Finite Group Databasebeta

← #14905

Subgroups — #14905

Order 256.

1023 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117C211yesmaximal, characteristic, complemented
2C2×#399C211yesmaximal, characteristic, complemented
3C2×#397C211yesmaximal, characteristic, complemented
4C2×#386C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1438C212yesmaximal
9#1437C212yesmaximal, complemented
10#1436C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×#117—24no—
15C2×#117—24no—
16C2×#117—24no—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2211yescharacteristic
27C22×C42C2212yes—
28#399—24no—
29#397—24no—
30#386—24no—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2311yescharacteristic
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—42no—
48C23×C4—24no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—24no—
52C23×C4D414yes—
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2312yes—
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62#117—232no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4D414yescomplemented
71C2×C4⋊C4D414yescomplemented
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C42—24no—
92C2×C42—24no—
93C2×C42—24no—
94#95—22no—
95#95—22no—
96#95—24no—
97#95—22no—
98#95—22no—
99#95—24no—
100#95—24no—
101#95—42no—
102#95—42no—
103#95—42no—
104C24—21no—
105C24—21no—
106C24—22no—
107C24—22no—
108C24—21no—
109C24—21no—
110C24—21no—
111C24—22no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—21no—
119C24—22no—
120C24—21no—
121C24—22no—
122C24—21no—
123C24C2×D412yes—
124C24#4112yes—
125C24#4114yes—
126C24C2411yescharacteristic
127C2×D4—28no—
128C2×D4—48no—
129C2×D4—48no—
130C2×D4—44no—
131C2×D4—44no—
132C22×C4—216no—
133C22×C4—216no—
134C22×C4—28no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—28no—
141C22×C4—28no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
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—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4C2×D414yes—
182C22×C4C2×D414yes—
183C22×C4C2×D412yes—
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4#4112yes—
188C22×C4#4114yes—
189C22×C4#4114yes—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—48no—
193C22⋊C4—216no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—48no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C42—216no—
201C23—22no—
202C23—22no—
203C23—24no—
204C23—24no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—22no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—24no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—42no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23#12314yes—
247C23C22×D411yescharacteristic
248C23C22×D411yescharacteristic
249C23C22×D411yescharacteristic
250C23C2×#4112yes—
251C23C2×#4112yes—
252C23C2×#4111yescharacteristic
253C23C2×#4111yescharacteristic
254C23C2×#4112yes—
255C23H(5,𝔽2)12yes—
256C23H(5,𝔽2)11yescharacteristic
257C232−1+411yescharacteristic
258D4—416no—
259C2×C4—232no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
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#12014yes—
312C2×C4#12314yes—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—44no—
327C22—44no—
328C22—42no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—82no—
333C22—82no—
334C22—81no—
335C22C2×#12011yescharacteristic
336C22C2×#12311yescharacteristic
337C22C2×#12311yescharacteristic
338C22#52512yes—
339C22#52512yes—
340C22#52511yescharacteristic
341C22#52912yes—
342C22#52912yes—
343C22#53211yescharacteristic
344C22#53511yescharacteristic
345C22#53711yescharacteristic
346C22#54012yes—
347C22#54212yes—
348C22#54611yescharacteristic
349C22#54612yes—
350C22#54711yescharacteristic
351C22#54811yescharacteristic
352C22#55011yescharacteristic
353C22#55012yes—
354C22#55112yes—
355C22#55112yes—
356C22#55112yes—
357C22#55212yes—
358C4—24no—
359C4—24no—
360C4—44no—
361C4—44no—
362C4—44no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—82no—
367C2—44no—
368C2—82no—
369C2#239511yescharacteristic
370C2#240711yescharacteristic
371C2#243111yescharacteristic
372C2#247112yes—
373C2#247112yes—
374C2#247712yes—
375C2#247912yes—
376C2#248211yescharacteristic
377C2#249811yescharacteristic
378C2#262411yescharacteristic
379C2#284911yescharacteristic
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×#117C2semidirect
2. C2×#399C2semidirect
3. C2×#397C2semidirect
5. #1445C2semidirect
6. #1445C2semidirect
7. #1439C2semidirect
9. #1437C2semidirect
10. #1436C2semidirect
11. C22⋊#95C2semidirect
70. C2×C4⋊C4D4semidirect
71. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#3992
C2×#3972
C2×#3862
#14452
#14392
#14382
#14372
#14362
C22⋊#952