Finite Group Databasebeta

← #35276

Subgroups — #35276

Order 256.

1233 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#540C211yesmaximal, characteristic, complemented
3#2861C211yesmaximal, characteristic, complemented
4#2858C211yesmaximal, characteristic, complemented
5#2856C211yesmaximal, characteristic, complemented
6#2850C211yesmaximal, characteristic, complemented
7#2660C211yesmaximal, characteristic, complemented
8#2657C211yesmaximal, characteristic, complemented
9#2620C211yesmaximal, characteristic, complemented
10#2619C211yesmaximal, characteristic, complemented
11#2555C214yesmaximal, complemented
12#2554C214yesmaximal, complemented
13#2464C214yesmaximal, complemented
14#2461C214yesmaximal, complemented
15#2328C211yesmaximal, characteristic, complemented
16#2326C211yesmaximal, characteristic, complemented
17C4×#127C211yesmaximal, characteristic, complemented
18C4×#124C211yesmaximal, characteristic, complemented
19C4×#117C211yesmaximal, characteristic, complemented
20#542—28no—
21#540—28no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#127C2211yescharacteristic
24C2×#127C2211yescharacteristic
25C2×#127C2211yescharacteristic
26C2×#126C2212yes—
27C2×#125C2211yescharacteristic
28C2×#124C2211yescharacteristic
29C2×#124C2211yescharacteristic
30C2×#124C2211yescharacteristic
31C2×#124C2212yes—
32C2×#124C2214yes—
33C2×#123C2212yes—
34C2×#123C2212yes—
35C2×#123C2212yes—
36C2×C22⋊Q8C2211yescharacteristic
37C2×C22⋊Q8C2212yes—
38C2×#121C2211yescharacteristic
39C2×#121C2211yescharacteristic
40C2×#121C2211yescharacteristic
41C2×#121C2212yes—
42C2×#121C2212yes—
43C2×#121C2214yes—
44C2×C4×Q8C2211yescharacteristic
45C2×C4×D4C2211yescharacteristic
46C2×C4×D4C2211yescharacteristic
47C2×C4×D4C2211yescharacteristic
48C2×C4×D4C2214yes—
49C2×#117C2211yescharacteristic
50C2×#117C2211yescharacteristic
51C22×C42C2211yescharacteristic
52#398C2214yes—
53#396C2214yes—
54#395C2214yes—
55#394C2214yes—
56#394C2214yes—
57#392C2214yes—
58#392C2214yes—
59#392C2214yes—
60#391C2211yescharacteristic
61#391C2211yescharacteristic
62#390C2211yescharacteristic
63#390C2211yescharacteristic
64#390C2214yes—
65#390C2211yescharacteristic
66#390C2211yescharacteristic
67#390C2211yescharacteristic
68#390C2211yescharacteristic
69#390C2214yes—
70#390C2214yes—
71#388C2214yes—
72#388C2214yes—
73#388C2214yes—
74#388C2214yes—
75#388C2214yes—
76#386C2214yes—
77#386C2214yes—
78#386C2214yes—
79#385C2214yes—
80#384C2211yescharacteristic
81#384C2211yescharacteristic
82#382C2211yescharacteristic
83#382C2211yescharacteristic
84C4×C4⋊C4C2212yes—
85C4×C4⋊C4C2212yes—
86C4×C22⋊C4C2212yes—
87C4×C22⋊C4C2212yes—
88C4×C22⋊C4C2214yes—
89#376C2211yescharacteristic
90#376C2211yescharacteristic
91C43C2211yescharacteristic
92C43C2211yescharacteristic
93C22×Q8C2311yescharacteristic
94C22×Q8C2312yes—
95C22×D4C2311yescharacteristic
96C22×D4C2311yescharacteristic
97C22×D4C2311yescharacteristic
98C22×D4C2314yes—
99C22×D4C2312yes—
100C22×D4C2312yes—
101C22×D4C2312yes—
102C22×D4C2312yes—
103C22×D4C2312yes—
104C22×D4C2314yes—
105C23×C4C2312yes—
106C23×C4C2311yescharacteristic
107C23×C4C2312yes—
108C23×C4C2312yes—
109C23×C4C2312yes—
110C4⋊Q8—24no—
111#127—24no—
112#127—24no—
113#127—24no—
114#126—28no—
115#125—24no—
116#124—28no—
117#124—24no—
118#124—24no—
119#124—24no—
120#124—48no—
121#123—28no—
122#123—28no—
123#123—28no—
124C22⋊Q8—24no—
125C22⋊Q8—28no—
126#121—24no—
127#121—24no—
128#121—24no—
129#121—28no—
130#121—28no—
131#121—48no—
132C4×Q8—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—48no—
137#117—24no—
138#117D418yescomplemented
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2314yes—
150C2×C4⋊C4C2314yes—
151C2×C4⋊C4C2311yescharacteristic
152C2×C4⋊C4C2311yescharacteristic
