Finite Group Databasebeta

← #40974

Subgroups — #40974

Order 256.

915 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2841C211yesmaximal, characteristic, complemented
2#2835C211yesmaximal, characteristic, complemented
3#2792C212yesmaximal, complemented
4#2779C212yesmaximal, complemented
5#2718C212yesmaximal, complemented
6#2695C212yesmaximal, complemented
7#2633C211yesmaximal, characteristic, complemented
8#2630C212yesmaximal, complemented
9#2630C212yesmaximal, complemented
10#2600C211yesmaximal, characteristic, complemented
11#2587C212yesmaximal, complemented
12#2573C212yesmaximal, complemented
13#2473C212yesmaximal, complemented
14#2466C212yesmaximal, complemented
15#2455C211yesmaximal, characteristic, complemented
16#2406C212yesmaximal, complemented
17#2388C211yesmaximal, characteristic
18D4×C4⋊C4C212yesmaximal, complemented
19#2322C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21C2×#126C2212yes—
22C2×#125C2211yescharacteristic
23C2×#123C2211yescharacteristic
24C2×#123C2212yes—
25C2×#123C2212yes—
26C2×C4×D4C2212yes—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2212yes—
29C2×C4×D4C2212yes—
30C22×C4⋊C4C2211yescharacteristic
31C22×C4⋊C4C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#399C2211yescharacteristic
37#399C2211yescharacteristic
38#398C2212yes—
39#398C2212yes—
40#398C2212yes—
41#398C2212yes—
42#398C2212yes—
43#397C2212yes—
44#397C2211yescharacteristic
45#397C2212yes—
46#397C2211yescharacteristic
47#396C2212yes—
48#396C2212yes—
49#396C2212yes—
50#396C2212yes—
51#395C2212yes—
52#395C2212yes—
53#395C2211yescharacteristic
54#394C2212yes—
55#394C2212yes—
56#394C2212yes—
57#394C2212yes—
58#394C2212yes—
59#394C2211yescharacteristic
60#394C2212yes—
61#393C2211yescharacteristic
62#392C2211yescharacteristic
63#392C2212yes—
64#389C2212yes—
65#389C2212yes—
66#389C2212yes—
67#389C2212yes—
68#389C2212yes—
69#389C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#387C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#386C2211yescharacteristic
83#386C2212yes—
84#386C2212yes—
85#385C2212yes—
86#385C2211yescharacteristic
87#385C2212yes—
88#385C2212yes—
89#385C2212yes—
90#385C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#384C2212yes—
97#383C2211yescharacteristic
98#381C2212yes—
99#381C2211yescharacteristic
100#380C2212yes—
101#380C2212yes—
102#380C2212yes—
103C4×C4⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2211yescharacteristic
106C4×C22⋊C4C2211yescharacteristic
107C2×#41—42no—
108C2×#41—44no—
109C22×Q8C2311yescharacteristic
110C22×D4C2312yes—
111C22×D4C2311yescharacteristic
112C22×D4C2311yescharacteristic
113C22×D4C2312yes—
114C23×C4C2312yes—
115C23×C4C2311yescharacteristic
116C23×C4C2312yes—
117C23×C4C2312yes—
118C23×C4C2311yescharacteristic
119C23×C4C2311yescharacteristic
120C23×C4C2311yescharacteristic
121#126—44no—
122#125—42no—
123#123—24no—
124#123—44no—
125#123—44no—
126C4×D4—28no—
127C4×D4—42no—
128C4×D4—44no—
129C4×D4—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
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⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2311yescharacteristic
150C2×C4⋊C4C2311yescharacteristic
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
155C2×C4⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2311yescharacteristic
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2311yescharacteristic
169C2×C22⋊C4C2311yescharacteristic
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2312yes—
174C2×C22⋊C4C2311yescharacteristic
175C2×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C42C2311yescharacteristic
178C2×C42C2312yes—
179C2×C42C2312yes—
180C2×C42C2312yes—
181C2×C42C2312yes—
182#95C2312yes—
183#95C2311yescharacteristic
184#95C2312yes—
185#95C2311yescharacteristic
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#95C2311yescharacteristic
200#95C2311yescharacteristic
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207C24C2411yescharacteristic
208C24C2411yescharacteristic
209C24C2412yes—
210#41—416no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4#4114yescomplemented
242C22×C4C2411yescharacteristic
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2411yescharacteristic
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2412yes—
255C22×C4C2412yes—
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C42—24no—
304C42—22no—
305C42—42no—
306C42—42no—
307C42—42no—
308C23—22no—
309C23—22no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—21no—
321C23—22no—
322C23—21no—
323C23C2×#4112yes—
324C23H(5,𝔽2)14yes—
325C23C2511yescharacteristic
326Q8—44no—
327D4—216no—
328D4—44no—
329D4—44no—
330D4—48no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
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—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4C2×#4114yes—
380C2×C4H(5,𝔽2)12yes—
381C2×C42−1+414yes—
382C22—22no—
383C22—24no—
384C22—24no—
385C22—24no—
386C22—24no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22C22×#4111yescharacteristic
400C22C2×H(5,𝔽2)11yescharacteristic
401C22C2×H(5,𝔽2)12yes—
402C22C2×2−1+412yes—
403C22#58511yescharacteristic
404C4—24no—
405C4—24no—
406C4—22no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C2—42no—
421C2—44no—
422C2—81no—
423C2#345911yescharacteristic
424C2#349212yes—
425C2#353211yescharacteristic
426C2#355011yescharacteristic
427C2#355912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2388C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2841C2semidirect
2. #2835C2semidirect
3. #2792C2semidirect
4. #2779C2semidirect
5. #2718C2semidirect
6. #2695C2semidirect
7. #2633C2semidirect
8. #2630C2semidirect
9. #2630C2semidirect
10. #2600C2semidirect
11. #2587C2semidirect
12. #2573C2semidirect
13. #2473C2semidirect
14. #2466C2semidirect
15. #2455C2semidirect
16. #2406C2semidirect
18. D4×C4⋊C4C2semidirect
19. #2322C2semidirect
241. C22×C4#41semidirect

Maximal subgroups

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

subgroupindex
#28412
#28352
#27922
#27792
#27182
#26952
#26332
#26302
#26002
#25872
#25732
#24732
#24662
#24552
#24062
#23882
D4×C4⋊C42
#23222