Finite Group Databasebeta

← #17026

Subgroups — #17026

Order 256.

779 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#390C211yesmaximal, characteristic
2C2×#387C211yesmaximal, characteristic, complemented
3#1444C211yesmaximal, characteristic, complemented
4#1442C211yesmaximal, characteristic, complemented
5#1439C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic
10#1438C211yesmaximal, characteristic
11#1434C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C42C2211yescharacteristic
24#390—24no—
25#387—24no—
26C2×#95C2211yescharacteristic
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2211yescharacteristic
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C25C2312yes—
38C22×D4—22no—
39C22×D4—22no—
40C22×D4—42no—
41C22×D4—42no—
42C22×D4Q814yes—
43C23×C4C2311yescharacteristic
44C23×C4C2312yes—
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
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—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C42—42no—
82C2×C42D414yes—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—24no—
89#95—22no—
90#95—42no—
91#95—42no—
92#95—42no—
93#95—42no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24C2×Q814yes—
109C24C2411yescharacteristic
110C2×D4—28no—
111C2×D4—48no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—48no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C22×C4C2×D412yes—
166C22×C4C2×Q812yes—
167C22×C4#4114yes—
168C22×C4#4112yes—
169C22×C4#4112yes—
170C22×C4#4112yes—
171C4⋊C4—28no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—48no—
175C22⋊C4—48no—
176C22⋊C4—48no—
177C22⋊C4—48no—
178C22⋊C4—48no—
179C22⋊C4—48no—
180C22⋊C4—48no—
181C42—44no—
182C23—24no—
183C23—24no—
184C23—24no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—24no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
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—
217C23C22×D411yescharacteristic
218C23C22×Q811yescharacteristic
219C23C2×#4111yescharacteristic
220C23C2×#4111yescharacteristic
221C23C2×#4111yescharacteristic
222C23C2×#4112yes—
223C23H(5,𝔽2)12yes—
224C23H(5,𝔽2)11yescharacteristic
225C23H(5,𝔽2)12yes—
226C23H(5,𝔽2)12yes—
227C23H(5,𝔽2)11yescharacteristic
228D4—416no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4C22⋊Q814yes—
285C22—44no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—44no—
291C22—44no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22C2×C22⋊Q811yescharacteristic
301C22#52511yescharacteristic
302C22#53211yescharacteristic
303C22#53411yescharacteristic
304C22#53511yescharacteristic
305C22#53511yescharacteristic
306C22#53511yescharacteristic
307C22#53812yes—
308C22#53812yes—
309C22#53812yes—
310C22#54012yes—
311C22#54012yes—
312C22#54312yes—
313C22#55112yes—
314C22#55312yes—
315C22#55412yes—
316C22#55411yescharacteristic
317C22#55411yescharacteristic
318C22#55512yes—
319C22#55911yescharacteristic
320C22#55911yescharacteristic
321C22#56012yes—
322C22#56012yes—
323C4—24no—
324C4—42no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—44no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—82no—
333C2—84no—
334C2#256611yescharacteristic
335C2#258011yescharacteristic
336C2#261011yescharacteristic
337C2#262611yescharacteristic
338C2#263511yescharacteristic
339C2#266111yescharacteristic
340C2#274812yes—
341C2#278412yes—
342C2#279712yes—
343C2#280312yes—
344C2#281811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#387C2semidirect
3. #1444C2semidirect
4. #1442C2semidirect
5. #1439C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
11. #1434C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
C2×#3872
#14442
#14422
#14392
#14382
#14342