Finite Group Databasebeta

← #38948

Subgroups — #38948

Order 256.

1189 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic, complemented
3#2861C211yesmaximal, characteristic, complemented
4#2702C214yesmaximal, complemented
5#2699C214yesmaximal, complemented
6#2620C211yesmaximal, characteristic, complemented
7#2619C211yesmaximal, characteristic, complemented
8#2565C211yesmaximal, characteristic, complemented
9#2544C211yesmaximal, characteristic, complemented
10#2503C211yesmaximal, characteristic, complemented
11#2500C211yesmaximal, characteristic, complemented
12#2459C214yesmaximal, complemented
13#2457C214yesmaximal, complemented
14#2436C211yesmaximal, characteristic, complemented
15#2431C211yesmaximal, characteristic, complemented
16#2311C211yesmaximal, characteristic, complemented
17#2310C211yesmaximal, characteristic, complemented
18C4×#124C211yesmaximal, characteristic, complemented
19#2293C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21C23×Q8C2211yescharacteristic
22#553—28no—
23#548—28no—
24C2×#127C2211yescharacteristic
25C2×#126C2212yes—
26C2×#125C2211yescharacteristic
27C2×#124C2211yescharacteristic
28C2×#124C2211yescharacteristic
29C2×#124C2212yes—
30C2×#124C2211yescharacteristic
31C2×#123C2212yes—
32C2×C22⋊Q8C2211yescharacteristic
33C2×C22⋊Q8C2212yes—
34C2×#121C2211yescharacteristic
35C2×#121C2211yescharacteristic
36C2×#121C2211yescharacteristic
37C2×#121C2212yes—
38C2×#121C2212yes—
39C2×C4×Q8C2211yescharacteristic
40C2×C4×D4C2212yes—
41C2×C4×D4C2212yes—
42C2×C4×D4C2211yescharacteristic
43C2×C4×D4C2211yescharacteristic
44C2×#117C2211yescharacteristic
45C2×#117C2211yescharacteristic
46C2×#117C2211yescharacteristic
47C22×C4⋊C4C2211yescharacteristic
48#400C2214yes—
49#398C2214yes—
50#396C2214yes—
51#394C2214yes—
52#394C2214yes—
53#394C2214yes—
54#394C2214yes—
55#393C2214yes—
56#392C2214yes—
57#392C2214yes—
58#391C2211yescharacteristic
59#391C2211yescharacteristic
60#391C2211yescharacteristic
61#391C2211yescharacteristic
62#391C2214yes—
63#391C2211yescharacteristic
64#391C2211yescharacteristic
65#390C2211yescharacteristic
66#390C2211yescharacteristic
67#390C2211yescharacteristic
68#390C2211yescharacteristic
69#390C2212yes—
70#389C2212yes—
71#388C2214yes—
72#388C2214yes—
73#388C2214yes—
74#388C2214yes—
75#388C2212yes—
76#388C2212yes—
77#387C2214yes—
78#386C2214yes—
79#386C2214yes—
80#386C2214yes—
81#386C2214yes—
82#384C2211yescharacteristic
83#382C2211yescharacteristic
84#382C2211yescharacteristic
85#381C2214yes—
86#381C2214yes—
87#380C2214yes—
88C4×C4⋊C4C2211yescharacteristic
89C4×C22⋊C4C2211yescharacteristic
90C4×C22⋊C4C2211yescharacteristic
91C4×C22⋊C4C2214yes—
92C43C2211yescharacteristic
93C2×#41—24no—
94C2×#41—24no—
95C2×#41—42no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—42no—
99C22×Q8C2311yescharacteristic
100C22×Q8C2312yes—
101C22×Q8C2311yescharacteristic
102C22×D4C2311yescharacteristic
103C22×D4C2311yescharacteristic
104C22×D4C2312yes—
105C22×D4C2312yes—
106C22×D4C2312yes—
107C22×D4C2312yes—
108C22×D4C2314yes—
109C22×D4C2311yescharacteristic
110C23×C4C2312yes—
111C23×C4C2312yes—
112C23×C4C2311yescharacteristic
113C23×C4C2311yescharacteristic
114C23×C4C2312yes—
115C23×C4C2312yes—
116#127—42no—
117#126—28no—
118#125—24no—
119#124—24no—
120#124—28no—
121#124—24no—
122#124—42no—
123#123—28no—
124C22⋊Q8—24no—
125C22⋊Q8—28no—
126#121—28no—
127#121—28no—
128#121—24no—
129#121—24no—
130#121—24no—
131C4×Q8—24no—
132C4×D4—28no—
133C4×D4—28no—
134C4×D4—24no—
