Finite Group Databasebeta

← #15070

Subgroups — #15070

Order 256.

1019 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic
2C2×#395C211yesmaximal, characteristic, complemented
3C2×#394C211yesmaximal, characteristic
4C2×#388C212yesmaximal, complemented
5C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
6#1446C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic
10#1437C212yesmaximal, complemented
11#1434C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C2×#124—28no—
15C2×#124—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C42C2211yescharacteristic
26C22×C42C2211yescharacteristic
27C22×C42C2211yescharacteristic
28#395—24no—
29#394—24no—
30#388—28no—
31C4×C4⋊C4—24no—
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2312yes—
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—42no—
47C22×D4—24no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—42no—
52C23×C4C2312yes—
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62#124—232no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4D414yes—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4D418yes—
90C2×C42—24no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—24no—
96C2×C42D414yes—
97#95—22no—
98#95—22no—
99#95—24no—
100#95—24no—
101#95—22no—
102#95—22no—
103#95—24no—
104#95—24no—
105#95—42no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24C2×D414yes—
121C24C2411yescharacteristic
122C2×Q8—28no—
123C2×Q8—48no—
124C2×Q8—48no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×D4—28no—
128C2×D4—48no—
129C2×D4—48no—
130C2×D4—44no—
131C2×D4—44no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
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×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4C2×D412yes—
177C22×C4C2×D414yes—
178C22×C4C2×D412yes—
179C22×C4C2×D414yes—
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4#4114yes—
185C22×C4#4112yes—
186C22×C4#4112yes—
187C22×C4#4112yes—
188C22×C4#4112yes—
189C22×C4#4112yes—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—48no—
193C22⋊C4—216no—
194C22⋊C4—216no—
195C22⋊C4—216no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C42—28no—
200C42—28no—
201C42—44no—
202C23—24no—
203C23—24no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
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—
237C23C22×D411yescharacteristic
238C23C22×D412yes—
239C23C22×D411yescharacteristic
240C23C2×#4111yescharacteristic
241C23C2×#4111yescharacteristic
242C23C2×#4111yescharacteristic
243C23C2×#4111yescharacteristic
244C23C2×#4111yescharacteristic
245C23C2×#4112yes—
246C23H(5,𝔽2)12yes—
247C232−1+412yes—
248Q8—416no—
249D4—416no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
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#12014yes—
306C2×C4#12114yes—
307C2×C4#12414yescomplemented
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22C2×#12011yescharacteristic
324C22C2×#12111yescharacteristic
325C22#52511yescharacteristic
326C22#52511yescharacteristic
327C22#52512yes—
328C22C2×#12411yescharacteristic
329C22#52911yescharacteristic
330C22#52911yescharacteristic
331C22#53211yescharacteristic
332C22#53211yescharacteristic
333C22#53212yes—
334C22#53511yescharacteristic
335C22#53611yescharacteristic
336C22#54212yes—
337C22#54212yes—
338C22D4212yes—
339C22#54612yes—
340C22#54712yes—
341C22#54712yes—
342C22#54812yes—
343C22#55112yes—
344C22#55512yes—
345C22#55612yes—
346C4—24no—
347C4—24no—
348C4—24no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—44no—
354C4—44no—
355C4—82no—
356C2—84no—
357C2#238811yescharacteristic
358C2#240611yescharacteristic
359C2#240811yescharacteristic
360C2#246112yes—
361C2#247412yes—
362C2#249112yes—
363C2#252112yes—
364C2#254511yescharacteristic
365C2#261011yescharacteristic
366C2#263211yescharacteristic
367C2#285011yescharacteristic
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×#395C2semidirect
4. C2×#388C2semidirect
5. C2×C4×C4⋊C4C2semidirect
6. #1446C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
10. #1437C2semidirect
11. #1434C2semidirect
307. C2×C4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3952
C2×#3942
C2×#3882
C2×C4×C4⋊C42
#14462
#14392
#14382
#14372
#14342