Finite Group Databasebeta

← #35508

Subgroups — #35508

Order 256.

950 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2610C211yesmaximal, characteristic, complemented
2#2606C211yesmaximal, characteristic, complemented
3#2600C212yesmaximal, complemented
4#2598C211yesmaximal, characteristic, complemented
5#2596C212yesmaximal, complemented
6#2560C212yesmaximal, complemented
7#2559C212yesmaximal, complemented
8#2543C212yesmaximal, complemented
9#2540C211yesmaximal, characteristic, complemented
10#2531C212yesmaximal, complemented
11#2479C212yesmaximal, complemented
12#2476C212yesmaximal, complemented
13#2438C212yesmaximal, complemented
14#2414C212yesmaximal, complemented
15#2337C211yesmaximal, characteristic, complemented
16#2333C212yesmaximal, complemented
17#2298C211yesmaximal, characteristic, complemented
18#2293C212yesmaximal, complemented
19#2275C211yesmaximal, characteristic, complemented
20C2×#126C2212yes—
21C2×#124C2211yescharacteristic
22C2×C22⋊Q8C2212yes—
23C2×#121C2212yes—
24C2×#121C2212yes—
25C2×#121C2212yes—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2212yes—
28C2×C4×D4C2212yes—
29C2×#117C2211yescharacteristic
30C2×#117C2212yes—
31C2×#117C2212yes—
32#400C2211yescharacteristic
33#400C2212yes—
34#399C2212yes—
35#399C2211yescharacteristic
36#398C2212yes—
37#398C2212yes—
38#398C2212yes—
39#397C2212yes—
40#397C2212yes—
41#395C2211yescharacteristic
42#394C2212yes—
43#394C2211yescharacteristic
44#394C2212yes—
45#394C2211yescharacteristic
46#394C2212yes—
47#394C2212yes—
48#393C2211yescharacteristic
49#392C2211yescharacteristic
50#392C2212yes—
51#392C2211yescharacteristic
52#391C2212yes—
53#390C2212yes—
54#390C2212yes—
55#390C2212yes—
56#390C2212yes—
57#390C2212yes—
58#389C2212yes—
59#389C2212yes—
60#389C2212yes—
61#388C2212yes—
62#388C2212yes—
63#388C2212yes—
64#388C2212yes—
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#386C2212yes—
81#386C2212yes—
82#386C2212yes—
83#386C2212yes—
84#386C2212yes—
85#386C2212yes—
86#386C2212yes—
87#385C2212yes—
88#385C2212yes—
89#385C2212yes—
90#383C2211yescharacteristic
91#381C2212yes—
92#380C2212yes—
93C4×C4⋊C4C2212yes—
94C4×C4⋊C4C2211yescharacteristic
95C4×C4⋊C4C2212yes—
96C4×C4⋊C4C2211yescharacteristic
97C4×C22⋊C4C2212yes—
98C4×C22⋊C4C2211yescharacteristic
99C4×C22⋊C4C2212yes—
100C4×C22⋊C4C2212yes—
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2211yescharacteristic
103#376C2211yescharacteristic
104#376C2211yescharacteristic
105C2×#95C2212yes—
106C43C2211yescharacteristic
107C22×Q8C2311yescharacteristic
108C22×D4C2311yescharacteristic
109C22×D4C2312yes—
110C22×D4C2311yescharacteristic
111C22×D4C2312yes—
112C22×D4C2312yes—
113C22×D4C2312yes—
114C23×C4C2312yes—
115C23×C4C2311yescharacteristic
116C23×C4C2312yes—
117C23×C4C2312yes—
118C23×C4C2312yes—
119#126—44no—
120#124—24no—
121C22⋊Q8—44no—
122#121—28no—
123#121—44no—
124#121—44no—
125C4×D4—28no—
126C4×D4—28no—
127C4×D4—24no—
128#117—24no—
129#117—44no—
130#117—44no—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2311yescharacteristic
150C2×C22⋊C4C2311yescharacteristic
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊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×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C42C2312yes—
172C2×C42C2311yescharacteristic
173C2×C42C2312yes—
174C2×C42C2312yes—
175C2×C42C2312yes—
176C2×C42C2312yes—
177C2×C42C2312yes—
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
181C2×C42C2311yescharacteristic
182C2×C42C2311yescharacteristic
183#95C2311yescharacteristic
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2311yescharacteristic
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#95C2311yescharacteristic
204#95C2312yes—
205#95C2312yes—
206C24C2411yescharacteristic
207C24C2412yes—
208C24C2412yes—
209C2×Q8—22no—
210C2×Q8—42no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4#4118yes—
242C22×C4C2411yescharacteristic
243C22×C4C2411yescharacteristic
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2412yes—
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—22no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42#4114yes—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—21no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—21no—
327C23—21no—
328C23C2×#4114yes—
329C23C2511yescharacteristic
330Q8—82no—
331D4—28no—
332D4—48no—
333D4—84no—
334D4—82no—
335D4—84no—
336D4—84no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
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—21no—
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—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4C2×#4112yes—
381C2×C4C2×#4114yes—
382C2×C4C2×#4114yes—
383C2×C4C2×#4112yes—
384C2×C4C2×#4112yes—
385C2×C4H(5,𝔽2)14yes—
386C22—24no—
387C22—24no—
388C22—24no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—42no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—41no—
405C22C22×#4111yescharacteristic
406C22C22×#4112yes—
407C22C2×H(5,𝔽2)12yes—
408C22#58511yescharacteristic
409C22#58511yescharacteristic
410C4—22no—
411C4—22no—
412C4—24no—
413C4—24no—
414C4—24no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—41no—
424C4—42no—
425C4—42no—
426C2—44no—
427C2—81no—
428C2—82no—
429C2#344411yescharacteristic
430C2#349311yescharacteristic
431C2#349412yes—
432C2#350512yes—
433C2#350811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2610C2semidirect
2. #2606C2semidirect
3. #2600C2semidirect
4. #2598C2semidirect
5. #2596C2semidirect
6. #2560C2semidirect
7. #2559C2semidirect
8. #2543C2semidirect
9. #2540C2semidirect
10. #2531C2semidirect
11. #2479C2semidirect
12. #2476C2semidirect
13. #2438C2semidirect
14. #2414C2semidirect
15. #2337C2semidirect
16. #2333C2semidirect
17. #2298C2semidirect
18. #2293C2semidirect
19. #2275C2semidirect

Maximal subgroups

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

subgroupindex
#26102
#26062
#26002
#25982
#25962
#25602
#25592
#25432
#25402
#25312
#24792
#24762
#24382
#24142
#23372
#23332
#22982
#22932
#22752