Finite Group Databasebeta

← #14723

Subgroups — #14723

Order 256.

1279 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×C4×D4C211yesmaximal, characteristic
3C2×#395C211yesmaximal, characteristic, complemented
4C2×#393C211yesmaximal, characteristic, complemented
5C2×#385C212yesmaximal, complemented
6#1441C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1439C211yesmaximal, characteristic, complemented
9#1438C212yesmaximal
10#1436C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16C2×C22⋊Q8—24no—
17C2×C22⋊Q8—28no—
18C2×C4×D4—24no—
19C2×C4×D4—24no—
20C2×C4×D4—24no—
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2212yes—
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2212yes—
29C22×C22⋊C4C2212yes—
30C22×C22⋊C4C2211yescharacteristic
31C22×C42C2211yescharacteristic
32#395—24no—
33#393—24no—
34#385—28no—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
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—22no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—22no—
56C22×D4Q814yescomplemented
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4D414yes—
67C23×C4C2312yes—
68C23×C4C2312yes—
69C23×C4C2312yes—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2312yes—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C22⋊Q8—232no—
77C4×D4—232no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4D414yes—
88C2×C4⋊C4D418yescomplemented
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4D414yes—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—22no—
110#95—24no—
111#95—24no—
112#95—24no—
113#95—22no—
114#95—22no—
115#95—24no—
116#95—24no—
117#95—24no—
118C24—22no—
119C24—22no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—22no—
125C24—22no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—22no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24C2×D412yes—
135C24C2×D412yes—
136C24C2×Q812yes—
137C24C2×Q812yes—
138C24#4112yes—
139C24#4112yes—
140C24C2411yescharacteristic
141C2×Q8—28no—
142C2×Q8—48no—
143C2×Q8—48no—
144C2×Q8—44no—
145C2×Q8—44no—
146C2×D4—28no—
147C2×D4—28no—
148C2×D4—28no—
149C2×D4—28no—
150C2×D4—28no—
151C2×D4—28no—
152C2×D4—28no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—28no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
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—28no—
190C22×C4—28no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4C2×D412yes—
206C22×C4C2×D414yes—
207C22×C4C2×D414yes—
208C22×C4C2×D412yes—
209C22×C4C2×D412yes—
210C22×C4C2×D412yes—
211C22×C4C2×D414yes—
212C22×C4C2×D412yes—
213C22×C4C2×D412yes—
214C22×C4C2×D412yes—
215C22×C4C2×Q812yes—
216C22×C4#4114yes—
217C22×C4#4112yes—
218C4⋊C4—28no—
219C4⋊C4—28no—
220C4⋊C4—216no—
221C4⋊C4—48no—
222C22⋊C4—216no—
223C22⋊C4—28no—
224C22⋊C4—216no—
225C22⋊C4—28no—
226C22⋊C4—48no—
227C22⋊C4—48no—
228C42—28no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—24no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—24no—
253C23—24no—
254C23—24no—
255C23—41no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23C22⋊Q814yes—
274C23C22⋊Q814yescomplemented
275C23C22×D411yescharacteristic
276C23C22×D411yescharacteristic
277C23C22×D411yescharacteristic
278C23C22×D412yes—
279C23C22×Q811yescharacteristic
280C23C2×#4111yescharacteristic
281C23C2×#4111yescharacteristic
282C23C2×#4111yescharacteristic
283C23C2×#4112yes—
284C23H(5,𝔽2)12yes—
285C232−1+412yes—
286Q8—416no—
287D4—232no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—416no—
316C2×C4—416no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4#12014yes—
343C2×C4#12014yes—
344C2×C4C22⋊Q814yescomplemented
345C22—24no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—28no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—44no—
367C22C2×#12011yescharacteristic
368C22C2×#12011yescharacteristic
369C22C2×C22⋊Q811yescharacteristic
370C22C2×C22⋊Q811yescharacteristic
371C22C2×C22⋊Q811yescharacteristic
372C22#52511yescharacteristic
373C22#52511yescharacteristic
374C22#52511yescharacteristic
375C22#52512yes—
376C22#53211yescharacteristic
377C22#53212yes—
378C22#53411yescharacteristic
379C22#53611yescharacteristic
380C22#53912yes—
381C22#54212yes—
382C22D4212yes—
383C22D4212yes—
384C22#54712yes—
385C22#54812yes—
386C22#54812yes—
387C22D4×Q812yes—
388C22#55112yes—
389C22#55412yes—
390C4—24no—
391C4—24no—
392C4—24no—
393C4—44no—
394C4—42no—
395C4—44no—
396C4—42no—
397C4—44no—
398C4—82no—
399C2—44no—
400C2—44no—
401C2#238811yescharacteristic
402C2#240311yescharacteristic
403C2#240911yescharacteristic
404C2#243211yescharacteristic
405C2#244912yes—
406C2#245212yes—
407C2#246112yes—
408C2#246312yes—
409C2#256411yescharacteristic
410C2#263811yescharacteristic
411C2#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
1. C22×C22⋊Q8C2semidirect
3. C2×#395C2semidirect
4. C2×#393C2semidirect
5. C2×#385C2semidirect
6. #1441C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
10. #1436C2semidirect
11. C22⋊#95C2semidirect
56. C22×D4Q8semidirect
88. C2×C4⋊C4D4semidirect
274. C23C22⋊Q8semidirect
344. C2×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C22×C4×D42
C2×#3952
C2×#3932
C2×#3852
#14412
#14392
#14382
#14362
C22⋊#952