Finite Group Databasebeta

← #46762

Subgroups — #46762

Order 256.

905 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2826C211yesmaximal, characteristic, complemented
2#2807C211yesmaximal, characteristic, complemented
3#2790C212yesmaximal, complemented
4#2777C212yesmaximal, complemented
5#2770C212yesmaximal, complemented
6#2753C212yesmaximal, complemented
7#2708C212yesmaximal, complemented
8#2675C211yesmaximal, characteristic
9#2598C211yesmaximal, characteristic, complemented
10#2598C211yesmaximal, characteristic, complemented
11#2505C212yesmaximal, complemented
12#2476C212yesmaximal, complemented
13#2455C212yesmaximal, complemented
14#2455C211yesmaximal, characteristic, complemented
15#2443C212yesmaximal, complemented
16#2425C212yesmaximal, complemented
17#2422C212yesmaximal, complemented
18#2325C212yesmaximal, complemented
19#2317C211yesmaximal, characteristic, complemented
20C2×#123C2212yes—
21C2×#123C2212yes—
22C2×#123C2212yes—
23C2×#121C2212yes—
24C2×#121C2212yes—
25C2×#121C2212yes—
26C2×C4×D4C2212yes—
27C2×C4×D4C2212yes—
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#400C2212yes—
34#400C2211yescharacteristic
35#400C2211yescharacteristic
36#400C2212yes—
37#400C2212yes—
38#399C2211yescharacteristic
39#399C2211yescharacteristic
40#399C2212yes—
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2212yes—
45#397C2212yes—
46#397C2212yes—
47#397C2212yes—
48#397C2212yes—
49#396C2212yes—
50#396C2212yes—
51#394C2212yes—
52#394C2212yes—
53#394C2211yescharacteristic
54#394C2211yescharacteristic
55#394C2212yes—
56#394C2212yes—
57#394C2212yes—
58#392C2211yescharacteristic
59#392C2211yescharacteristic
60#392C2212yes—
61#390C2212yes—
62#389C2212yes—
63#389C2212yes—
64#389C2212yes—
65#389C2212yes—
66#389C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#387C2212yes—
76#387C2212yes—
77#387C2212yes—
78#387C2212yes—
79#386C2212yes—
80#386C2212yes—
81#386C2212yes—
82#386C2212yes—
83#386C2212yes—
84#386C2212yes—
85#386C2212yes—
86#385C2211yescharacteristic
87#385C2211yescharacteristic
88#385C2211yescharacteristic
89#385C2211yescharacteristic
90#385C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#385C2212yes—
97#385C2212yes—
98#385C2212yes—
99#384C2212yes—
100#383C2211yescharacteristic
101#383C2211yescharacteristic
102#382C2212yes—
103#380C2212yes—
104#380C2212yes—
105C4×C22⋊C4C2212yes—
106C2×#95C2212yes—
107C22×D4C2311yescharacteristic
108C22×D4C2312yes—
109C22×D4C2312yes—
110C22×D4C2312yes—
111C22×D4C2312yes—
112C22×D4C2311yescharacteristic
113C23×C4C2311yescharacteristic
114C23×C4C2312yes—
115C23×C4C2312yes—
116C23×C4C2311yescharacteristic
117C23×C4C2311yescharacteristic
118C23×C4C2312yes—
119C23×C4C2312yes—
120#123—44no—
121#123—44no—
122#123—44no—
123#121—28no—
124#121—44no—
125#121—44no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—44no—
129C4×D4—44no—
130#117—24no—
131#117—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C22⋊C4C2311yescharacteristic
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2312yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2311yescharacteristic
165C2×C22⋊C4C2311yescharacteristic
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C2×C42C2312yes—
181C2×C42C2312yes—
182C2×C42C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2311yescharacteristic
199#95C2311yescharacteristic
200#95C2311yescharacteristic
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207C24C2412yes—
208C24C2411yescharacteristic
209C24C2412yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4C2411yescharacteristic
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2411yescharacteristic
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2412yes—
255C22×C4C2412yes—
256C22×C4C2412yes—
257C22×C4C2411yescharacteristic
258C22×C4C2412yes—
259C22×C4C2411yescharacteristic
260C22×C4C2411yescharacteristic
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C22⋊C4—22no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4#4114yes—
306C42—24no—
307C42—22no—
308C42—22no—
309C42—42no—
310C42—42no—
311C23—22no—
312C23—22no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23C2×#4112yes—
329C23H(5,𝔽2)14yes—
330C23C2511yescharacteristic
331D4—48no—
332D4—48no—
333D4—44no—
334D4—44no—
335D4—84no—
336D4—84no—
337C2×C4—28no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4C2×#4114yes—
386C2×C4H(5,𝔽2)12yes—
387C2×C42−1+412yes—
388C22—22no—
389C22—24no—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—42no—
397C22—41no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22C22×#4111yescharacteristic
408C22C2×H(5,𝔽2)11yescharacteristic
409C22C2×H(5,𝔽2)12yes—
410C22C2×2−1+411yescharacteristic
411C22#58512yes—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—41no—
423C4—42no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4—42no—
428C2—42no—
429C2—44no—
430C2—82no—
431C2#349211yescharacteristic
432C2#351312yes—
433C2#353711yescharacteristic
434C2#354011yescharacteristic
435C2#355912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2675C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2826C2semidirect
2. #2807C2semidirect
3. #2790C2semidirect
4. #2777C2semidirect
5. #2770C2semidirect
6. #2753C2semidirect
7. #2708C2semidirect
9. #2598C2semidirect
10. #2598C2semidirect
11. #2505C2semidirect
12. #2476C2semidirect
13. #2455C2semidirect
14. #2455C2semidirect
15. #2443C2semidirect
16. #2425C2semidirect
17. #2422C2semidirect
18. #2325C2semidirect
19. #2317C2semidirect

Maximal subgroups

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

subgroupindex
#28262
#28072
#27902
#27772
#27702
#27532
#27082
#26752
#25982
#25052
#24762
#24552
#24432
#24252
#24222
#23252
#23172