Finite Group Databasebeta

← #23586

Subgroups — #23586

Order 256.

529 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#541C211yesmaximal, characteristic
2C2×#441C211yesmaximal, characteristic, complemented
3C2×C4×Q16C211yesmaximal, characteristic, complemented
4#2323C211yesmaximal, characteristic
5#2059C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic
8#2035C211yesmaximal, characteristic
9#2026C211yesmaximal, characteristic, complemented
10#2016C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic
12#2001C211yesmaximal, characteristic
13#1763C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18#541—24no—
19#541—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126—21no—
22C2×#126—21no—
23C2×#125—21no—
24C2×#125—21no—
25C2×#125C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×#117C2211yescharacteristic
33#441—24no—
34C4×Q16—24no—
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×#97C2211yescharacteristic
48C2×C4×C8C2211yescharacteristic
49#391C2211yescharacteristic
50#389—21no—
51#389C2211yescharacteristic
52#388—21no—
53#388—21no—
54#387—21no—
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79#126—28no—
80#126—28no—
81#125—28no—
82#125—28no—
83#125—22no—
84#125—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8D414yes—
96C4×Q8D414yes—
97#117—22no—
98#117—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129#107—22no—
130#106—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141#97—22no—
142C4×C8—22no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24C2×D411yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8C2×D412yes—
165C2×Q8C2×D412yes—
166C2×Q8#4112yes—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2411yescharacteristic
187Q16—28no—
188Q16—28no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8#4112yes—
200C2×C8#4112yes—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4C2×D412yes—
222C4⋊C4C2×D412yes—
223C4⋊C4#4112yes—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—81no—
230C42—24no—
231C42—24no—
232C42—21no—
233C42—21no—
234C42—42no—
235C42—81no—
236C42C2×D412yes—
237C42C2×D412yes—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23C22×D411yescharacteristic
246Q8—22no—
247Q8—22no—
248Q8—22no—
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4#12012yes—
300C2×C4#12012yes—
301C2×C4C22×D411yescharacteristic
302C2×C4C22×D411yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C8—22no—
308C8—22no—
309C8—41no—
310C8—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22C2×#12011yescharacteristic
319C22#52511yescharacteristic
320C22#53211yescharacteristic
321C22#57711yescharacteristic
322C22#57811yescharacteristic
323C22#57811yescharacteristic
324C22#57811yescharacteristic
325C4—22no—
326C4—22no—
327C4—42no—
328C4—41no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—41no—
333C4—42no—
334C4—41no—
335C4—42no—
336C4—81no—
337C4—81no—
338C4#52511yescharacteristic
339C4#52511yescharacteristic
340C4#53211yescharacteristic
341C4#53211yescharacteristic
342C2—81no—
343C2#238811yescharacteristic
344C2#301111yescharacteristic
345C2#301311yescharacteristic
346C2#312011yescharacteristic
347C2#312711yescharacteristic
348C2#317711yescharacteristic
349C2#317811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#441C2semidirect
3. C2×C4×Q16C2semidirect
5. #2059C2semidirect
6. #2043C2semidirect
9. #2026C2semidirect
10. #2016C2semidirect
13. #1763C2semidirect
14. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4412
C2×C4×Q162
#23232
#20592
#20432
#20362
#20352
#20262
#20162
#20012
#17632
C4×#1032
#17502