Finite Group Databasebeta

← #14621

Subgroups — #14621

Order 256.

1009 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#392C211yesmaximal, characteristic
2C2×#386C212yesmaximal, complemented
3#1443C212yesmaximal, complemented
4#1440C212yesmaximal, complemented
5#1439C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1438C212yesmaximal, complemented
8#1438C212yesmaximal, complemented
9C22⋊#95C211yesmaximal, characteristic, complemented
10C23×D4C2212yes—
11C23×D4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4C2211yescharacteristic
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22#392C418yescomplemented
23#386—28no—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C25C2312yes—
34C25C2311yescharacteristic
35C22×D4—24no—
36C22×D4—24no—
37C22×D4—42no—
38C22×D4—42no—
39C22×D4—42no—
40C22×D4—42no—
41C22×D4—42no—
42C22×D4—42no—
43C22×D4—41no—
44C22×D4—41no—
45C22×D4C2×C418yes—
46C22×D4C2×C414yescomplemented
47C23×C4—42no—
48C23×C4—42no—
49C23×C4—41no—
50C23×C4—41no—
51C23×C4D414yescomplemented
52C23×C4C2312yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C23×C4C2311yescharacteristic
58C23×C4C2312yes—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—42no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—41no—
66C2×C22⋊C4—41no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4C2×C414yes—
84C2×C22⋊C4C2×C418yescomplemented
85#95—24no—
86#95—24no—
87#95—42no—
88#95—41no—
89#95—42no—
90#95—42no—
91#95—42no—
92#95—42no—
93#95C2×C414yescomplemented
94C24—21no—
95C24—21no—
96C24—21no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—21no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24C22×C414yes—
117C24C22×C412yes—
118C24C2×D412yes—
119C24#4112yes—
120C24C2411yescharacteristic
121C2×D4—216no—
122C2×D4—48no—
123C2×D4—48no—
124C2×D4—44no—
125C2×D4—48no—
126C2×D4—48no—
127C2×D4—48no—
128C2×D4—48no—
129C2×D4—48no—
130C2×D4—48no—
131C2×D4—44no—
132C2×D4—44no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
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—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—44no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4C22×C412yes—
180C22×C4C22×C414yes—
181C22×C4C22×C412yes—
182C22×C4C2×D414yes—
183C4⋊C4—44no—
184C22⋊C4—44no—
185C22⋊C4—48no—
186C22⋊C4—48no—
187C22⋊C4—48no—
188C22⋊C4—48no—
189C22⋊C4—44no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C23—22no—
193C23—24no—
194C23—24no—
195C23—22no—
196C23—24no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—42no—
212C23—41no—
213C23—41no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C4×D414yes—
250C23C23×C411yescharacteristic
251C23C22×D411yescharacteristic
252C23C2×#4111yescharacteristic
253C23H(5,𝔽2)12yes—
254C23H(5,𝔽2)12yes—
255C23H(5,𝔽2)11yescharacteristic
256C23H(5,𝔽2)12yes—
257C23H(5,𝔽2)12yes—
258C23H(5,𝔽2)11yescharacteristic
259C23H(5,𝔽2)11yescharacteristic
260C232−1+411yescharacteristic
261D4—432no—
262D4—416no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—416no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C22—28no—
310C22—24no—
311C22—42no—
312C22—44no—
313C22—42no—
314C22—42no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—44no—
319C22—42no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22C2×C4×D411yescharacteristic
336C22#51812yes—
337C22#51812yes—
338C22#51811yescharacteristic
339C22#52011yescharacteristic
340C22#53412yes—
341C22#53411yescharacteristic
342C22#53512yes—
343C22#53711yescharacteristic
344C22#53812yes—
345C22#53811yescharacteristic
346C22#53812yes—
347C22#53911yescharacteristic
348C22#56011yescharacteristic
349C22#56012yes—
350C22#56012yes—
351C22#56012yes—
352C22#56012yes—
353C22#56011yescharacteristic
354C22#56011yescharacteristic
355C22H(3,𝔽4)11yescharacteristic
356C22#56212yes—
357C22#56212yes—
358C4—44no—
359C4—42no—
360C4—42no—
361C4—44no—
362C4—82no—
363C4—82no—
364C4—82no—
365C2—44no—
366C2—84no—
367C2#232112yes—
368C2#232511yescharacteristic
369C2#237511yescharacteristic
370C2#237712yes—
371C2#238011yescharacteristic
372C2#281211yescharacteristic
373C2#281312yes—
374C2#281511yescharacteristic
375C2#282411yescharacteristic
376C2#282512yes—
377C2#282611yescharacteristic
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
2. C2×#386C2semidirect
3. #1443C2semidirect
4. #1440C2semidirect
5. #1439C2semidirect
6. #1439C2semidirect
7. #1438C2semidirect
8. #1438C2semidirect
9. C22⋊#95C2semidirect
22. #392C4semidirect
46. C22×D4C2×C4semidirect
51. C23×C4D4semidirect
84. C2×C22⋊C4C2×C4semidirect
93. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3922
C2×#3862
#14432
#14402
#14392
#14382
C22⋊#952