Finite Group Databasebeta

← #35534

Subgroups — #35534

Order 256.

1303 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#540C211yesmaximal, characteristic, complemented
2C2×#520C211yesmaximal, characteristic, complemented
3#2566C211yesmaximal, characteristic, complemented
4#2565C211yesmaximal, characteristic, complemented
5#2548C211yesmaximal, characteristic, complemented
6#2541C211yesmaximal, characteristic, complemented
7#2504C211yesmaximal, characteristic, complemented
8#2501C211yesmaximal, characteristic, complemented
9#2471C214yesmaximal, complemented
10#2464C214yesmaximal, complemented
11#2456C214yesmaximal, complemented
12#2443C214yesmaximal, complemented
13#2431C211yesmaximal, characteristic, complemented
14#2431C211yesmaximal, characteristic, complemented
15#2328C211yesmaximal, characteristic, complemented
16#2326C211yesmaximal, characteristic, complemented
17C4×#127C211yesmaximal, characteristic, complemented
18C4×#124C211yesmaximal, characteristic, complemented
19#2298C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21#540—28no—
22C2×#127C2211yescharacteristic
23C2×#127C2211yescharacteristic
24C2×#127C2211yescharacteristic
25C2×#125C2211yescharacteristic
26C2×#124C2211yescharacteristic
27C2×#124C2211yescharacteristic
28C2×#123C2212yes—
29C2×#123C2212yes—
30C2×#123C2214yes—
31C2×#121C2211yescharacteristic
32C2×#121C2211yescharacteristic
33C2×#121C2212yes—
34C2×#121C2212yes—
35C2×#121C2214yes—
36#520—28no—
37C2×C4×Q8C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic
43C2×C4×D4C2212yes—
44C2×C4×D4C2212yes—
45C2×C4×D4C2214yes—
46C2×#117C2212yes—
47C2×#117C2211yescharacteristic
48C22×C4⋊C4C2212yes—
49C22×C4⋊C4C2211yescharacteristic
50#399C2214yes—
51#398C2214yes—
52#397C2214yes—
53#396C2214yes—
54#394C2214yes—
55#392C2214yes—
56#392C2214yes—
57#392C2214yes—
58#391C2211yescharacteristic
59#391C2211yescharacteristic
60#390C2211yescharacteristic
61#390C2214yes—
62#390C2211yescharacteristic
63#390C2211yescharacteristic
64#390C2211yescharacteristic
65#390C2211yescharacteristic
66#390C2214yes—
67#390C2211yescharacteristic
68#388C2214yes—
69#388C2214yes—
70#388C2212yes—
71#388C2212yes—
72#387C2214yes—
73#387C2212yes—
74#387C2212yes—
75#386C2214yes—
76#386C2214yes—
77#386C2214yes—
78#386C2214yes—
79#386C2214yes—
80#385C2214yes—
81#385C2214yes—
82#384C2211yescharacteristic
83#384C2211yescharacteristic
84#382C2211yescharacteristic
85#382C2211yescharacteristic
86C4×C4⋊C4C2211yescharacteristic
87C4×C4⋊C4C2211yescharacteristic
88C4×C22⋊C4C2211yescharacteristic
89C4×C22⋊C4C2211yescharacteristic
90C4×C22⋊C4C2214yes—
91#376C2211yescharacteristic
92C43C2211yescharacteristic
93C2×#41—24no—
94C2×#41—24no—
95C2×#41—24no—
96C22×Q8C2311yescharacteristic
97C22×D4C2312yes—
98C22×D4C2311yescharacteristic
99C22×D4C2311yescharacteristic
100C22×D4C2314yes—
101C22×D4C2311yescharacteristic
102C22×D4C2312yes—
103C22×D4C2312yes—
104C22×D4C2312yes—
105C22×D4C2312yes—
106C22×D4C2314yes—
107C23×C4C2312yes—
108C23×C4C2311yescharacteristic
109C23×C4C2312yes—
110C23×C4C2312yes—
111C23×C4C2312yes—
112C23×C4C2312yes—
113C23×C4C2312yes—
114#127—24no—
115#127—24no—
116#127—24no—
117#125—24no—
118#124—24no—
119#124—24no—
120#123—28no—
121#123—28no—
122#123—48no—
123#121—24no—
124#121—24no—
125#121—28no—
126#121—28no—
127#121—48no—
128C4×Q8—24no—
129C4×Q8—24no—
130C4×D4—28no—
131C4×D4—28no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—48no—
137#117—28no—
138#117D418yescomplemented
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2311yescharacteristic
