Finite Group Databasebeta

← #512209

Subgroups — #512209

Order 512.

675 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
1C2×#2243C211yesmaximal, characteristic, complemented
2C2×C4×M32C211yesmaximal, characteristic, complemented
3#33209C211yesmaximal, characteristic, complemented
4#33173C211yesmaximal, characteristic
5#13283C211yesmaximal, characteristic, complemented
6#13205C211yesmaximal, characteristic, complemented
7#13148C212yesmaximal, complemented
8#13147C212yesmaximal, complemented
9#13062C212yesmaximal, complemented
10#13061C212yesmaximal
11#12890C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#500C2211yescharacteristic
14C2×#495C2211yescharacteristic
15C2×#467C2211yescharacteristic
16C2×#466C2211yescharacteristic
17C2×#465—21no—
18C22×C4×C8C2211yescharacteristic
19#2857—21no—
20#2850—21no—
21#2243—24no—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C4×M32—24no—
27C2×#346C2211yescharacteristic
28C2×#346C2211yescharacteristic
29C2×C4×C16C2212yes—
30#1990—21no—
31#1970C2211yescharacteristic
32#1969C2211yescharacteristic
33#1968—21no—
34#1958—21no—
35#1942—21no—
36#1942C2212yes—
37#1846C2211yescharacteristic
38#1845C2211yescharacteristic
39#1808—21no—
40#1808C2212yes—
41#1380C2212yes—
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45C22×Q16C2311yescharacteristic
46C22×SD16—21no—
47C22×D8C2311yescharacteristic
48C23×C8D411yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—21no—
51C2×#124—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—41no—
54C2×#121—21no—
55C22×C42D411yescharacteristic
56C2×M32—24no—
57C2×M32—24no—
58C2×M32—24no—
59C22×C16C2312yes—
60C22×C16C2312yes—
61#500—22no—
62#500—22no—
63#495—22no—
64#495—22no—
65#467—24no—
66#466—24no—
67#465—44no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107C2312yes—
71C2×#107C2312yes—
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#106—21no—
75C2×#103—21no—
76C2×#103—21no—
77C2×#102—21no—
78C2×#102—21no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—21no—
86#391—41no—
87#390—21no—
88#389—22no—
89#389—41no—
90#388—41no—
91#388—41no—
92#386—41no—
93#386—41no—
94#384—21no—
95#382—21no—
96#380—22no—
97#380—41no—
98#358—24no—
99#358—24no—
100#357—24no—
101#357—24no—
102#346—22no—
103#346D414yescomplemented
104C4×C16—24no—
105#340—22no—
106#340—22no—
107C22×Q8—21no—
108C22×Q8—41no—
109C22×D4—21no—
110C22×D4—41no—
111C23×C4D811yescharacteristic
112C23×C4D811yescharacteristic
113C23×C4C2×D411yescharacteristic
114C2×Q16—21no—
115C2×Q16—22no—
116C2×Q16—22no—
117C2×Q16—21no—
118C2×SD16—42no—
119C2×SD16—42no—
120C2×SD16—42no—
121C2×D8—21no—
122C2×D8—22no—
123C2×D8—22no—
124C2×D8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8SD1614yes—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8#4114yes—
135C22×C8C2411yescharacteristic
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138#124—42no—
139#124—42no—
140C22⋊Q8—42no—
141C22⋊Q8—42no—
142C22⋊Q8—84no—
143#121—42no—
144#121—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42D811yescharacteristic
164C2×C42D811yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167M32—216no—
168C2×C16—22no—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16C2×D414yes—
173C2×C16#4114yes—
174#107—22no—
175#107—22no—
176#107—22no—
177#107—22no—
178#107—22no—
179#107—22no—
180#107—42no—
181#107—42no—
182#106—42no—
183#103—42no—
184#103—42no—
185#102—42no—
186#102—42no—
187C4×C8—22no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—22no—
191C4×C8C2×D412yes—
192C4×C8#4112yes—
193#95—42no—
194#95—42no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199C24—41no—
200C24C2×D811yescharacteristic
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—82no—
217C2×D4—82no—
218C2×D4—82no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#12412yes—
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×SD1612yes—
241C22×C4C22×D411yescharacteristic
242Q16—42no—
243Q16—42no—
244SD16—84no—
245D8—42no—
246D8—42no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—28no—
254C2×C8#12112yes—
255C2×C8#12412yes—
256C2×C8C2×SD1612yes—
257C2×C8C2×SD1612yes—
258C2×C8#13512yes—
259C2×C8C22×D411yescharacteristic
260C2×C8C2×#4112yes—
261C2×C8C2×#4112yes—
262C2×C8C2×#4111yescharacteristic
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C22⋊C4—84no—
277C22⋊C4—82no—
278C22⋊C4—82no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C22⋊C4—82no—
283C42—21no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42#12112yes—
288C42#13512yes—
289C16—24no—
290C16—24no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23#48612yes—
305C23C22×D811yescharacteristic
306Q8—81no—
307Q8—81no—
308Q8—81no—
309Q8—81no—
310Q8—164no—
311D4—81no—
312D4—81no—
313D4—81no—
314D4—81no—
315D4—164no—
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—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4#46612yes—
348C2×C4#46612yes—
349C2×C4#48612yes—
350C2×C4#50912yes—
351C2×C4#51012yes—
352C2×C4C2×#12111yescharacteristic
353C2×C4C2×#12411yescharacteristic
354C2×C4#53211yescharacteristic
355C2×C4C22×SD1611yescharacteristic
356C2×C4C2×#13511yescharacteristic
357C2×C4C2×#13511yescharacteristic
358C8—22no—
359C8—22no—
360C8#46514yes—
361C8#52512yes—
362C8#52912yes—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22C2×#46611yescharacteristic
378C22C2×#48611yescharacteristic
379C22#316911yescharacteristic
380C22C2×#50911yescharacteristic
381C22C2×#51011yescharacteristic
382C22#341411yescharacteristic
383C22#341411yescharacteristic
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—161no—
390C4—162no—
391C4—161no—
392C4—161no—
393C4—162no—
394C4#285011yescharacteristic
395C4C2×#46511yescharacteristic
396C4#313111yescharacteristic
397C4#316711yescharacteristic
398C2—42no—
399C2—321no—
400C2#3304211yescharacteristic
401C2#3360511yescharacteristic
402C2#3360611yescharacteristic
403C2#3370611yescharacteristic
404C2#3370711yescharacteristic
405C2#3373711yescharacteristic
406C2#3373811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2243C2semidirect
2. C2×C4×M32C2semidirect
3. #33209C2semidirect
5. #13283C2semidirect
6. #13205C2semidirect
7. #13148C2semidirect
8. #13147C2semidirect
9. #13062C2semidirect
11. #12890C2semidirect
103. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
C2×C4×M322
#332092
#331732
#132832
#132052
#131482
#131472
#130622
#130612
#128902