Finite Group Databasebeta

← #38467

Subgroups — #38467

Order 256.

900 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2855C211yesmaximal, characteristic, complemented
2#2808C211yesmaximal, characteristic, complemented
3#2806C211yesmaximal, characteristic, complemented
4#2805C211yesmaximal, characteristic, complemented
5#2800C212yesmaximal, complemented
6#2774C212yesmaximal, complemented
7#2727C211yesmaximal, characteristic, complemented
8#2652C212yesmaximal, complemented
9#2589C212yesmaximal, complemented
10#2587C212yesmaximal, complemented
11#2498C212yesmaximal, complemented
12#2488C212yesmaximal, complemented
13#2481C212yesmaximal, complemented
14#2413C212yesmaximal, complemented
15#2406C212yesmaximal
16#2344C212yesmaximal, complemented
17#2334C212yesmaximal, complemented
18#2297C211yesmaximal, characteristic
19C2×#389C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21C2×#126C2212yes—
22C2×#123C2212yes—
23C2×#123C2212yes—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C2×#117C2212yes—
27C22×C4⋊C4C2212yes—
28C22×C4⋊C4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C42C2211yescharacteristic
31#400C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#399C2212yes—
37#398C2212yes—
38#398C2212yes—
39#398C2212yes—
40#397C2212yes—
41#397C2212yes—
42#397C2212yes—
43#396C2212yes—
44#396C2212yes—
45#396C2212yes—
46#395C2212yes—
47#395C2212yes—
48#394C2212yes—
49#394C2212yes—
50#393C2212yes—
51#391C2211yescharacteristic
52#391C2212yes—
53#391C2211yescharacteristic
54#391C2211yescharacteristic
55#391C2211yescharacteristic
56#390C2211yescharacteristic
57#390C2211yescharacteristic
58#389C2211yescharacteristic
59#389C2212yes—
60#389C2212yes—
61#389C2211yescharacteristic
62#389C2212yes—
63#389C2211yescharacteristic
64#389C2212yes—
65#389C2212yes—
66#389C2211yescharacteristic
67#389C2211yescharacteristic
68#389C2212yes—
69#389C2211yescharacteristic
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2212yes—
84#387C2212yes—
85#387C2212yes—
86#387C2212yes—
87#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#386C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#382C2212yes—
97#381C2212yes—
98#381C2212yes—
99#380C2211yescharacteristic
100#380C2212yes—
101C4×C4⋊C4C2212yes—
102C4×C4⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106C2×#95C2212yes—
107C2×#41—24no—
108C2×#41—42no—
109C2×#41—42no—
110C22×Q8C2311yescharacteristic
111C22×Q8C2311yescharacteristic
112C22×D4C2312yes—
113C22×D4C2311yescharacteristic
114C23×C4C2312yes—
115C23×C4C2311yescharacteristic
116C23×C4C2312yes—
117C23×C4C2311yescharacteristic
118C23×C4C2311yescharacteristic
119C23×C4C2312yes—
120#126—28no—
121#123—44no—
122#123—44no—
123C22⋊Q8—42no—
124#117—42no—
125#117—44no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2311yescharacteristic
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2311yescharacteristic
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×C42—42no—
171C2×C42C2312yes—
172C2×C42C2312yes—
173C2×C42C2312yes—
174C2×C42C2312yes—
175C2×C42C2311yescharacteristic
176C2×C42C2311yescharacteristic
177C2×C42C2312yes—
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180#95C2312yes—
181#95C2312yes—
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—
207C24C2412yes—
208C24C2411yescharacteristic
209#41—416no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—41no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4#4114yes—
240C22×C4C2411yescharacteristic
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4#4114yescomplemented
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—24no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23C2×#4112yes—
319C23H(5,𝔽2)12yes—
320C23H(5,𝔽2)12yes—
321C23C2511yescharacteristic
322Q8—44no—
323Q8—82no—
324D4—44no—
325D4—48no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
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—21no—
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—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4C2×#4114yes—
378C2×C4C2×#4112yes—
379C2×C4C2×#4112yes—
380C2×C4C2×#4112yes—
381C2×C42−1+412yes—
382C2×C42−1+412yes—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22#53814yes—
397C22C22×#4111yescharacteristic
398C22C22×#4111yescharacteristic
399C22C2×H(5,𝔽2)11yescharacteristic
400C22C2×H(5,𝔽2)11yescharacteristic
401C22C2×2−1+411yescharacteristic
402C22C2×2−1+411yescharacteristic
403C22#58511yescharacteristic
404C4—22no—
405C4—22no—
406C4—24no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—41no—
418C4—42no—
419C4—42no—
420C4—42no—
421C2—24no—
422C2—82no—
423C2C2×#53811yescharacteristic
424C2#347511yescharacteristic
425C2#349211yescharacteristic
426C2#350911yescharacteristic
427C2#352211yescharacteristic
428C2#353511yescharacteristic
429C2#355911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC22×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2855C2semidirect
2. #2808C2semidirect
3. #2806C2semidirect
4. #2805C2semidirect
5. #2800C2semidirect
6. #2774C2semidirect
7. #2727C2semidirect
8. #2652C2semidirect
9. #2589C2semidirect
10. #2587C2semidirect
11. #2498C2semidirect
12. #2488C2semidirect
13. #2481C2semidirect
14. #2413C2semidirect
16. #2344C2semidirect
17. #2334C2semidirect
19. C2×#389C2semidirect
282. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
#28552
#28082
#28062
#28052
#28002
#27742
#27272
#26522
#25892
#25872
#24982
#24882
#24812
#24132
#24062
#23442
#23342
#22972
C2×#3892