Finite Group Databasebeta

← #14299

Subgroups — #14299

Order 256.

1283 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic, complemented
2C2×#392C211yesmaximal, characteristic
3C2×#388C211yesmaximal, characteristic, complemented
4C2×#386C212yesmaximal, complemented
5#1442C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1439C211yesmaximal, characteristic, complemented
8#1438C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic, complemented
10#1434C211yesmaximal, characteristic, complemented
11C22⋊#95C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C23×D4C2212yes—
14C24×C4C2212yes—
15C2×C4×D4—28no—
16C2×C4×D4—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C42C2211yescharacteristic
28#392C418yescomplemented
29#388—24no—
30#386—28no—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2311yescharacteristic
41C25C2312yes—
42C22×D4—24no—
43C22×D4—24no—
44C22×D4—24no—
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—42no—
49C22×D4—42no—
50C22×D4—42no—
51C22×D4—42no—
52C22×D4C2×C414yes—
53C22×D4C2×C418yescomplemented
54C23×C4—24no—
55C23×C4—24no—
56C23×C4—42no—
57C23×C4—42no—
58C23×C4—42no—
59C23×C4—42no—
60C23×C4D418yescomplemented
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2312yes—
64C23×C4C2311yescharacteristic
65C23×C4C2312yes—
66C23×C4C2311yescharacteristic
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C4×D4—232no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—42no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4C2×C414yescomplemented
96C2×C22⋊C4C2×C418yescomplemented
97C2×C42—24no—
98C2×C42D414yescomplemented
99#95—24no—
100#95—24no—
101#95—22no—
102#95—22no—
103#95—22no—
104#95—42no—
105#95—42no—
106#95—42no—
107#95C2×C414yescomplemented
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—21no—
121C24—22no—
122C24—21no—
123C24—22no—
124C24—21no—
125C24—22no—
126C24—22no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24C22×C412yes—
131C24C22×C414yes—
132C24C2×D414yes—
133C24#4114yes—
134C24C2411yescharacteristic
135C2×D4—28no—
136C2×D4—28no—
137C2×D4—216no—
138C2×D4—216no—
139C2×D4—216no—
140C2×D4—28no—
141C2×D4—48no—
142C2×D4—48no—
143C2×D4—48no—
144C2×D4—48no—
145C2×D4—48no—
146C2×D4—48no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—216no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4C22×C414yes—
197C22×C4C22×C412yes—
198C22×C4C22×C412yes—
199C22×C4C2×D414yes—
200C22×C4C2×D412yes—
201C22×C4C2×D412yes—
202C22×C4C2×D414yes—
203C22×C4C2×D412yes—
204C22×C4#4112yes—
205C4⋊C4—28no—
206C4⋊C4—44no—
207C22⋊C4—216no—
208C22⋊C4—28no—
209C22⋊C4—48no—
210C22⋊C4—48no—
211C22⋊C4—44no—
212C22⋊C4—48no—
213C22⋊C4—48no—
214C22⋊C4—44no—
215C42—28no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—22no—
235C23—24no—
236C23—22no—
237C23—24no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—41no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—41no—
271C23—42no—
272C23—41no—
273C23C4×D418yes—
274C23C23×C411yescharacteristic
275C23C22×D412yes—
276C23C22×D411yescharacteristic
277C23C2×#4112yes—
278C23C2×#4111yescharacteristic
279C23H(5,𝔽2)11yescharacteristic
280C23H(5,𝔽2)12yes—
281C23H(5,𝔽2)11yescharacteristic
282C23H(5,𝔽2)12yes—
283C23H(5,𝔽2)11yescharacteristic
284C232−1+411yescharacteristic
285D4—232no—
286D4—432no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—432no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4C4×D414yes—
338C2×C4#12014yes—
339C22—28no—
340C22—28no—
341C22—28no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—44no—
357C22—42no—
358C22—42no—
359C22—81no—
360C22—81no—
361C22—82no—
362C22—81no—
363C22—82no—
364C22—81no—
365C22C2×C4×D412yes—
366C22C2×C4×D411yescharacteristic
367C22#51811yescharacteristic
368C22#51812yes—
369C22#52011yescharacteristic
370C22C2×#12011yescharacteristic
371C22#52512yes—
372C22#52511yescharacteristic
373C22#53412yes—
374C22#53412yes—
375C22#53512yes—
376C22#53511yescharacteristic
377C22#53511yescharacteristic
378C22#53511yescharacteristic
379C22#53711yescharacteristic
380C22#53712yes—
381C22#53812yes—
382C22#53812yes—
383C22#53812yes—
384C22#53912yes—
385C22#54011yescharacteristic
386C22#54012yes—
387C22#54211yescharacteristic
388C4—24no—
389C4—24no—
390C4—44no—
391C4—42no—
392C4—44no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—82no—
397C2—48no—
398C2—82no—
399C2C4×#12011yescharacteristic
400C2#232112yes—
401C2#232411yescharacteristic
402C2#232512yes—
403C2#232811yescharacteristic
404C2#261311yescharacteristic
405C2#261511yescharacteristic
406C2#263012yes—
407C2#263111yescharacteristic
408C2#263512yes—
409C2#263611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC2×#392C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×D4C2semidirect
3. C2×#388C2semidirect
4. C2×#386C2semidirect
5. #1442C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
10. #1434C2semidirect
11. C22⋊#95C2semidirect
28. #392C4semidirect
53. C22×D4C2×C4semidirect
60. C23×C4D4semidirect
95. C2×C22⋊C4C2×C4semidirect
96. C2×C22⋊C4C2×C4semidirect
98. C2×C42D4semidirect
107. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#3922
C2×#3882
C2×#3862
#14422
#14392
#14382
#14342
C22⋊#952