Finite Group Databasebeta

← #14915

Subgroups — #14915

Order 256.

1235 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic, complemented
2C2×#393C211yesmaximal, characteristic, complemented
3C2×#392C211yesmaximal, characteristic
4C2×#386C212yesmaximal, complemented
5C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
6#1446C211yesmaximal, characteristic, complemented
7#1440C212yesmaximal, complemented
8#1438C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic, complemented
10#1437C212yesmaximal, complemented
11#1434C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C23×D4C2212yes—
15C24×C4C2211yescharacteristic
16C2×#124—28no—
17C2×#124—24no—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C42C2211yescharacteristic
27C22×C42C2211yescharacteristic
28#393—24no—
29#392—24no—
30#386—28no—
31C4×C22⋊C4—24no—
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2312yes—
42C25C2311yescharacteristic
43C22×Q8—24no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×D4—24no—
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—24no—
52C22×D4—24no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—24no—
56C22×D4—22no—
57C22×D4—42no—
58C22×D4—42no—
59C22×D4—42no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—24no—
64C23×C4—42no—
65C23×C4C2312yes—
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2312yes—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74#124—232no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
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⋊C4D414yes—
98C2×C22⋊C4D418yescomplemented
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—42no—
102C2×C42D414yescomplemented
103#95—22no—
104#95—22no—
105#95—24no—
106#95—24no—
107#95—22no—
108#95—24no—
109#95—22no—
110#95—24no—
111#95—42no—
112C24—21no—
113C24—22no—
114C24—21no—
115C24—21no—
116C24—21no—
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—
134C24C2×D414yes—
135C24C2×D412yes—
136C24#4112yes—
137C24#4112yes—
138C24C2411yescharacteristic
139C2×Q8—28no—
140C2×Q8—48no—
141C2×Q8—48no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×D4—28no—
145C2×D4—216no—
146C2×D4—48no—
147C2×D4—48no—
148C2×D4—48no—
149C2×D4—48no—
150C2×D4—44no—
151C2×D4—48no—
152C2×D4—48no—
153C2×D4—48no—
154C2×D4—48no—
155C2×D4—44no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—28no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—21no—
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×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4C2×D412yes—
202C22×C4C2×D412yes—
203C22×C4C2×D414yes—
204C22×C4C2×D414yes—
205C22×C4C2×D412yes—
206C22×C4C2×D412yes—
207C22×C4C2×D412yes—
208C22×C4#4114yes—
209C22×C4#4112yes—
210C22×C4#4112yes—
211C22×C4#4112yes—
212C22⋊C4—28no—
213C22⋊C4—216no—
214C22⋊C4—216no—
215C22⋊C4—28no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C42—28no—
221C42—44no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—41no—
245C23—41no—
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—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23#12414yescomplemented
280C23C22×D411yescharacteristic
281C23C22×D412yes—
282C23C22×D411yescharacteristic
283C23C2×#4111yescharacteristic
284C23C2×#4111yescharacteristic
285C23C2×#4111yescharacteristic
286C23C2×#4111yescharacteristic
287C23C2×#4111yescharacteristic
288C23C2×#4112yes—
289C23H(5,𝔽2)12yes—
290C23H(5,𝔽2)12yes—
291Q8—416no—
292D4—416no—
293D4—432no—
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—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—416no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
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—42no—
343C2×C4—42no—
344C2×C4#12014yes—
345C2×C4#12114yes—
346C22—24no—
347C22—28no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—42no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—44no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22C2×#12011yescharacteristic
373C22C2×#12111yescharacteristic
374C22#52511yescharacteristic
375C22#52511yescharacteristic
376C22#52512yes—
377C22C2×#12411yescharacteristic
378C22#52911yescharacteristic
379C22#52911yescharacteristic
380C22#53211yescharacteristic
381C22#53211yescharacteristic
382C22#53212yes—
383C22#53511yescharacteristic
384C22#53511yescharacteristic
385C22#53812yes—
386C22#53812yes—
387C22D4212yes—
388C22#54612yes—
389C22#54612yes—
390C22#54612yes—
391C22#54612yes—
392C22#55112yes—
393C22#55512yes—
394C22#55912yes—
395C4—24no—
396C4—24no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—44no—
401C4—44no—
402C4—42no—
403C4—82no—
404C2—44no—
405C2—84no—
406C2#238811yescharacteristic
407C2#240411yescharacteristic
408C2#240811yescharacteristic
409C2#245912yes—
410C2#247212yes—
411C2#254411yescharacteristic
412C2#255512yes—
413C2#255712yes—
414C2#261011yescharacteristic
415C2#263111yescharacteristic
416C2#285011yescharacteristic
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×#124C2semidirect
2. C2×#393C2semidirect
4. C2×#386C2semidirect
5. C2×C4×C22⋊C4C2semidirect
6. #1446C2semidirect
7. #1440C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
10. #1437C2semidirect
11. #1434C2semidirect
98. C2×C22⋊C4D4semidirect
102. C2×C42D4semidirect
279. C23#124semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3932
C2×#3922
C2×#3862
C2×C4×C22⋊C42
#14462
#14402
#14382
#14372
#14342