Finite Group Databasebeta

← #15909

Subgroups — #15909

Order 256.

1075 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#126C211yesmaximal, characteristic, complemented
2C2×#394C211yesmaximal, characteristic, complemented
3C2×#392C211yesmaximal, characteristic
4C2×#390C211yesmaximal, characteristic, complemented
5#1445C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1438C212yesmaximal, complemented
8#1438C212yesmaximal, complemented
9#1437C212yesmaximal, complemented
10#1437C211yesmaximal, characteristic, complemented
11#1436C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C23×D4C2212yes—
14C2×#126—24no—
15C2×#126—24no—
16C2×#126—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2212yes—
29C22×C42C2211yescharacteristic
30C22×C42C2211yescharacteristic
31#394—24no—
32#392—24no—
33#390—24no—
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2312yes—
43C22×D4—24no—
44C22×D4—24no—
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—24no—
50C22×D4—42no—
51C22×D4—42no—
52C22×D4—42no—
53C22×D4—42no—
54C22×D4—42no—
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64#126—232no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D414yescomplemented
73C2×C4⋊C4D414yescomplemented
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4D414yes—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—42no—
106#95—22no—
107#95—22no—
108#95—24no—
109#95—42no—
110#95—42no—
111#95—42no—
112#95—42no—
113C24—21no—
114C24—21no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—22no—
129C24—22no—
130C24—22no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24C2×D412yes—
136C24#4114yes—
137C24C2411yescharacteristic
138C2×D4—28no—
139C2×D4—48no—
140C2×D4—48no—
141C2×D4—48no—
142C2×D4—48no—
143C2×D4—44no—
144C2×D4—48no—
145C2×D4—48no—
146C2×D4—48no—
147C2×D4—48no—
148C2×D4—48no—
149C2×D4—44no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
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×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4C2×D412yes—
192C22×C4C2×D412yes—
193C22×C4C2×D412yes—
194C22×C4C2×D412yes—
195C22×C4C2×D412yes—
196C22×C4C2×D412yes—
197C22×C4C2×D412yes—
198C22×C4C2×D412yes—
199C22×C4#4114yes—
200C22×C4#4114yes—
201C4⋊C4—28no—
202C4⋊C4—28no—
203C4⋊C4—28no—
204C22⋊C4—28no—
205C22⋊C4—28no—
206C22⋊C4—28no—
207C22⋊C4—48no—
208C22⋊C4—48no—
209C22⋊C4—48no—
210C22⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—48no—
213C42—28no—
214C42—44no—
215C23—22no—
216C23—24no—
217C23—24no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—42no—
228C23—41no—
229C23—42no—
230C23—42no—
231C23—41no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—41no—
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—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23C22×D411yescharacteristic
272C23C22×D411yescharacteristic
273C23C22×D411yescharacteristic
274C23C2×#4112yes—
275C23C2×#4112yes—
276C23C2×#4112yes—
277C23H(5,𝔽2)12yes—
278C23H(5,𝔽2)12yes—
279C23H(5,𝔽2)12yes—
280D4—416no—
281D4—432no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
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—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4#12014yes—
333C22—44no—
334C22—42no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22C2×#12011yescharacteristic
358C22#52511yescharacteristic
359C22#52912yes—
360C22#53211yescharacteristic
361C22#53211yescharacteristic
362C22#53511yescharacteristic
363C22#53511yescharacteristic
364C22#53511yescharacteristic
365C22#53812yes—
366C22#53812yes—
367C22#53812yes—
368C22#54612yes—
369C22#54612yes—
370C22#54612yes—
371C22#54612yes—
372C22#55012yes—
373C22#55012yes—
374C22#55112yes—
375C22#55312yes—
376C22#55312yes—
377C22#56012yes—
378C4—24no—
379C4—42no—
380C4—44no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—44no—
386C4—82no—
387C2—84no—
388C2—82no—
389C2#240411yescharacteristic
390C2#243111yescharacteristic
391C2#243111yescharacteristic
392C2#246712yes—
393C2#255412yes—
394C2#255912yes—
395C2#268512yes—
396C2#271412yes—
397C2#271412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC2×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#126C2semidirect
2. C2×#394C2semidirect
4. C2×#390C2semidirect
5. #1445C2semidirect
6. #1439C2semidirect
7. #1438C2semidirect
8. #1438C2semidirect
9. #1437C2semidirect
10. #1437C2semidirect
11. #1436C2semidirect
72. C2×C4⋊C4D4semidirect
73. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1262
C2×#3942
C2×#3922
C2×#3902
#14452
#14392
#14382
#14372
#14362