Finite Group Databasebeta

← #4123498

Subgroups — #4123498

Order 512.

3562 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2449C216yesmaximal, complemented
2#17085C211yesmaximal, characteristic, complemented
3#15961C213yesmaximal, complemented
4#15778C213yesmaximal, complemented
5#15614C216yesmaximal, complemented
6#15429C213yesmaximal, complemented
7#14230C213yesmaximal, complemented
8#14175C213yesmaximal
9#14130C213yesmaximal, complemented
10C22×C22⋊Q8C2216yes—
11C22×C4×D4C2213yes—
12C23×C4⋊C4C2213yes—
13#2449—248no—
14C2×#399C2213yes—
15C2×#398C2216yes—
16C2×#396C2216yes—
17C2×#395C2216yes—
18C2×#392C2211yescharacteristic
19C2×#389C2216yes—
20C2×#389C2213yes—
21C2×#387C2216yes—
22C2×#386C2216yes—
23C2×#386C2216yes—
24C2×#385C2216yes—
25C2×#385C2216yes—
26C2×#385C2216yes—
27C2×#380C2213yes—
28C2×C4×C22⋊C4C2213yes—
29#1446C2212yes—
30#1445C2213yes—
31#1444C2211yescharacteristic
32#1444C2213yes—
33#1444C2213yes—
34#1442C2213yes—
35#1442C2213yes—
36#1442C2213yes—
37#1441C2216yes—
38#1441C2216yes—
39#1439C2216yes—
40#1439C2213yes—
41#1439C2213yes—
42#1438C2213yes—
43#1437C2216yes—
44C4⋊#95C2216yes—
45#1434C2213yes—
46C22⋊#95C2216yes—
47C4×#95C2211yescharacteristic
48C23×Q8C2312yes—
49C23×D4C2313yes—
50C24×C4C2313yes—
51C24×C4C2316yes—
52C2×C22⋊Q8—224no—
53C2×C22⋊Q8—224no—
54C2×C22⋊Q8—412no—
55C2×C4×D4—224no—
56C2×C4×D4—46no—
57C22×C4⋊C4—224no—
58C22×C4⋊C4—46no—
59C22×C4⋊C4C2316yes—
60C22×C4⋊C4C2313yes—
61C22×C4⋊C4C2313yes—
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2316yes—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2316yes—
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2313yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C2316yes—
70C22×C22⋊C4C2316yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2313yes—
73C22×C22⋊C4C2313yes—
74C22×C42C2313yes—
75C22×C42C2313yes—
76#399D4124yescomplemented
77#398—224no—
78#396—224no—
79#395—224no—
80#392Q818yescomplemented
81#389—224no—
82#389—46no—
83#387—224no—
84#386—224no—
85#386—224no—
86#385—224no—
87#385—224no—
88#385—224no—
89#380—212no—
90C4×C22⋊C4—46no—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2313yes—
99C2×#95C2311yescharacteristic
100C2×#95C2313yes—
101C2×#95C2316yes—
102C2×#95C2313yes—
103C2×#95C2313yes—
104C2×#95C2313yes—
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107C25C2413yes—
108C22×Q8—212no—
109C22×Q8—46no—
110C22×Q8—42no—
111C22×D4—212no—
112C22×D4—212no—
113C22×D4—43no—
114C22×D4—43no—
115C22×D4C2×Q8112yes—
116C23×C4—26no—
117C23×C4—26no—
118C23×C4—26no—
119C23×C4—212no—
120C23×C4—212no—
121C23×C4—212no—
122C23×C4—212no—
123C23×C4—46no—
124C23×C4—46no—
125C23×C4—46no—
126C23×C4—46no—
127C23×C4—46no—
128C23×C4C2416yes—
129C23×C4C2413yes—
130C23×C4C2416yes—
131C23×C4C2413yes—
132C23×C4C2411yescharacteristic
133C23×C4C2413yes—
134C23×C4C2413yes—
135C23×C4C2413yes—
136C22⋊Q8—496no—
137C4×D4—448no—
138C2×C4⋊C4—212no—
139C2×C4⋊C4—212no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—212no—
142C2×C4⋊C4—212no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4—212no—
145C2×C4⋊C4—212no—
146C2×C4⋊C4—212no—
147C2×C4⋊C4—212no—
148C2×C4⋊C4—212no—
149C2×C4⋊C4—212no—
150C2×C4⋊C4—448no—
151C2×C4⋊C4—43no—
152C2×C4⋊C4—43no—
153C2×C4⋊C4—43no—
154C2×C4⋊C4—46no—
155C2×C4⋊C4—43no—
156C2×C4⋊C4—46no—
157C2×C4⋊C4—43no—
158C2×C4⋊C4—46no—
159C2×C4⋊C4—43no—
160C2×C4⋊C4C2×D4124yes—
161C2×C4⋊C4C2×D4112yes—
162C2×C22⋊C4—212no—
163C2×C22⋊C4—212no—
164C2×C22⋊C4—212no—
165C2×C22⋊C4—212no—
166C2×C22⋊C4—212no—
167C2×C22⋊C4—212no—
168C2×C22⋊C4—212no—
169C2×C22⋊C4—212no—
170C2×C22⋊C4—43no—
171C2×C22⋊C4—43no—
172C2×C22⋊C4—46no—
173C2×C22⋊C4—46no—
174C2×C22⋊C4—46no—
175C2×C22⋊C4—43no—
176C2×C22⋊C4—43no—
177C2×C22⋊C4C2×D4124yes—
178C2×C22⋊C4C2×D4112yes—
179C2×C22⋊C4C2×Q8112yes—
180C2×C42—212no—
181C2×C42—26no—