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2314yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2314yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2314yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2314yes—
167C2×C22⋊C4C2314yes—
168C2×C22⋊C4C2314yes—
169C2×C22⋊C4C2314yes—
170C2×C22⋊C4C2314yes—
171C2×C22⋊C4C2311yescharacteristic
172C2×C22⋊C4C2311yescharacteristic
173C2×C42—24no—
174C2×C42C2311yescharacteristic
175C2×C42C2312yes—
176C2×C42C2312yes—
177C2×C42C2312yes—
178C2×C42C2311yescharacteristic
179C2×C42C2312yes—
180C2×C42C2312yes—
181C2×C42C2312yes—
182C2×C42C2311yescharacteristic
183C2×C42C2311yescharacteristic
184C2×C42C2311yescharacteristic
185C2×C42C2311yescharacteristic
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2314yes—
191#95C2314yes—
192#95C2314yes—
193#95C2314yes—
194C24C2412yes—
195C24C2412yes—
196C24C2412yes—
197C24C2411yescharacteristic
198C2×Q8—22no—
199C2×Q8—24no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—44no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—28no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—28no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4C2×D414yes—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2411yescharacteristic
247C22×C4C2411yescharacteristic
248C22×C4C2411yescharacteristic
249C22×C4C2411yescharacteristic
250C22×C4C2411yescharacteristic
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2411yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4C2×D414yes—
268C4⋊C4C2×D414yes—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—28no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4C2×D414yes—
288C22⋊C4C2×D414yes—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—24no—
295C42—42no—
296C42—42no—
297C42C2×D414yes—
298C42C2×D414yes—
299C42#4114yes—
300C42#4114yes—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23C22×D412yes—
321C23C2511yescharacteristic
322Q8—44no—
323Q8—84no—
324D4—416no—
325D4—44no—
326D4—44no—
327D4—44no—
328D4—88no—
329D4—84no—
330D4—84no—
331D4—84no—
332D4—84no—
333D4—84no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—28no—
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—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—48no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4C22×D412yes—
366C2×C4C22×D412yes—
367C2×C4C22×D412yes—
368C2×C4C22×D412yes—
369C2×C4C22×D412yes—
370C2×C4C22×D412yes—
371C2×C4C2×#4114yes—
372C2×C4C2×#4114yes—
373C2×C4C2×#4112yes—
374C2×C4C2×#4112yes—
375C2×C4H(5,𝔽2)14yes—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—44no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22C23×D411yescharacteristic
397C22C22×#4111yescharacteristic
398C22C22×#4111yescharacteristic
399C22C2×H(5,𝔽2)12yes—
400C22#58512yes—
401C4—22no—
402C4—24no—
403C4—24no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4#53214yes—
416C2—42no—
417C2—82no—
418C2—82no—
419C2—82no—
420C2C2×#53211yescharacteristic
421C2#346012yes—
422C2#349412yes—
423C2#349811yescharacteristic
424C2#350011yescharacteristic
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×#542C2semidirect
2. C2×#540C2semidirect
3. #2861C2semidirect
4. #2858C2semidirect
5. #2856C2semidirect
6. #2850C2semidirect
7. #2660C2semidirect
8. #2657C2semidirect
9. #2620C2semidirect
10. #2619C2semidirect
11. #2555C2semidirect
12. #2554C2semidirect
13. #2464C2semidirect
14. #2461C2semidirect
15. #2328C2semidirect
16. #2326C2semidirect
17. C4×#127C2semidirect
18. C4×#124C2semidirect
19. C4×#117C2semidirect
138. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#5402
#28612
#28582
#28562
#28502
#26602
#26572
#26202
#26192
#25552
#25542
#24642
#24612
#23282
#23262
C4×#1272
C4×#1242
C4×#1172