Finite Group Databasebeta

← #48781

Subgroups — #48781

Order 256.

999 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
1C2×#555C211yesmaximal, characteristic
2C2×#539C211yesmaximal, characteristic, complemented
3#2717C212yesmaximal, complemented
4#2715C212yesmaximal, complemented
5#2713C212yesmaximal, complemented
6#2708C212yesmaximal, complemented
7#2701C212yesmaximal, complemented
8#2700C212yesmaximal, complemented
9#2638C211yesmaximal, characteristic, complemented
10#2625C211yesmaximal, characteristic, complemented
11#2579C211yesmaximal, characteristic, complemented
12#2484C211yesmaximal, characteristic, complemented
13#2478C212yesmaximal, complemented
14#2475C212yesmaximal, complemented
15#2447C212yesmaximal, complemented
16#2444C212yesmaximal, complemented
17#2438C212yesmaximal, complemented
18#2419C212yesmaximal, complemented
19#2377C211yesmaximal, characteristic, complemented
20#555—28no—
21#539—28no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#126C2212yes—
24C2×#125C2212yes—
25C2×#124C2212yes—
26C2×#124C2212yes—
27C2×#123C2212yes—
28C2×#123C2212yes—
29C2×C22⋊Q8C2211yescharacteristic
30C2×C22⋊Q8C2212yes—
31C2×#121C2212yes—
32C2×#121C2211yescharacteristic
33C2×#121C2211yescharacteristic
34C2×C4×D4C2211yescharacteristic
35C2×C4×D4C2211yescharacteristic
36C2×C4×D4C2212yes—
37C2×#117C2211yescharacteristic
38C2×#117C2212yes—
39C2×#117C2211yescharacteristic
40C22×C4⋊C4C2211yescharacteristic
41#400C2212yes—
42#400C2212yes—
43#400C2212yes—
44#398C2212yes—
45#398C2212yes—
46#398C2211yescharacteristic
47#398C2212yes—
48#398C2212yes—
49#397C2212yes—
50#397C2212yes—
51#397C2212yes—
52#397C2211yescharacteristic
53#397C2212yes—
54#397C2212yes—
55#397C2212yes—
56#397C2212yes—
57#397C2212yes—
58#396C2212yes—
59#396C2212yes—
60#395C2211yescharacteristic
61#395C2212yes—
62#395C2212yes—
63#394C2212yes—
64#394C2212yes—
65#394C2212yes—
66#393C2211yescharacteristic
67#391C2212yes—
68#391C2212yes—
69#391C2212yes—
70#390C2212yes—
71#389C2212yes—
72#389C2212yes—
73#389C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2212yes—
84#387C2212yes—
85#386C2211yescharacteristic
86#386C2211yescharacteristic
87#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#386C2212yes—
91#385C2211yescharacteristic
92#385C2212yes—
93#385C2211yescharacteristic
94#385C2212yes—
95#385C2212yes—
96#385C2212yes—
97#385C2212yes—
98#385C2212yes—
99#385C2212yes—
100#381C2211yescharacteristic
101#381C2212yes—
102#381C2212yes—
103#381C2212yes—
104#380C2211yescharacteristic
105#380C2212yes—
106#380C2212yes—
107C4×C4⋊C4C2212yes—
108C2×#95C2212yes—
109C22×Q8C2311yescharacteristic
110C22×Q8C2312yes—
111C22×D4C2311yescharacteristic
112C22×D4C2312yes—
113C22×D4C2311yescharacteristic
114C22×D4C2312yes—
115C23×C4C2312yes—
116C23×C4C2311yescharacteristic
117C23×C4C2311yescharacteristic
118C23×C4C2311yescharacteristic
119C23×C4C2312yes—
120C23×C4C2312yes—
121C23×C4C2311yescharacteristic
122C4⋊Q8—24no—
123#126—28no—
124#125—28no—
125#124—28no—
126#124—28no—
127#123—28no—
128#123—28no—
129C22⋊Q8—24no—
130C22⋊Q8—28no—
131#121—24no—
132#121—28no—
133#121D418yes—
134C4×D4—28no—
135C4×D4—24no—
136C4×D4—24no—
137#117—24no—
138#117—24no—
139#117—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2311yescharacteristic
170C2×C22⋊C4C2311yescharacteristic
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2311yescharacteristic
174C2×C22⋊C4C2311yescharacteristic
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×C42C2312yes—
185C2×C42C2312yes—
186C2×C42C2311yescharacteristic
187C2×C42C2312yes—
188C2×C42C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2311yescharacteristic
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2311yescharacteristic
208#95C2312yes—
209#95C2312yes—
210#95C2312yes—
211#95C2312yes—
212#95C2312yes—
213#95C2312yes—
214#95C2312yes—
215C24C2411yescharacteristic
216C24C2412yes—
217C24C2411yescharacteristic
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×D4—22no—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4C2×D414yes—
233C2×D4C2×D418yes—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C2×D414yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2412yes—
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2412yes—
259C22×C4C2412yes—
260C22×C4C2411yescharacteristic
261C22×C4C2412yes—
262C22×C4C2412yes—
263C22×C4C2411yescharacteristic
264C22×C4C2412yes—
265C22×C4C2411yescharacteristic
266C22×C4C2411yescharacteristic
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4C2×D414yes—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—22no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C22⋊C4C2×D418yes—
310C42—24no—
311C42—24no—
312C42—22no—
313C42—24no—
314C42—42no—
315C23—22no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—22no—
330C23C22×D414yes—
331C23C22×D412yes—
332C23C2511yescharacteristic
333Q8—82no—
334Q8—84no—
335D4—44no—
336D4—48no—
337D4—44no—
338D4—84no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
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—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—44no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4C22×D412yes—
387C2×C4C22×D414yes—
388C2×C4C22×D412yes—
389C2×C42−1+412yes—
390C22—22no—
391C22—22no—
392C22—24no—
393C22—24no—
394C22—24no—
395C22—22no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22—42no—
407C22C23×D411yescharacteristic
408C22C2×2−1+411yescharacteristic
409C22#58512yes—
410C22#58512yes—
411C22#58511yescharacteristic
412C4—22no—
413C4—22no—
414C4—24no—
415C4—22no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—41no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—42no—
428C2—42no—
429C2—44no—
430C2—81no—
431C2#348511yescharacteristic
432C2#348511yescharacteristic
433C2#348911yescharacteristic
434C2#356512yes—
435C2#356612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#555C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#539C2semidirect
3. #2717C2semidirect
4. #2715C2semidirect
5. #2713C2semidirect
6. #2708C2semidirect
7. #2701C2semidirect
8. #2700C2semidirect
9. #2638C2semidirect
10. #2625C2semidirect
11. #2579C2semidirect
12. #2484C2semidirect
13. #2478C2semidirect
14. #2475C2semidirect
15. #2447C2semidirect
16. #2444C2semidirect
17. #2438C2semidirect
18. #2419C2semidirect
19. #2377C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#5392
#27172
#27152
#27132
#27082
#27012
#27002
#26382
#26252
#25792
#24842
#24782
#24752
#24472
#24442
#24382
#24192
#23772