Finite Group Databasebeta

← #51211

Subgroups — #51211

Order 256.

973 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#564C211yesmaximal, characteristic, complemented
2C2×#553C211yesmaximal, characteristic
3#2721C212yesmaximal, complemented
4#2711C212yesmaximal, complemented
5#2711C212yesmaximal, complemented
6#2711C212yesmaximal, complemented
7#2707C212yesmaximal, complemented
8#2694C212yesmaximal, complemented
9#2633C211yesmaximal, characteristic, complemented
10#2633C211yesmaximal, characteristic, complemented
11#2566C211yesmaximal, characteristic, complemented
12#2562C211yesmaximal, characteristic, complemented
13#2519C212yesmaximal, complemented
14#2490C212yesmaximal, complemented
15#2475C212yesmaximal, complemented
16#2473C212yesmaximal, complemented
17#2460C212yesmaximal, complemented
18#2454C212yesmaximal, complemented
19#2336C211yesmaximal, characteristic, complemented
20#564—28no—
21#553—28no—
22C2×#126C2212yes—
23C2×#125C2211yescharacteristic
24C2×#125C2211yescharacteristic
25C2×#125C2212yes—
26C2×#125C2212yes—
27C2×#125C2212yes—
28C2×#125C2212yes—
29C2×#125C2212yes—
30C2×#125C2212yes—
31C2×#125C2211yescharacteristic
32C2×#123C2212yes—
33C2×#121C2211yescharacteristic
34C2×#121C2211yescharacteristic
35C2×#121C2212yes—
36C2×C4×D4C2211yescharacteristic
37C2×C4×D4C2211yescharacteristic
38C2×C4×D4C2212yes—
39C2×#117C2211yescharacteristic
40C22×C4⋊C4C2211yescharacteristic
41#400C2212yes—
42#399C2211yescharacteristic
43#399C2211yescharacteristic
44#399C2212yes—
45#399C2212yes—
46#399C2212yes—
47#398C2212yes—
48#398C2212yes—
49#398C2212yes—
50#398C2212yes—
51#397C2211yescharacteristic
52#397C2211yescharacteristic
53#397C2212yes—
54#397C2212yes—
55#396C2212yes—
56#396C2212yes—
57#396C2212yes—
58#396C2212yes—
59#396C2212yes—
60#394C2212yes—
61#394C2212yes—
62#394C2212yes—
63#394C2212yes—
64#394C2212yes—
65#394C2212yes—
66#394C2212yes—
67#390C2211yescharacteristic
68#390C2211yescharacteristic
69#390C2212yes—
70#390C2212yes—
71#390C2212yes—
72#390C2212yes—
73#389C2212yes—
74#389C2212yes—
75#389C2212yes—
76#388C2212yes—
77#388C2212yes—
78#388C2212yes—
79#388C2212yes—
80#388C2212yes—
81#388C2212yes—
82#388C2212yes—
83#388C2212yes—
84#387C2212yes—
85#387C2212yes—
86#387C2212yes—
87#387C2212yes—
88#387C2212yes—
89#387C2212yes—
90#387C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#385C2212yes—
97#385C2212yes—
98#385C2212yes—
99#384C2211yescharacteristic
100#383C2211yescharacteristic
101#382C2211yescharacteristic
102#380C2212yes—
103#380C2212yes—
104#380C2212yes—
105C4×C4⋊C4C2211yescharacteristic
106C4×C4⋊C4C2212yes—
107C4×C4⋊C4C2212yes—
108C4×C22⋊C4C2212yes—
109C22×D4C2311yescharacteristic
110C22×D4C2312yes—
111C22×D4C2311yescharacteristic
112C22×D4C2312yes—
113C23×C4C2312yes—
114C23×C4C2311yescharacteristic
115C23×C4C2311yescharacteristic
116C23×C4C2312yes—
117#126—28no—
118#125—28no—
119#125—28no—
120#125—28no—
121#125—28no—
122#125—28no—
123#125—28no—
124#125—24no—
125#125—24no—
126#125D418yes—
127#123—28no—
128#121—28no—
129#121—24no—
130#121—24no—
131C4×D4—28no—
132C4×D4—24no—
133C4×D4—24no—
134#117—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
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×C4⋊C4C2312yes—
155C2×C4⋊C4C2312yes—
156C2×C4⋊C4C2312yes—
157C2×C4⋊C4C2312yes—
