups — #507754

Order 512.

477 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21725C211yesmaximal, characteristic, complemented
2#21282C211yesmaximal, characteristic
3#13507C211yesmaximal, characteristic, complemented
4#13482C211yesmaximal, characteristic
5#13452C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic
7#13131C211yesmaximal, characteristic
8#13131C211yesmaximal, characteristic
9#13111C211yesmaximal, characteristic, complemented
10#13111C211yesmaximal, characteristic, complemented
11#13110C211yesmaximal, characteristic, complemented
12#13109C211yesmaximal, characteristic, complemented
13#12937C211yesmaximal, characteristic
14#12935C211yesmaximal, characteristic
15#12895C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2331—21no—
19#2319—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28#1993—21no—
29#1970C2211yescharacteristic
30#1970C2211yescharacteristic
31#1943—21no—
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1936—21no—
37#1936—21no—
38#1846C2211yescharacteristic
39#1776—21no—
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43#1762—21no—
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×C4×D4—21no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66#500—22no—
67#500—22no—
68C4×D8—22no—
69C4×D8—22no—
70C2×#107—21no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#106—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#97D411yescharacteristic
83C2×#97D411yescharacteristic
84C2×#97D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#391—41no—
89#391—41no—
90#390—21no—
91#390—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#388—41no—
97#388—41no—
98#387—41no—
99#387—41no—
100#382—21no—
101#381—41no—
102#380—41no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106#376D411yescharacteristic
107#367—22no—
108#367—22no—
109#367—22no—
110#366—22no—
111#357—22no—
112#357—22no—
113#357—22no—
114#357—22no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122C22×Q8—41no—
123C22×D4—21no—
124C23×C4—41no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8#4112yes—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2411yescharacteristic
135C4⋊Q8—24no—
136C4⋊Q8—42no—
137C4×D4—24no—
138C4×D4—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×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C16—21no—
163C2×C16—21no—
164C2×C16—21no—
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—
174#107—24no—
175#107—21no—
176#107—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#4112yes—
187#107#4112yes—
188#107#4112yes—
189#106—42no—
190#102—24no—
191#102—42no—
192#102—42no—
193#97—22no—
194#97—41no—
195#97—41no—
196C4×C8—21no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95D811yescharacteristic
212#95D811yescharacteristic
213#95D811yescharacteristic
214#95D811yescharacteristic
215C24—41no—
216C2×Q8—44no—
217C2×Q8—161no—
218C2×Q8—161no—
219C2×D4—22no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C22×C4—44no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—161no—
234C22×C4—161no—
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C22×D411yescharacteristic
242D8—22no—
243D8—22no—
244D8—41no—
245D8—41no—
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×C8#13712yes—
255C2×C8C2×#4111yescharacteristic
256C2×C8C2×#4111yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C82−1+411yescharacteristic
260C2×C82−1+411yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—44no—
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—161no—
277C4⋊C4—161no—
278C4⋊C4—161no—
279C4⋊C4—161no—
280C22⋊C4—44no—
281C22⋊C4—161no—
282C22⋊C4—161no—
283C22⋊C4—161no—
284C22⋊C4—161no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—161no—
292C42—161no—
293C42—161no—
294C16—41no—
295C16—41no—
296C16—41no—
297C16—41no—
298C23—42no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23C22×D811yescharacteristic
306Q8—164no—
307D4—42no—
308D4—42no—
309D4—81no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
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—42no—
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—164no—
351C2×C4#52511yescharacteristic
352C2×C4#52511yescharacteristic
353C2×C4#53611yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4#57511yescharacteristic
356C2×C4#57511yescharacteristic
357C8—22no—
358C8—41no—
359C8—41no—
360C8#55211yescharacteristic
361C8#55211yescharacteristic
362C8#55611yescharacteristic
363C8#55611yescharacteristic
364C22—82no—
365C22—82no—
366C22—82no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22#310011yescharacteristic
372C22#310011yescharacteristic
373C22#322611yescharacteristic
374C22#341511yescharacteristic
375C22#341611yescharacteristic
376C22#341611yescharacteristic
377C22#341611yescharacteristic
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4#259911yescharacteristic
392C4#311611yescharacteristic
393C4#311711yescharacteristic
394C4#324011yescharacteristic
395C2—162no—
396C2#2938811yescharacteristic
397C2#3367811yescharacteristic
398C2#3367811yescharacteristic
399C2#3368511yescharacteristic
400C2#3368611yescharacteristic
401C2#3380711yescharacteristic
402C2#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. #21725C2semidirect
3. #13507C2semidirect
5. #13452C2semidirect
9. #13111C2semidirect
10. #13111C2semidirect
11. #13110C2semidirect
12. #13109C2semidirect
15. #12895C2semidirect

Maximal subgroups

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

subgroupindex
#217252
#212822
#135072
#134822
#134522
#133632
#131312
#131112
#131102
#131092
#129372
#129352
#128952