Finite Group Databasebeta

← #4464028

Subgroups — #4464028

Order 512.

2512 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2643C212yesmaximal, complemented
2#17230C211yesmaximal, characteristic, complemented
3#17213C212yesmaximal, complemented
4#17209C213yesmaximal, complemented
5#17135C211yesmaximal, characteristic, complemented
6#17124C216yesmaximal, complemented
7#17121C213yesmaximal, complemented
8#15014C213yesmaximal, complemented
9#14913C213yesmaximal
10#14729C216yesmaximal, complemented
11C2×#1444C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2213yes—
14#2643—216no—
15C2×#401C2212yes—
16C2×#400C2216yes—
17C2×#397C2216yes—
18C2×#396C2212yes—
19C2×#396C2216yes—
20C2×#396C2216yes—
21C2×#386C2216yes—
22C2×#386C2216yes—
23C2×#385C2216yes—
24C2×#380C2213yes—
25C22×#95C2213yes—
26C22×#95C2211yescharacteristic
27#1445C2213yes—
28#1445C2213yes—
29#1444C2213yes—
30#1444C2211yescharacteristic
31#1444C2211yescharacteristic
32#1444C2213yes—
33#1444C2213yes—
34#1444C2213yes—
35#1442C2213yes—
36#1442C2213yes—
37#1442C2216yes—
38#1442C2216yes—
39#1441C2216yes—
40#1441C2216yes—
41#1441C2216yes—
42#1441C2216yes—
43#1440C2216yes—
44#1440C2216yes—
45#1439C2216yes—
46#1439C2216yes—
47#1438C2213yes—
48#1438C2213yes—
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C23×D4C2312yes—
52C23×D4C2311yescharacteristic
53C24×C4C2313yes—
54C24×C4C2311yescharacteristic
55C24×C4C2313yes—
56C2×#121—46no—
57C22×C4⋊C4—412no—
58C22×C4⋊C4—46no—
59C22×C4⋊C4C2313yes—
60C22×C4⋊C4C2316yes—
61C22×C4⋊C4C2313yes—
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2316yes—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2313yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4C2316yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C2316yes—
70C22×C22⋊C4C2316yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2312yes—
73#401—28no—
74#400—224no—
75#397—224no—
76#396—28no—
77#396—412no—
78#396—412no—
79#386—224no—
80#386—412no—
81#385—224no—
82#380—46no—
83C2×#95—46no—
84C2×#95D418yescomplemented
85C2×#95C2313yes—
86C2×#95C2316yes—
87C2×#95C2316yes—
88C2×#95C2316yes—
89C2×#95C2316yes—
90C2×#95C2313yes—
91C2×#95C2316yes—
92C2×#95C2313yes—
93C2×#95C2313yes—
94C2×#95C2316yes—
95C2×#95C2313yes—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2313yes—
99C2×#95C2313yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2311yescharacteristic
103C2×#95C2313yes—
104C25C2412yes—
105C25C2411yescharacteristic
106C22×D4—24no—
107C22×D4—212no—
108C22×D4—212no—
109C22×D4—22no—
110C22×D4—43no—
111C22×D4—43no—
112C23×C4—26no—
113C23×C4—26no—
114C23×C4—212no—
115C23×C4—26no—
116C23×C4—26no—
117C23×C4—26no—
118C23×C4—22no—
119C23×C4—46no—
120C23×C4—46no—
121C23×C4—46no—
122C23×C4—43no—
123C23×C4—43no—
124C23×C4C2×D4112yes—
125C23×C4C2416yes—
126C23×C4C2416yes—
127C23×C4C2413yes—
128C23×C4C2411yescharacteristic
129C23×C4C2412yes—
130C23×C4C2413yes—
131C23×C4C2413yes—
132C23×C4C2413yes—
133C23×C4C2411yescharacteristic
134#121—416no—
135C2×C4⋊C4—212no—
136C2×C4⋊C4—212no—
137C2×C4⋊C4—212no—
138C2×C4⋊C4—212no—
139C2×C4⋊C4—26no—
140C2×C4⋊C4—43no—
141C2×C4⋊C4—46no—
142C2×C4⋊C4—46no—
143C2×C4⋊C4—43no—
144C2×C4⋊C4—46no—
145C2×C4⋊C4—43no—
146C2×C4⋊C4—448no—
147C2×C4⋊C4—43no—
148C2×C4⋊C4—46no—
149C2×C4⋊C4—46no—
150C2×C4⋊C4—43no—
151C2×C4⋊C4—43no—
152C2×C4⋊C4—46no—
153C2×C4⋊C4—46no—
154C2×C4⋊C4—46no—
155C2×C22⋊C4—212no—
156C2×C22⋊C4—212no—
157C2×C22⋊C4—212no—
158C2×C22⋊C4—212no—
159C2×C22⋊C4—212no—
160C2×C22⋊C4—212no—
161C2×C22⋊C4—46no—
162C2×C22⋊C4—46no—
163C2×C22⋊C4—46no—
164C2×C22⋊C4—46no—
165C2×C22⋊C4—46no—
166C2×C22⋊C4—46no—
167C2×C22⋊C4—46no—
168C2×C22⋊C4—46no—
169C2×C22⋊C4—46no—
170C2×C22⋊C4—46no—
171#95—212no—
172#95—212no—
173#95—212no—
174#95—212no—
175#95—26no—
176#95—26no—
177#95—43no—
178#95—43no—
179#95—43no—
180#95—43no—
181#95—43no—
182#95—43no—
183#95—46no—
184#95—46no—
185#95—46no—
186#95—46no—
187#95—46no—