158C2×C4⋊C4C2312yes—
159C2×C4⋊C4C2312yes—
160C2×C4⋊C4C2312yes—
161C2×C4⋊C4C2312yes—
162C2×C4⋊C4C2311yescharacteristic
163C2×C4⋊C4C2311yescharacteristic
164C2×C4⋊C4C2312yes—
165C2×C4⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2311yescharacteristic
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2311yescharacteristic
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2312yes—
178C2×C22⋊C4C2312yes—
179C2×C22⋊C4C2312yes—
180C2×C22⋊C4C2312yes—
181C2×C22⋊C4C2312yes—
182C2×C22⋊C4C2312yes—
183C2×C22⋊C4C2312yes—
184C2×C22⋊C4C2312yes—
185C2×C22⋊C4C2312yes—
186C2×C22⋊C4C2312yes—
187C2×C22⋊C4C2312yes—
188C2×C42C2312yes—
189C2×C42C2312yes—
190C2×C42C2312yes—
191C2×C42C2312yes—
192C2×C42C2311yescharacteristic
193C2×C42C2312yes—
194C2×C42C2311yescharacteristic
195C2×C42C2311yescharacteristic
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208#95C2311yescharacteristic
209#95C2311yescharacteristic
210#95C2312yes—
211#95C2312yes—
212#95C2312yes—
213#95C2312yes—
214#95C2312yes—
215C24C2411yescharacteristic
216C24C2412yes—
217C24C2411yescharacteristic
218C2×D4—22no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—24no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2411yescharacteristic
245C22×C4C2412yes—
246C22×C4C2411yescharacteristic
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2411yescharacteristic
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2411yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4C2×D414yes—
282C4⋊C4C2×D414yes—
283C4⋊C4C2×D418yes—
284C4⋊C4C2×D418yes—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—41no—
311C42—41no—
312C42—42no—
313C42—42no—
314C42C2×D414yes—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23—21no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C232−1+412yes—
332C23C2511yescharacteristic
333D4—44no—
334D4—48no—
335D4—44no—
336D4—84no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
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—44no—
377C2×C4—44no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4C22×D414yes—
381C2×C4C22×D414yes—
382C2×C4C22×D412yes—
383C2×C4C22×D412yes—
384C2×C4C22×D412yes—
385C22—22no—
386C22—24no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22C23×D411yescharacteristic
403C22C2×2−1+411yescharacteristic
404C22#58512yes—
405C22#58511yescharacteristic
406C22#58512yes—
407C4—22no—
408C4—22no—
409C4—24no—
410C4—22no—
411C4—24no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C2—42no—
424C2—81no—
425C2—82no—
426C2#348711yescharacteristic
427C2#348911yescharacteristic
428C2#350011yescharacteristic
429C2#355312yes—
430C2#356712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#553C2
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×#564C2semidirect
3. #2721C2semidirect
4. #2711C2semidirect
5. #2711C2semidirect
6. #2711C2semidirect
7. #2707C2semidirect
8. #2694C2semidirect
9. #2633C2semidirect
10. #2633C2semidirect
11. #2566C2semidirect
12. #2562C2semidirect
13. #2519C2semidirect
14. #2490C2semidirect
15. #2475C2semidirect
16. #2473C2semidirect
17. #2460C2semidirect
18. #2454C2semidirect
19. #2336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5642
C2×#5532
#27212
#27112
#27072
#26942
#26332
#25662
#25622
#25192
#24902
#24752
#24732
#24602
#24542
#23362