Finite Group Databasebeta

← #14825

Subgroups — #14825

Order 256.

1099 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic
2C2×#400C211yesmaximal, characteristic, complemented
3C2×#399C211yesmaximal, characteristic, complemented
4C2×#380C211yesmaximal, characteristic, complemented
5#1441C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1439C211yesmaximal, characteristic, complemented
8#1438C212yesmaximal
9#1436C212yesmaximal, complemented
10C22⋊#95C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C2×C4×D4—24no—
16C2×C4×D4—24no—
17C2×C4×D4—24no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C42C2211yescharacteristic
29#400—24no—
30#399—24no—
31#380—24no—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2311yescharacteristic
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4Q814yes—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—24no—
56C23×C4—24no—
57C23×C4—42no—
58C23×C4—42no—
59C23×C4D414yes—
60C23×C4C2312yes—
61C23×C4C2312yes—
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2312yes—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C4×D4—232no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4D414yes—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
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⋊C4D414yes—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102#95—24no—
103#95—22no—
104#95—22no—
105#95—24no—
106#95—24no—
107#95—24no—
108#95—24no—
109#95—42no—
110#95—42no—
111C24—21no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24C2×D412yes—
128C24C2×D412yes—
129C24C2×Q812yes—
130C24C2×Q812yes—
131C24#4112yes—
132C24#4112yes—
133C24C2411yescharacteristic
134C2×D4—28no—
135C2×D4—28no—
136C2×D4—28no—
137C2×D4—28no—
138C2×D4—28no—
139C2×D4—28no—
140C2×D4—28no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—28no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
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—28no—
166C22×C4—28no—
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—44no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4C2×D412yes—
196C22×C4C2×D412yes—
197C22×C4C2×D412yes—
198C22×C4C2×D412yes—
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C22×C4C2×D412yes—
202C22×C4C2×Q812yes—
203C22×C4#4114yes—
204C22×C4#4112yes—
205C22×C4#4114yes—
206C4⋊C4—28no—
207C4⋊C4—48no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C22⋊C4—28no—
211C22⋊C4—28no—
212C22⋊C4—48no—
213C22⋊C4—48no—
214C22⋊C4—48no—
215C22⋊C4—48no—
216C42—28no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23C22⋊Q814yes—
262C23C22⋊Q814yes—
263C23C22×D411yescharacteristic
264C23C22×D411yescharacteristic
265C23C22×D411yescharacteristic
266C23C22×Q811yescharacteristic
267C23C2×#4111yescharacteristic
268C23C2×#4111yescharacteristic
269C23C2×#4111yescharacteristic
270C23C2×#4112yes—
271C23C2×#4112yes—
272C23H(5,𝔽2)12yes—
273C23H(5,𝔽2)12yes—
274D4—232no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—416no—
297C2×C4—416no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
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#12014yes—
330C2×C4C22⋊Q814yes—
331C2×C4#12314yes—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—28no—
336C22—24no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22—42no—
354C22C2×#12011yescharacteristic
355C22C2×C22⋊Q811yescharacteristic
356C22C2×C22⋊Q811yescharacteristic
357C22C2×C22⋊Q811yescharacteristic
358C22C2×#12311yescharacteristic
359C22#52511yescharacteristic
360C22#52511yescharacteristic
361C22#52511yescharacteristic
362C22#52912yes—
363C22#52912yes—
364C22#53211yescharacteristic
365C22#53411yescharacteristic
366C22#53511yescharacteristic
367C22#53812yes—
368C22#54012yes—
369C22#54612yes—
370C22#54612yes—
371C22#54612yes—
372C22#55012yes—
373C22#55112yes—
374C22#55312yes—
375C22#55412yes—
376C22#55412yes—
377C4—24no—
378C4—24no—
379C4—24no—
380C4—42no—
381C4—42no—
382C4—44no—
383C4—82no—
384C4—82no—
385C4—82no—
386C2—44no—
387C2—44no—
388C2#239511yescharacteristic
389C2#240511yescharacteristic
390C2#243111yescharacteristic
391C2#245012yes—
392C2#245412yes—
393C2#245712yes—
394C2#246712yes—
395C2#248711yescharacteristic
396C2#256411yescharacteristic
397C2#263511yescharacteristic
398C2#284811yescharacteristic
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×#400C2semidirect
3. C2×#399C2semidirect
4. C2×#380C2semidirect
5. #1441C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
9. #1436C2semidirect
10. C22⋊#95C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#4002
C2×#3992
C2×#3802
#14412
#14392
#14382
#14362
C22⋊#952