182C2×C42—26no—
183C2×C42—46no—
184#95—212no—
185#95—212no—
186#95—212no—
187#95—212no—
188#95—212no—
189#95—212no—
190#95—212no—
191#95—26no—
192#95—41no—
193#95—43no—
194#95—43no—
195#95—43no—
196#95—46no—
197#95C2×D4112yes—
198#95C2×Q814yes—
199#95#4114yes—
200C24—26no—
201C24—23no—
202C24—26no—
203C24—23no—
204C24—26no—
205C24—23no—
206C24—23no—
207C24—26no—
208C24C22×D416yes—
209C24C22×Q816yes—
210C242−1+416yes—
211C24C2511yescharacteristic
212C2×Q8—424no—
213C2×Q8—812no—
214C2×Q8—84no—
215C2×D4—248no—
216C2×D4—412no—
217C2×D4—412no—
218C2×D4—424no—
219C2×D4—412no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—26no—
223C22×C4—224no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—23no—
232C22×C4—23no—
233C22×C4—23no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—23no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—23no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—424no—
256C22×C4—424no—
257C22×C4—424no—
258C22×C4—412no—
259C22×C4—424no—
260C22×C4—424no—
261C22×C4—412no—
262C22×C4—812no—
263C22×C4—812no—
264C22×C4—812no—
265C22×C4—812no—
266C22×C4C22×D416yes—
267C22×C4C22×D4112yes—
268C22×C4C22×D416yes—
269C22×C4C22×D4112yes—
270C22×C4C22×Q816yes—
271C22×C4C22×Q812yes—
272C22×C4C2×#4116yes—
273C22×C4H(5,𝔽2)16yes—
274C4⋊C4—248no—
275C4⋊C4—412no—
276C4⋊C4—424no—
277C4⋊C4—412no—
278C4⋊C4—412no—
279C4⋊C4—412no—
280C4⋊C4—424no—
281C4⋊C4—424no—
282C4⋊C4—86no—
283C22⋊C4—248no—
284C22⋊C4—412no—
285C22⋊C4—412no—
286C22⋊C4—424no—
287C22⋊C4—424no—
288C22⋊C4—424no—
289C42—412no—
290C42—86no—
291C23—212no—
292C23—212no—
293C23—26no—
294C23—26no—
295C23—212no—
296C23—212no—
297C23—26no—
298C23—26no—
299C23—26no—
300C23—212no—
301C23—212no—
302C23—26no—
303C23—43no—
304C23—46no—
305C23—46no—
306C23—46no—
307C23—46no—
308C23—43no—
309C23—46no—
310C23—43no—
311C23—43no—
312C23—46no—
313C23D4×Q8112yes—
314C23C23×D413yes—
315C23C23×Q811yescharacteristic
316C23C22×#4111yescharacteristic
317C23C2×H(5,𝔽2)13yes—
318C23C2×2−1+413yes—
319C23#58513yes—
320C23#58511yescharacteristic
321Q8—816no—
322D4—448no—
323C2×C4—212no—
324C2×C4—212no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—212no—
328C2×C4—212no—
329C2×C4—212no—
330C2×C4—212no—
331C2×C4—212no—
332C2×C4—212no—
333C2×C4—26no—
334C2×C4—26no—
335C2×C4—26no—
336C2×C4—212no—
337C2×C4—26no—
338C2×C4—26no—
339C2×C4—448no—
340C2×C4—43no—
341C2×C4—43no—
342C2×C4—43no—
343C2×C4—46no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—46no—
347C2×C4—46no—
348C2×C4—46no—
349C2×C4—43no—
350C2×C4—43no—
351C2×C4—43no—
352C2×C4—43no—
353C2×C4—43no—
354C2×C4—41no—
355C2×C4—43no—
356C2×C4—46no—
357C2×C4—46no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—46no—
364C2×C4—46no—
365C2×C4—848no—
366C2×C4D42112yes—
367C22—224no—
368C22—212no—
369C22—46no—
370C22—412no—
371C22—46no—
372C22—46no—
373C22—412no—
374C22—46no—
375C22—46no—
376C22—412no—
377C22—46no—
378C22C2×D4213yes—
379C22C2×D4×Q813yes—
380C22D4×#4113yes—
381C22Q8×#4111yescharacteristic
382C22#348916yes—
383C22#349213yes—
384C22#349613yes—
385C22#350113yes—
386C22#353313yes—
387C22#353813yes—
388C22#354911yescharacteristic
389C22#355313yes—
390C4—212no—
391C4—46no—
392C4—46no—
393C4—46no—
394C4—412no—
395C4—42no—
396C4—46no—
397C4—86no—
398C2—412no—
399C2#4683513yes—
400C2#4780411yescharacteristic
401C2#4781811yescharacteristic
402C2#5899313yes—
403C2#5899813yes—
404C2#5981611yescharacteristic
405C2#5983013yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2449C2semidirect
2. #17085C2semidirect
3. #15961C2semidirect
4. #15778C2semidirect
5. #15614C2semidirect
6. #15429C2semidirect
7. #14230C2semidirect
9. #14130C2semidirect
76. #399D4semidirect
80. #392Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24492
#170852
#159612
#157782
#156142
#154292
#142302
#141752
#141302