Finite Group Databasebeta

← #314607

Subgroups — #314607

Order 512.

623 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21210C211yesmaximal, characteristic
2#20782C211yesmaximal, characteristic, complemented
3#10665C212yesmaximal, complemented
4#10629C212yesmaximal, complemented
5#9578C212yesmaximal, complemented
6#9577C212yesmaximal, complemented
7#8862C211yesmaximal, characteristic, complemented
8#8861C211yesmaximal, characteristic, complemented
9#8731C211yesmaximal, characteristic, complemented
10#8409C211yesmaximal, characteristic
11#8409C211yesmaximal, characteristic
12#3449C2211yescharacteristic
13#2905C2211yescharacteristic
14C2×#421—21no—
15C22×#104—21no—
16C4×#117C2211yescharacteristic
17#1963—21no—
18#1963C2212yes—
19#1962—21no—
20#1962C2212yes—
21#1840—21no—
22#1832C2211yescharacteristic, complemented
23#1832C2211yescharacteristic, complemented
24#1818—21no—
25#1798C2211yescharacteristic
26#1798C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic, complemented
28C4×C4⋊C8C2211yescharacteristic, complemented
29C4×C22⋊C8—21no—
30#1744—21no—
31#1571C2212yescomplemented
32#1570C2212yescomplemented
33#1540C2212yes—
34#1540C2212yes—
35#1480C2212yes—
36#1480C2212yes—
37#1458C2211yescharacteristic
38#1458C2211yescharacteristic
39#1301C2212yescomplemented
40#1301C2212yescomplemented
41#1276C2212yescomplemented
42#1274C2212yescomplemented
43C22×#41—21no—
44C22×M16—21no—
45#537—21no—
46#537—21no—
47#536—22no—
48#535—22no—
49#534—21no—
50#534—21no—
51#532—21no—
52#532C2312yes—
53#525—42no—
54C2×#117D411yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C42D411yescharacteristic
57#431C2312yes—
58#431C2312yes—
59#424—42no—
60#423—21no—
61#423C2312yes—
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66#421—44no—
67C2×#104—22no—
68C2×#104—22no—
69C2×#104—22no—
70C2×#104—22no—
71C2×#104—44no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80#376D411yescharacteristic
81#376C2311yescharacteristic
82C43D411yescharacteristic
83C43C2311yescharacteristic
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#335—42no—
93#334—42no—
94#332—22no—
95#332—22no—
96#331—22no—
97#331—22no—
98D4⋊C8—42no—
99D4⋊C8—42no—
100#324—21no—
101#324—21no—
102#323—21no—
103#323—21no—
104C2×#41—22no—
105C2×#41—22no—
106C2×#41—41no—
107C2×#41—42no—
108C2×#41—42no—
109C2×#41—41no—
110C2×#41—42no—
111C2×#41—42no—
112C22×Q8—21no—
113C22×D4—21no—
114C22×D4—41no—
115C23×C4—21no—
116C23×C4—41no—
117C23×C4C2×D411yescharacteristic
118C2×M16—22no—
119C2×M16—41no—
120C2×M16—42no—
121C2×M16—42no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—41no—
127C4⋊Q8—21no—
128C4⋊Q8#4112yes—
129#127—21no—
130#127#4112yes—
131#124—22no—
132#124—22no—
133#123—42no—
134#123—42no—
135C22⋊Q8—42no—
136C22⋊Q8—42no—
137#121—22no—
138#121—22no—
139#121—42no—
140#121—42no—
141#121—41no—
142#121—41no—
143#120—42no—
144#120—42no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—41no—
148C4×D4—41no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4#4111yescharacteristic
155C2×C4⋊C4#4111yescharacteristic
156C2×C22⋊C4—21no—
157C2×C22⋊C4#4111yescharacteristic
158C2×C22⋊C4#4111yescharacteristic
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—41no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2411yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—42no—
181#104—44no—
182#104—44no—
183#104—44no—
184#104—44no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—42no—
192C4×C8—42no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197C24—41no—
198C24#12311yescharacteristic
199#41—44no—
200#41—44no—
201#41—42no—
202#41—42no—
203#41—42no—
204#41—42no—
205#41—82no—
206#41—82no—
207#41—82no—
208#41—82no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—82no—
224C2×D4—82no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#13512yes—
245C22×C4AGL(1,ℤ/8)12yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249M16—42no—
250M16—42no—
251M16—82no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—82no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42AGL(1,ℤ/8)12yes—
297C42C2×#4112yes—
298C42H(5,𝔽2)12yes—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—82no—
305C23—161no—
306C23—161no—
307C23#48212yes—
308C23C2×#12311yescharacteristic
309Q8—42no—
310Q8—42no—
311Q8—82no—
312D4—42no—
313D4—42no—
314D4—82no—
315D4—82no—
316D4—82no—
317D4—84no—
318D4—82no—
319D4—82no—
320D4—161no—
321D4—161no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#48212yes—
353C2×C4#52511yescharacteristic
354C2×C4#53511yescharacteristic
355C2×C4#55112yes—
356C2×C4#55112yes—
357C2×C4C2×#13511yescharacteristic
358C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
359C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
360C2×C4#57511yescharacteristic
361C8—82no—
362C8—82no—
363C8—81no—
364C8—81no—
365C8—81no—
366C8—81no—
367C8—82no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—164no—
379C22#248211yescharacteristic
380C22C2×#48211yescharacteristic
381C22#309411yescharacteristic
382C4—24no—
383C4—22no—
384C4—22no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—82no—
392C4#310511yescharacteristic
393C4#311411yescharacteristic
394C4#321111yescharacteristic
395C4#321711yescharacteristic
396C2—21no—
397C2—21no—
398C2—42no—
399C2—162no—
400C2#2758811yescharacteristic
401C2#2759811yescharacteristic
402C2#2759811yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2482
prime radical#3449C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20782C2semidirect
3. #10665C2semidirect
4. #10629C2semidirect
5. #9578C2semidirect
6. #9577C2semidirect
7. #8862C2semidirect
8. #8861C2semidirect
9. #8731C2semidirect
22. #1832C22semidirect
23. #1832C22semidirect
27. C4×C4⋊C8C22semidirect
28. C4×C4⋊C8C22semidirect
31. #1571C22semidirect
32. #1570C22semidirect
39. #1301C22semidirect
40. #1301C22semidirect
41. #1276C22semidirect
42. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#212102
#207822
#106652
#106292
#95782
#95772
#88622
#88612
#87312
#84092