188#95C2×D414yes—
189#95C2×D4112yes—
190C24—23no—
191C24—21no—
192C24—23no—
193C24—21no—
194C24—22no—
195C24—26no—
196C24—26no—
197C24—26no—
198C24—22no—
199C24—22no—
200C24—26no—
201C24C22×D412yes—
202C24H(5,𝔽2)16yes—
203C242−1+416yes—
204C24C2511yescharacteristic
205C2×D4—44no—
206C2×D4—424no—
207C2×D4—48no—
208C2×D4—424no—
209C2×D4—86no—
210C2×D4—86no—
211C22×C4—224no—
212C22×C4—23no—
213C22×C4—23no—
214C22×C4—23no—
215C22×C4—23no—
216C22×C4—26no—
217C22×C4—26no—
218C22×C4—26no—
219C22×C4—26no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—23no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—22no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—23no—
231C22×C4—26no—
232C22×C4—26no—
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—26no—
242C22×C4—26no—
243C22×C4—21no—
244C22×C4—26no—
245C22×C4—23no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—23no—
251C22×C4—424no—
252C22×C4—424no—
253C22×C4—412no—
254C22×C4—412no—
255C22×C4—412no—
256C22×C4—412no—
257C22×C4—424no—
258C22×C4—44no—
259C22×C4—412no—
260C22×C4—86no—
261C22×C4—86no—
262C22×C4—812no—
263C22×C4C22×D416yes—
264C22×C4C22×D416yes—
265C22×C4H(5,𝔽2)16yes—
266C22×C42−1+412yes—
267C4⋊C4—412no—
268C4⋊C4—424no—
269C4⋊C4—412no—
270C4⋊C4—44no—
271C4⋊C4—812no—
272C4⋊C4—812no—
273C4⋊C4—86no—
274C4⋊C4—86no—
275C22⋊C4—48no—
276C22⋊C4—424no—
277C22⋊C4—812no—
278C22⋊C4—812no—
279C22⋊C4—812no—
280C22⋊C4—812no—
281C23—26no—
282C23—22no—
283C23—26no—
284C23—26no—
285C23—26no—
286C23—26no—
287C23—212no—
288C23—26no—
289C23—26no—
290C23—46no—
291C23—46no—
292C23—42no—
293C23—42no—
294C23—46no—
295C23—46no—
296C23—46no—
297C23—46no—
298C23—46no—
299C23—412no—
300C23—46no—
301C23—46no—
302C23#537112yes—
303C23#56014yes—
304C23#563112yes—
305C23C23×D411yescharacteristic
306C23C2×H(5,𝔽2)13yes—
307C23C2×H(5,𝔽2)13yes—
308C23C2×2−1+413yes—
309C23C2×2−1+411yescharacteristic
310C23#58513yes—
311C23#58511yescharacteristic
312D4—88no—
313D4—816no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—212no—
317C2×C4—212no—
318C2×C4—212no—
319C2×C4—26no—
320C2×C4—26no—
321C2×C4—448no—
322C2×C4—41no—
323C2×C4—43no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—43no—
327C2×C4—43no—
328C2×C4—43no—
329C2×C4—43no—
330C2×C4—46no—
331C2×C4—46no—
332C2×C4—46no—
333C2×C4—46no—
334C2×C4—46no—
335C2×C4—46no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—46no—
339C2×C4—46no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—43no—
343C2×C4—46no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—43no—
347C2×C4—46no—
348C2×C4—43no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—43no—
352C2×C4—46no—
353C2×C4—43no—
354C2×C4—46no—
355C2×C4—46no—
356C2×C4—46no—
357C2×C4—46no—
358C2×C4—88no—
359C2×C4—824no—
360C22—24no—
361C22—212no—
362C22—212no—
363C22—24no—
364C22—212no—
365C22—212no—
366C22—82no—
367C22—86no—
368C22—82no—
369C22—86no—
370C22—82no—
371C22—86no—
372C22—86no—
373C22#282318yes—
374C22C2×#53713yes—
375C22#349613yes—
376C22#349711yescharacteristic
377C22C2×#56011yescharacteristic
378C22C2×#56313yes—
379C22#353213yes—
380C22#353313yes—
381C22#353413yes—
382C22#353513yes—
383C22#354116yes—
384C22#355213yes—
385C22#355313yes—
386C4—42no—
387C4—46no—
388C4—46no—
389C4—46no—
390C4—81no—
391C4—86no—
392C4—82no—
393C4—86no—
394C4—83no—
395C2—28no—
396C2—162no—
397C2C2×#282311yescharacteristic
398C2#3673813yes—
399C2#3713013yes—
400C2#5239811yescharacteristic
401C2#5241413yes—
402C2#5245313yes—
403C2#5274211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2643C2semidirect
2. #17230C2semidirect
3. #17213C2semidirect
4. #17209C2semidirect
5. #17135C2semidirect
6. #17124C2semidirect
7. #17121C2semidirect
8. #15014C2semidirect
10. #14729C2semidirect
11. C2×#1444C2semidirect
84. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26432
#172302
#172132
#172092
#171352
#171242
#171212
#150142
#149132
#147292
C2×#14442