Finite Group Databasebeta

← #48265

Subgroups — #48265

Order 256.

908 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2796C212yesmaximal, complemented
2#2786C212yesmaximal, complemented
3#2751C212yesmaximal, complemented
4#2723C212yesmaximal, complemented
5#2717C211yesmaximal, characteristic, complemented
6#2709C211yesmaximal, characteristic
7#2700C212yesmaximal, complemented
8#2638C212yesmaximal, complemented
9#2586C212yesmaximal, complemented
10#2560C212yesmaximal, complemented
11#2544C211yesmaximal, characteristic
12#2520C212yesmaximal, complemented
13#2468C212yesmaximal, complemented
14#2449C212yesmaximal, complemented
15#2409C211yesmaximal, characteristic, complemented
16#2409C212yesmaximal, complemented
17#2379C211yesmaximal, characteristic, complemented
18#2366C211yesmaximal, characteristic
19#2322C211yesmaximal, characteristic, complemented
20C23×Q8C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#126C2212yes—
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123C2211yescharacteristic
26C2×C22⋊Q8C2212yes—
27C2×C22⋊Q8C2212yes—
28C2×C22⋊Q8C2212yes—
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2212yes—
31C2×C4×D4C2212yes—
32C22×C4⋊C4C2211yescharacteristic
33#400C2212yes—
34#399C2212yes—
35#398C2212yes—
36#398C2212yes—
37#398C2212yes—
38#397C2212yes—
39#397C2212yes—
40#397C2212yes—
41#397C2212yes—
42#397C2212yes—
43#397C2212yes—
44#396C2212yes—
45#395C2212yes—
46#395C2212yes—
47#395C2212yes—
48#395C2212yes—
49#394C2212yes—
50#394C2212yes—
51#394C2212yes—
52#394C2212yes—
53#394C2212yes—
54#393C2212yes—
55#393C2212yes—
56#393C2212yes—
57#391C2212yes—
58#391C2211yescharacteristic
59#391C2212yes—
60#391C2212yes—
61#391C2212yes—
62#390C2212yes—
63#390C2212yes—
64#390C2211yescharacteristic
65#389C2212yes—
66#389C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#387C2212yes—
74#387C2212yes—
75#387C2212yes—
76#387C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#386C2212yes—
83#386C2211yescharacteristic
84#386C2211yescharacteristic
85#386C2212yes—
86#386C2211yescharacteristic
87#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#385C2211yescharacteristic
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2211yescharacteristic
95#385C2211yescharacteristic
96#385C2212yes—
97#385C2212yes—
98#382C2212yes—
99#381C2211yescharacteristic
100#381C2212yes—
101#380C2212yes—
102#380C2211yescharacteristic
103C4×C4⋊C4C2212yes—
104C4×C4⋊C4C2212yes—
105C4×C4⋊C4C2211yescharacteristic
106C4×C22⋊C4C2211yescharacteristic
107C22×Q8—44no—
108C22×Q8—42no—
109C22×Q8C2312yes—
110C22×Q8C2312yes—
111C22×Q8C2311yescharacteristic
112C22×D4C2312yes—
113C22×D4C2312yes—
114C22×D4C2311yescharacteristic
115C22×D4C2311yescharacteristic
116C23×C4C2311yescharacteristic
117C23×C4C2311yescharacteristic
118C23×C4C2311yescharacteristic
119C23×C4C2312yes—
120C23×C4C2311yescharacteristic
121C23×C4C2311yescharacteristic
122C23×C4C2311yescharacteristic
123#127—24no—
124#126—44no—
125#124—24no—
126#124—42no—
127#123—24no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C4×D4—42no—
132C4×D4—44no—
133C4×D4—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⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2311yescharacteristic
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2311yescharacteristic
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2311yescharacteristic
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2311yescharacteristic
172C2×C22⋊C4C2311yescharacteristic
173C2×C22⋊C4C2311yescharacteristic
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2312yes—
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#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—
208C24C2412yes—
209C24C2411yescharacteristic
210C24C2411yescharacteristic
211C2×Q8—22no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—416no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—41no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#4114yes—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C2411yescharacteristic
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2412yes—
256C22×C4C2412yes—
257C22×C4C2411yescharacteristic
258C22×C4C2412yes—
259C22×C4C2412yes—
260C22×C4C2411yescharacteristic
261C22×C4C2411yescharacteristic
262C22×C4C2412yes—
263C22×C4C2411yescharacteristic
264C22×C4C2412yes—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
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—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23C2×#4112yes—
329C23C2×#4112yes—
330C23C2511yescharacteristic
331Q8—48no—
332Q8—48no—
333Q8—82no—
334D4—44no—
335D4—48no—
336D4—48no—
337D4—82no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
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—21no—
361C2×C4—21no—
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—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4C2×#4112yes—
386C2×C4H(5,𝔽2)12yes—
387C2×C4H(5,𝔽2)14yes—
388C2×C42−1+414yes—
389C22—22no—
390C22—22no—
391C22—24no—
392C22—24no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22C22×#4111yescharacteristic
407C22C2×H(5,𝔽2)11yescharacteristic
408C22C2×H(5,𝔽2)12yes—
409C22C2×2−1+412yes—
410C22#58511yescharacteristic
411C4—22no—
412C4—24no—
413C4—24no—
414C4—22no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—41no—
422C4—42no—
423C4—41no—
424C4—42no—
425C4—42no—
426C4—42no—
427C2—42no—
428C2—42no—
429C2—82no—
430C2#350511yescharacteristic
431C2#352012yes—
432C2#353211yescharacteristic
433C2#355011yescharacteristic
434C2#355412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2796C2semidirect
2. #2786C2semidirect
3. #2751C2semidirect
4. #2723C2semidirect
5. #2717C2semidirect
7. #2700C2semidirect
8. #2638C2semidirect
9. #2586C2semidirect
10. #2560C2semidirect
12. #2520C2semidirect
13. #2468C2semidirect
14. #2449C2semidirect
15. #2409C2semidirect
16. #2409C2semidirect
17. #2379C2semidirect
19. #2322C2semidirect

Maximal subgroups

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

subgroupindex
#27962
#27862
#27512
#27232
#27172
#27092
#27002
#26382
#25862
#25602
#25442
#25202
#24682
#24492
#24092
#23792
#23662
#23222