Finite Group Databasebeta

← #43487

Subgroups — #43487

Order 256.

912 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2772C212yesmaximal, complemented
2#2753C212yesmaximal, complemented
3#2752C211yesmaximal, characteristic, complemented
4#2691C212yesmaximal, complemented
5#2675C211yesmaximal, characteristic
6#2600C212yesmaximal, complemented
7#2600C211yesmaximal, characteristic, complemented
8#2598C211yesmaximal, characteristic, complemented
9#2573C212yesmaximal, complemented
10#2559C212yesmaximal, complemented
11#2517C211yesmaximal, characteristic, complemented
12#2505C212yesmaximal, complemented
13#2484C211yesmaximal, characteristic, complemented
14#2444C212yesmaximal, complemented
15#2420C212yesmaximal, complemented
16#2392C211yesmaximal, characteristic, complemented
17#2364C212yesmaximal, complemented
18#2361C212yesmaximal, complemented
19#2334C212yesmaximal, complemented
20C2×#126C2212yes—
21C2×#121C2212yes—
22C2×#121C2212yes—
23C2×#121C2212yes—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2212yes—
27C2×C4×D4C2212yes—
28C2×C4×D4C2212yes—
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#400C2212yes—
32#400C2212yes—
33#400C2211yescharacteristic
34#400C2212yes—
35#399C2211yescharacteristic
36#398C2211yescharacteristic
37#398C2212yes—
38#397C2212yes—
39#397C2211yescharacteristic
40#397C2212yes—
41#397C2212yes—
42#397C2212yes—
43#396C2212yes—
44#396C2212yes—
45#394C2211yescharacteristic
46#394C2212yes—
47#394C2212yes—
48#394C2212yes—
49#394C2212yes—
50#394C2211yescharacteristic
51#394C2212yes—
52#392C2211yescharacteristic
53#392C2212yes—
54#392C2211yescharacteristic
55#390C2212yes—
56#390C2212yes—
57#389C2212yes—
58#389C2212yes—
59#389C2212yes—
60#388C2212yes—
61#388C2212yes—
62#388C2212yes—
63#388C2212yes—
64#388C2212yes—
65#388C2212yes—
66#388C2212yes—
67#388C2212yes—
68#387C2212yes—
69#387C2212yes—
70#387C2212yes—
71#387C2212yes—
72#387C2212yes—
73#387C2212yes—
74#387C2212yes—
75#387C2212yes—
76#386C2212yes—
77#386C2212yes—
78#386C2212yes—
79#386C2212yes—
80#386C2212yes—
81#386C2212yes—
82#386C2212yes—
83#386C2212yes—
84#385C2212yes—
85#385C2212yes—
86#385C2212yes—
87#385C2212yes—
88#385C2212yes—
89#385C2212yes—
90#385C2211yescharacteristic
91#385C2211yescharacteristic
92#383C2211yescharacteristic
93#382C2212yes—
94#382C2212yes—
95#381C2212yes—
96#381C2211yescharacteristic
97#381C2211yescharacteristic
98#381C2211yescharacteristic
99#380C2212yes—
100C4×C4⋊C4C2212yes—
101C4×C4⋊C4C2211yescharacteristic
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106C2×#95C2212yes—
107C22×D4C2311yescharacteristic
108C22×D4C2312yes—
109C22×D4C2312yes—
110C22×D4C2312yes—
111C22×D4C2312yes—
112C22×D4C2311yescharacteristic
113C23×C4C2311yescharacteristic
114C23×C4C2311yescharacteristic
115C23×C4C2311yescharacteristic
116C23×C4C2312yes—
117C23×C4C2312yes—
118C23×C4C2312yes—
119C23×C4C2312yes—
120#126—28no—
121#121—28no—
122#121—28no—
123#121—44no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—44no—
127C4×D4—44no—
128C4×D4—44no—
129#117—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2311yescharacteristic
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2311yescharacteristic
156C2×C22⋊C4C2311yescharacteristic
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2312yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2311yescharacteristic
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C42C2311yescharacteristic
173C2×C42C2311yescharacteristic
174C2×C42C2312yes—
175C2×C42C2312yes—
176C2×C42C2312yes—
177C2×C42C2312yes—
178C2×C42C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2311yescharacteristic
191#95C2312yes—
192#95C2311yescharacteristic
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2311yescharacteristic
203#95C2311yescharacteristic
204#95C2312yes—
205C24C2412yes—
206C24C2412yes—
207C24C2411yescharacteristic
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—41no—
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—41no—
243C22×C4—42no—
244C22×C4C2411yescharacteristic
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2411yescharacteristic
250C22×C4C2412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2412yes—
259C22×C4C2411yescharacteristic
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—22no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4#4118yes—
301C42—24no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23C2×#4114yes—
326C23H(5,𝔽2)12yes—
327C23C2511yescharacteristic
328D4—48no—
329D4—48no—
330D4—44no—
331D4—48no—
332D4—44no—
333D4—84no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
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—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
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—44no—
376C2×C4—44no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4C2×#4114yes—
383C2×C4C2×#4114yes—
384C2×C4H(5,𝔽2)12yes—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—24no—
389C22—22no—
390C22—41no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22C22×#4112yes—
405C22C2×H(5,𝔽2)11yescharacteristic
406C22C2×H(5,𝔽2)11yescharacteristic
407C22#58511yescharacteristic
408C22#58512yes—
409C4—24no—
410C4—24no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—41no—
424C4—41no—
425C2—44no—
426C2—42no—
427C2—82no—
428C2#348111yescharacteristic
429C2#351212yes—
430C2#351312yes—
431C2#353411yescharacteristic
432C2#354011yescharacteristic
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. #2772C2semidirect
2. #2753C2semidirect
3. #2752C2semidirect
4. #2691C2semidirect
6. #2600C2semidirect
7. #2600C2semidirect
8. #2598C2semidirect
9. #2573C2semidirect
10. #2559C2semidirect
11. #2517C2semidirect
12. #2505C2semidirect
13. #2484C2semidirect
14. #2444C2semidirect
15. #2420C2semidirect
16. #2392C2semidirect
17. #2364C2semidirect
18. #2361C2semidirect
19. #2334C2semidirect

Maximal subgroups

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

subgroupindex
#27722
#27532
#27522
#26912
#26752
#26002
#25982
#25732
#25592
#25172
#25052
#24842
#24442
#24202
#23922
#23642
#23612
#23342