Finite Group Databasebeta

← #59948

Subgroups — #59948

Order 256.

949 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#2809C211yesmaximal, characteristic, complemented
2#2775C212yesmaximal, complemented
3#2731C211yesmaximal, characteristic, complemented
4#2612C212yesmaximal, complemented
5#2574C212yesmaximal, complemented
6#2532C212yesmaximal, complemented
7#2488C212yesmaximal, complemented
8#2474C212yesmaximal, complemented
9#2439C212yesmaximal, complemented
10#2436C211yesmaximal, characteristic
11#2434C211yesmaximal, characteristic, complemented
12#2425C212yesmaximal
13#2413C212yesmaximal, complemented
14#2372C211yesmaximal, characteristic, complemented
15#2369C211yesmaximal, characteristic, complemented
16#2359C212yesmaximal, complemented
17#2343C212yesmaximal, complemented
18C2×#391C211yesmaximal, characteristic, complemented
19#2290C212yesmaximal, complemented
20C23×Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123C2212yes—
26C2×#123C2212yes—
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×D4C2212yes—
30C2×#117C2212yes—
31C2×#117C2212yes—
32C22×C4⋊C4C2211yescharacteristic
33C22×C42C2211yescharacteristic
34#400C2212yes—
35#400C2212yes—
36#398C2212yes—
37#398C2212yes—
38#398C2212yes—
39#397C2212yes—
40#397C2212yes—
41#396C2212yes—
42#396C2212yes—
43#396C2212yes—
44#395C2212yes—
45#395C2212yes—
46#394C2212yes—
47#393C2212yes—
48#393C2212yes—
49#393C2212yes—
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#391C2212yes—
53#391C2211yescharacteristic
54#391C2211yescharacteristic
55#391C2212yes—
56#391C2211yescharacteristic
57#391C2211yescharacteristic
58#391C2212yes—
59#391C2211yescharacteristic
60#391C2211yescharacteristic
61#389C2212yes—
62#389C2212yes—
63#389C2212yes—
64#389C2212yes—
65#388C2212yes—
66#388C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#387C2212yes—
75#387C2212yes—
76#387C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#386C2212yes—
83#386C2212yes—
84#386C2212yes—
85#386C2212yes—
86#385C2212yes—
87#385C2212yes—
88#385C2212yes—
89#385C2212yes—
90#383C2212yes—
91#382C2212yes—
92#382C2212yes—
93#381C2212yes—
94#380C2212yes—
95C4×C4⋊C4C2212yes—
96C4×C4⋊C4C2212yes—
97C4×C4⋊C4C2212yes—
98C4×C22⋊C4C2212yes—
99C4×C22⋊C4C2211yescharacteristic
100C4×C22⋊C4C2212yes—
101C4×C22⋊C4C2211yescharacteristic
102C4×C22⋊C4C2212yes—
103#376C2212yes—
104#376C2212yes—
105C2×#95C2212yes—
106C2×#95C2212yes—
107C22×Q8—24no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8C2311yescharacteristic
111C22×Q8C2311yescharacteristic
112C22×Q8C2311yescharacteristic
113C22×Q8C2311yescharacteristic
114C22×Q8C2311yescharacteristic
115C22×D4C2312yes—
116C22×D4C2311yescharacteristic
117C23×C4C2312yes—
118C23×C4C2311yescharacteristic
119C23×C4C2312yes—
120C23×C4C2312yes—
121C23×C4C2311yescharacteristic
122C23×C4C2311yescharacteristic
123C4⋊Q8—24no—
124C4⋊Q8—42no—
125#124—24no—
126#124—42no—
127#123—28no—
128#123—44no—
129C22⋊Q8—42no—
130#121—42no—
131C4×D4—28no—
132#117—28no—
133#117—44no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2311yescharacteristic
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2311yescharacteristic
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
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⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C42—42no—
170C2×C42C2312yes—
171C2×C42C2312yes—
172C2×C42C2312yes—
173C2×C42C2312yes—
174C2×C42C2311yescharacteristic
175C2×C42C2311yescharacteristic
176C2×C42C2312yes—
177C2×C42C2312yes—
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
181C2×C42C2311yescharacteristic
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
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#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208#95C2312yes—
209#95C2312yes—
210C24C2412yes—
211C24C2411yescharacteristic
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—416no—
227C2×Q8#4114yes—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2412yes—
259C22×C4C2412yes—
260C22×C4C2412yes—
261C22×C4C2412yes—
262C22×C4C2411yescharacteristic
263C22×C4C2411yescharacteristic
264C22×C4C2412yes—
265C22×C4C2412yes—
266C22×C4C2411yescharacteristic
267C22×C4C2411yescharacteristic
268C22×C4C2412yes—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4#4118yes—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—22no—
307C42—22no—
308C42—24no—
309C42—24no—
310C42—41no—
311C42—41no—
312C42—42no—
313C42—41no—
314C42—41no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23C2×#4114yes—
326C23H(5,𝔽2)12yes—
327C23C2511yescharacteristic
328Q8—44no—
329Q8—44no—
330Q8—44no—
331Q8—44no—
332Q8—82no—
333D4—48no—
334D4—82no—
335C2×C4—28no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
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—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4C2×#4114yes—
386C2×C4C2×#4112yes—
387C2×C4C2×#4114yes—
388C2×C4C2×#4112yes—
389C2×C4C2×#4112yes—
390C2×C4H(5,𝔽2)12yes—
391C2×C42−1+414yes—
392C22—24no—
393C22—22no—
394C22—24no—
395C22—22no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22#55514yes—
404C22C22×#4112yes—
405C22C22×#4111yescharacteristic
406C22C2×H(5,𝔽2)11yescharacteristic
407C22C2×H(5,𝔽2)11yescharacteristic
408C22C2×2−1+412yes—
409C4—24no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—24no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C2—24no—
427C2—82no—
428C2C2×#55511yescharacteristic
429C2#348212yes—
430C2#349011yescharacteristic
431C2#352012yes—
432C2#354411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2809C2semidirect
2. #2775C2semidirect
3. #2731C2semidirect
4. #2612C2semidirect
5. #2574C2semidirect
6. #2532C2semidirect
7. #2488C2semidirect
8. #2474C2semidirect
9. #2439C2semidirect
11. #2434C2semidirect
13. #2413C2semidirect
14. #2372C2semidirect
15. #2369C2semidirect
16. #2359C2semidirect
17. #2343C2semidirect
18. C2×#391C2semidirect
19. #2290C2semidirect

Maximal subgroups

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

subgroupindex
#28092
#27752
#27312
#26122
#25742
#25322
#24882
#24742
#24392
#24362
#24342
#24252
#24132
#23722
#23692
#23592
#23432
C2×#3912
#22902