Finite Group Databasebeta

← #499609

Subgroups — #499609

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#499C211yesmaximal, characteristic, complemented
2#21275C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13515C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13474C211yesmaximal, characteristic
9#13453C211yesmaximal, characteristic, complemented
10#13365C211yesmaximal, characteristic
11C4×#367C211yesmaximal, characteristic, complemented
12C4×#366C211yesmaximal, characteristic, complemented
13C4×C4⋊C16C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C4×#125—21no—
19#2293—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×C4×C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#1991—21no—
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1941—21no—
40#1934—21no—
41#1934—21no—
42#1846C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#106—21no—
47C4×#102—21no—
48C42×C8C2211yescharacteristic
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—
62C2×#117—41no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#499—22no—
68#499—22no—
69C4×D8—22no—
70C4×D8Q814yes—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#106—21no—
79C2×#106—21no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#390—21no—
90#390—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#388—41no—
96#388—41no—
97#387—41no—
98#387—41no—
99#382—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—41no—
104C4×C4⋊C4—41no—
105C4×C22⋊C4—41no—
106C43D411yescharacteristic
107#367—22no—
108#367—22no—
109#366—22no—
110#366—22no—
111C4⋊C16—22no—
112C4⋊C16—22no—
113#357—22no—
114#357—22no—
115#357—22no—
116#357—22no—
117C4×C16—22no—
118C4×C16—22no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
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—24no—
139#125—42no—
140C4×D4—24no—
141C4×D4—42no—
142#117—84no—
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—41no—
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⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42D811yescharacteristic
165C2×C42D811yescharacteristic
166C2×C42D811yescharacteristic
167C2×C42D811yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
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×C16#4112yes—
182C2×C16#4112yes—
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#107C2×Q812yes—
195#107#4112yes—
196#107#4112yes—
197#106—24no—
198#106—42no—
199#102—24no—
200#102—42no—
201#102—42no—
202C4×C8—22no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—22no—
208C4×C8—41no—
209C4×C8C2×Q812yes—
210C4×C8#4112yes—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220C24—41no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4C22×D411yescharacteristic
244D8—22no—
245D8—22no—
246D8—41no—
247D8—41no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8C22⋊Q812yes—
256C2×C8#13512yes—
257C2×C8C22×Q811yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8C2×#4111yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C2×C8C2×#4111yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—44no—
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—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C22⋊C4—44no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—41no—
296C42—82no—
297C42—82no—
298C42—82no—
299C42C22⋊Q812yes—
300C42#13712yes—
301C16—22no—
302C16—22no—
303C16—41no—
304C16—41no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23C22×D811yescharacteristic
313D4—42no—
314D4—42no—
315D4—81no—
316D4—81no—
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—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—88no—
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—81no—
352C2×C4#47412yes—
353C2×C4#47412yes—
354C2×C4#50812yes—
355C2×C4#50812yes—
356C2×C4#50912yes—
357C2×C4C2×C22⋊Q811yescharacteristic
358C2×C4#52511yescharacteristic
359C2×C4#53211yescharacteristic
360C2×C4C2×#13511yescharacteristic
361C2×C4C2×#13711yescharacteristic
362C2×C4#57511yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8#52911yescharacteristic
367C8#52911yescharacteristic
368C8#53311yescharacteristic
369C8#53311yescharacteristic
370C22—82no—
371C22—82no—
372C22—82no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22C2×#47411yescharacteristic
378C22#310011yescharacteristic
379C22#316911yescharacteristic
380C22C2×#50811yescharacteristic
381C22C2×#50811yescharacteristic
382C22C2×#50911yescharacteristic
383C22#341411yescharacteristic
384C4—22no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—82no—
390C4—82no—
391C4—82no—
392C4—161no—
393C4—161no—
394C4—161no—
395C4—161no—
396C4#222814yes—
397C4#239611yescharacteristic
398C4#307611yescharacteristic
399C4#310811yescharacteristic
400C4#317211yescharacteristic
401C2—162no—
402C2#2377411yescharacteristic
403C2C2×#222811yescharacteristic
404C2#3362611yescharacteristic
405C2#3366211yescharacteristic
406C2#3366611yescharacteristic
407C2#3373511yescharacteristic
408C2#3373711yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutator, Frattini, cosocleC22×C8C24
socleC23C22×D8
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#499C2semidirect
3. #13539C2semidirect
5. #13516C2semidirect
6. #13515C2semidirect
9. #13453C2semidirect
11. C4×#367C2semidirect
12. C4×#366C2semidirect
13. C4×C4⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4992
#212752
#135392
#135282
#135162
#135152
#134762
#134742
#134532
#133652
C4×#3672
C4×#3662
C4×C4⋊C162
C4×#3572