Finite Group Databasebeta

← #16251

Subgroups — #16251

Order 256.

1123 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C22×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#395C211yesmaximal, characteristic, complemented
4C2×#394C212yesmaximal
5C2×#393C211yesmaximal, characteristic, complemented
6#1444C212yesmaximal, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1440C212yesmaximal, complemented
9#1439C212yesmaximal, complemented
10#1439C211yesmaximal, characteristic, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C2×#123—24no—
16C2×#123—24no—
17C2×#123—24no—
18C2×C22⋊Q8—24no—
19C2×C22⋊Q8—28no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2212yes—
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2212yes—
24C22×C4⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2212yes—
29C22×C22⋊C4C2212yes—
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2212yes—
32#395—24no—
33#394—28no—
34#393—24no—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C25C2311yescharacteristic
44C25C2311yescharacteristic
45C22×Q8—24no—
46C22×Q8—22no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—42no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—24no—
53C22×D4—22no—
54C22×D4—42no—
55C23×C4—24no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—42no—
59C23×C4D414yes—
60C23×C4C2312yes—
61C23×C4C2312yes—
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69#123—232no—
70C22⋊Q8—232no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4D414yes—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4D418yes—
105C2×C22⋊C4D414yes—
106#95—24no—
107#95—22no—
108#95—24no—
109#95—22no—
110#95—24no—
111#95—24no—
112#95—42no—
113#95—42no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—21no—
118C24—22no—
119C24—21no—
120C24—21no—
121C24—22no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—22no—
126C24—22no—
127C24—21no—
128C24—22no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—22no—
133C24C2×D412yes—
134C24C2×D412yes—
135C24C2×D414yes—
136C24C2411yescharacteristic
137C2×Q8—28no—
138C2×Q8—48no—
139C2×Q8—48no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×D4—28no—
143C2×D4—48no—
144C2×D4—48no—
145C2×D4—44no—
146C2×D4—44no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—28no—
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—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—28no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4C2×D412yes—
198C22×C4C2×D412yes—
199C22×C4C2×D414yes—
200C22×C4C2×D414yes—
201C22×C4C2×D412yes—
202C22×C4C2×D412yes—
203C22×C4C2×D412yes—
204C22×C4C2×D412yes—
205C22×C4C2×D412yes—
206C22×C4#4114yes—
207C4⋊C4—28no—
208C4⋊C4—216no—
209C4⋊C4—28no—
210C4⋊C4—28no—
211C4⋊C4—48no—
212C22⋊C4—28no—
213C22⋊C4—28no—
214C22⋊C4—216no—
215C22⋊C4—28no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C22⋊C4—48no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—24no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—24no—
235C23—24no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23#12014yes—
265C23C22×D411yescharacteristic
266C23C22×D412yes—
267C23C22×D411yescharacteristic
268C23C22×D411yescharacteristic
269C23C2×#4112yes—
270C23H(5,𝔽2)12yes—
271C23H(5,𝔽2)11yescharacteristic
272C232−1+411yescharacteristic
273C232−1+411yescharacteristic
274C232−1+412yes—
275C232−1+411yescharacteristic
276Q8—416no—
277D4—416no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
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—416no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
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—42no—
333C2×C4#12014yes—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—42no—
338C22—42no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—44no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—44no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—82no—
353C22—82no—
354C22—81no—
355C22—81no—
356C22C2×#12011yescharacteristic
357C22C2×#12011yescharacteristic
358C22#53211yescharacteristic
359C22#53411yescharacteristic
360C22#53611yescharacteristic
361C22#53612yes—
362C22#53612yes—
363C22#53611yescharacteristic
364C22#53611yescharacteristic
365C22#53611yescharacteristic
366C22#53711yescharacteristic
367C22#53711yescharacteristic
368C22#53711yescharacteristic
369C22#53912yes—
370C22#54212yes—
371C22D4212yes—
372C22D4212yes—
373C22#54612yes—
374C22#54712yes—
375C22#54712yes—
376C22#54812yes—
377C22#56212yes—
378C22#56312yes—
379C4—24no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4—42no—
387C4—82no—
388C2—44no—
389C2—82no—
390C2#240311yescharacteristic
391C2#243011yescharacteristic
392C2#243211yescharacteristic
393C2#261411yescharacteristic
394C2#261611yescharacteristic
395C2#266311yescharacteristic
396C2#267212yes—
397C2#267412yes—
398C2#268012yes—
399C2#268112yes—
400C2#281411yescharacteristic
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. C22×C22⋊Q8C2semidirect
3. C2×#395C2semidirect
5. C2×#393C2semidirect
6. #1444C2semidirect
7. #1441C2semidirect
8. #1440C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C22×C22⋊Q82
C2×#3952
C2×#3942
C2×#3932
#14442
#14412
#14402
#14392
C22⋊#952