Finite Group Databasebeta

← #14567

Subgroups — #14567

Order 256.

1037 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1037 conjugacy classes of subgroups — too many to draw. Only its 219 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#392C211yesmaximal, characteristic
2C2×#390C212yesmaximal, complemented
3C2×#388C212yesmaximal, complemented
4#1440C211yesmaximal, characteristic, complemented
5#1438C212yesmaximal, complemented
6#1438C212yesmaximal, complemented
7#1437C211yesmaximal, characteristic, complemented
8#1437C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C23×D4C2212yes—
11C23×D4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C22⋊C4C2212yes—
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C42C2211yescharacteristic
23C22×C42C2212yes—
24#392C418yescomplemented
25#390—28no—
26#388—28no—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C25C2312yes—
35C25C2311yescharacteristic
36C22×D4—24no—
37C22×D4—24no—
38C22×D4—42no—
39C22×D4—42no—
40C22×D4—42no—
41C22×D4—42no—
42C22×D4—42no—
43C22×D4—42no—
44C22×D4—41no—
45C22×D4—41no—
46C22×D4C2×C418yes—
47C22×D4C2×C414yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2312yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4D418yescomplemented
58C2×C22⋊C4—24no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—41no—
70C2×C22⋊C4—41no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4C2×C418yes—
82C2×C22⋊C4C2×C414yes—
83C2×C42—24no—
84C2×C42—24no—
85C2×C42—41no—
86C2×C42—42no—
87C2×C42—42no—
88#95—24no—
89#95—24no—
90#95—24no—
91#95—42no—
92#95—41no—
93#95—42no—
94#95C2×C414yes—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24—21no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24C22×C414yes—
118C24C22×C412yes—
119C24C2411yescharacteristic
120C2×D4—216no—
121C2×D4—48no—
122C2×D4—48no—
123C2×D4—44no—
124C2×D4—48no—
125C2×D4—48no—
126C2×D4—48no—
127C2×D4—48no—
128C2×D4—48no—
129C2×D4—48no—
130C2×D4—44no—
131C2×D4—44no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4C22×C412yes—
170C22×C4C22×C412yes—
171C22×C4C22×C414yes—
172C22×C4C2×D414yes—
173C22×C4C2×D414yes—
174C22×C4C2×D414yes—
175C22×C4#4114yes—
176C22×C4#4114yes—
177C4⋊C4—48no—
178C22⋊C4—216no—
179C22⋊C4—44no—
180C22⋊C4—48no—
181C22⋊C4—44no—
182C22⋊C4—48no—
183C22⋊C4—48no—
184C22⋊C4—48no—
185C22⋊C4—48no—
186C22⋊C4—44no—
187C42—44no—
188C42—48no—
189C23—24no—
190C23—24no—
191C23—24no—
192C23—22no—
193C23—24no—
194C23—22no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—41no—
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—41no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23C23×C411yescharacteristic
246C23C22×D412yes—
247C23C2×#4112yes—
248C23C2×#4112yes—
249C23H(5,𝔽2)12yes—
250C23H(5,𝔽2)12yes—
251C23H(5,𝔽2)12yes—
252C23H(5,𝔽2)11yescharacteristic
253C23H(5,𝔽2)11yescharacteristic
254D4—432no—
255D4—416no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4C4×D418yes—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—42no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22—42no—
311C22—44no—
312C22—42no—
313C22—82no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22C2×C4×D412yes—
326C22C4×#4111yescharacteristic
327C22#51811yescharacteristic
328C22#51812yes—
329C22#51811yescharacteristic
330C22#53412yes—
331C22#53512yes—
332C22#53812yes—
333C22#53812yes—
334C22#53812yes—
335C22#54012yes—
336C22D4211yescharacteristic
337C22#54612yes—
338C22#55012yes—
339C22#55012yes—
340C22#55312yes—
341C22#55512yes—
342C22#55911yescharacteristic
343C22#56012yes—
344C22#56011yescharacteristic
345C22H(3,𝔽4)11yescharacteristic
346C4—28no—
347C4—42no—
348C4—42no—
349C4—44no—
350C4—44no—
351C4—44no—
352C4—82no—
353C2—82no—
354C2—84no—
355C2#232112yes—
356C2#233311yescharacteristic
357C2#236312yes—
358C2#236611yescharacteristic
359C2#237511yescharacteristic
360C2#267312yes—
361C2#271412yes—
362C2#273312yes—
363C2#280312yes—
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×#390C2semidirect
3. C2×#388C2semidirect
4. #1440C2semidirect
5. #1438C2semidirect
6. #1438C2semidirect
7. #1437C2semidirect
8. #1437C2semidirect
9. #1436C2semidirect
24. #392C4semidirect
57. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3922
C2×#3902
C2×#3882
#14402
#14382
#14372
#14362