148C2×C4⋊C4C2314yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2314yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2311yescharacteristic
155C2×C4⋊C4C2311yescharacteristic
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2314yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2314yes—
165C2×C22⋊C4C2314yes—
166C2×C22⋊C4C2314yes—
167C2×C22⋊C4C2314yes—
168C2×C22⋊C4C2314yes—
169C2×C22⋊C4C2314yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2314yes—
172C2×C22⋊C4C2311yescharacteristic
173C2×C22⋊C4C2311yescharacteristic
174C2×C42C2312yes—
175C2×C42C2312yes—
176C2×C42C2312yes—
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
181C2×C42C2312yes—
182C2×C42C2311yescharacteristic
183C2×C42C2311yescharacteristic
184#95C2312yes—
185#95C2312yes—
186#95C2314yes—
187#95C2312yes—
188#95C2312yes—
189#95C2314yes—
190#95C2312yes—
191#95C2312yes—
192#95C2314yes—
193#95C2314yes—
194C24C2412yes—
195C24C2412yes—
196C24C2411yescharacteristic
197C24C2412yes—
198#41—232no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8#4114yes—
202C2×D4—24no—
203C2×D4—24no—
204C2×D4—28no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—28no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4#4114yes—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4C2×D414yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2411yescharacteristic
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2412yes—
257C22×C4C2412yes—
258C22×C4C2411yescharacteristic
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—28no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4C2×D414yes—
273C4⋊C4C2×D414yes—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4C2×D414yes—
291C22⋊C4C2×D414yes—
292C42—24no—
293C42—24no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—24no—
299C42—42no—
300C42C2×D414yes—
301C42C2×D414yes—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—22no—
307C23—24no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23C22×D412yes—
321C23C2×#4114yes—
322C23C2511yescharacteristic
323Q8—28no—
324D4—216no—
325D4—28no—
326D4—416no—
327D4—44no—
328D4—44no—
329D4—84no—
330D4—84no—
331D4—84no—
332D4—84no—
333D4—88no—
334C2×C4—216no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4C22×D412yes—
369C2×C4C22×D412yes—
370C2×C4C22×D412yes—
371C2×C4C22×D412yes—
372C2×C4C22×D412yes—
373C2×C4C22×D412yes—
374C2×C4C2×#4114yes—
375C2×C4C2×#4112yes—
376C2×C4C2×#4112yes—
377C2×C4H(5,𝔽2)14yes—
378C22—22no—
379C22—24no—
380C22—22no—
381C22—22no—
382C22—28no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—44no—
398C22C23×D411yescharacteristic
399C22C22×#4111yescharacteristic
400C22C22×#4111yescharacteristic
401C22C2×H(5,𝔽2)12yes—
402C22#58512yes—
403C4—22no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—24no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4#52514yes—
418C2—42no—
419C2—44no—
420C2—82no—
421C2—82no—
422C2C2×#52511yescharacteristic
423C2#349811yescharacteristic
424C2#350011yescharacteristic
425C2#351312yes—
426C2#351712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#540C2semidirect
2. C2×#520C2semidirect
3. #2566C2semidirect
4. #2565C2semidirect
5. #2548C2semidirect
6. #2541C2semidirect
7. #2504C2semidirect
8. #2501C2semidirect
9. #2471C2semidirect
10. #2464C2semidirect
11. #2456C2semidirect
12. #2443C2semidirect
13. #2431C2semidirect
14. #2431C2semidirect
15. #2328C2semidirect
16. #2326C2semidirect
17. C4×#127C2semidirect
18. C4×#124C2semidirect
19. #2298C2semidirect
138. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5402
C2×#5202
#25662
#25652
#25482
#25412
#25042
#25012
#24712
#24642
#24562
#24432
#24312
#23282
#23262
C4×#1272
C4×#1242
#22982