135C4×D4D418yescomplemented
136#117—24no—
137#117—42no—
138#117—42no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2314yes—
153C2×C4⋊C4C2311yescharacteristic
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⋊C4C2314yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2314yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2314yes—
169C2×C22⋊C4C2314yes—
170C2×C22⋊C4C2314yes—
171C2×C22⋊C4C2314yes—
172C2×C22⋊C4C2311yescharacteristic
173C2×C22⋊C4C2311yescharacteristic
174C2×C42C2311yescharacteristic
175C2×C42C2312yes—
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2312yes—
179C2×C42C2311yescharacteristic
180C2×C42C2312yes—
181C2×C42C2312yes—
182C2×C42C2311yescharacteristic
183#95C2314yes—
184#95C2314yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2314yes—
189#95C2314yes—
190#95C2314yes—
191#95C2314yes—
192#95C2314yes—
193#95C2314yes—
194C24C2412yes—
195C24C2412yes—
196C24C2411yescharacteristic
197C24C2411yescharacteristic
198#41—416no—
199C2×Q8—22no—
200C2×Q8—24no—
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—42no—
207C2×Q8—416no—
208C2×Q8#4114yes—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—42no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4C2×D414yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4C2×D414yes—
244C22×C4C2×D414yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2411yescharacteristic
258C22×C4C2412yes—
259C22×C4C2411yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4C2×D414yes—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—28no—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4C2×D414yes—
293C22⋊C4C2×D414yes—
294C42—24no—
295C42—24no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—42no—
301C42—42no—
302C42C2×D414yes—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23C22×D412yes—
322C23C22×D412yes—
323C23C2511yescharacteristic
324Q8—28no—
325Q8—48no—
326Q8—44no—
327D4—44no—
328D4—48no—
329D4—48no—
330D4—48no—
331D4—44no—
332D4—44no—
333D4—84no—
334D4—88no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
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—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4C22×D412yes—
373C2×C4C22×D412yes—
374C2×C4C22×D412yes—
375C2×C4C22×D412yes—
376C2×C4C22×D412yes—
377C2×C4C2×#4114yes—
378C2×C4C2×#4112yes—
379C2×C4H(5,𝔽2)12yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—44no—
400C22C23×D411yescharacteristic
401C22C22×#4111yescharacteristic
402C22C2×H(5,𝔽2)11yescharacteristic
403C22#58512yes—
404C22#58512yes—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—24no—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4#54614yes—
420C2—42no—
421C2—42no—
422C2—82no—
423C2—82no—
424C2#344911yescharacteristic
425C2C2×#54611yescharacteristic
426C2#350011yescharacteristic
427C2#352412yes—
428C2#355112yes—
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. C2×#553C2semidirect
2. C2×#548C2semidirect
3. #2861C2semidirect
4. #2702C2semidirect
5. #2699C2semidirect
6. #2620C2semidirect
7. #2619C2semidirect
8. #2565C2semidirect
9. #2544C2semidirect
10. #2503C2semidirect
11. #2500C2semidirect
12. #2459C2semidirect
13. #2457C2semidirect
14. #2436C2semidirect
15. #2431C2semidirect
16. #2311C2semidirect
17. #2310C2semidirect
18. C4×#124C2semidirect
19. #2293C2semidirect
135. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#5482
#28612
#27022
#26992
#26202
#26192
#25652
#25442
#25032
#25002
#24592
#24572
#24362
#24312
#23112
#23102
C4×#1242
#22932