Finite Group Databasebeta

← #37391

Subgroups — #37391

Order 256.

1143 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#536C211yesmaximal, characteristic, complemented
3#2862C211yesmaximal, characteristic, complemented
4#2858C211yesmaximal, characteristic, complemented
5#2723C214yesmaximal, complemented
6#2710C214yesmaximal, complemented
7#2706C214yesmaximal, complemented
8#2692C214yesmaximal, complemented
9#2658C211yesmaximal, characteristic, complemented
10#2640C211yesmaximal, characteristic, complemented
11#2547C211yesmaximal, characteristic, complemented
12#2546C211yesmaximal, characteristic, complemented
13#2545C211yesmaximal, characteristic, complemented
14#2544C211yesmaximal, characteristic, complemented
15#2436C211yesmaximal, characteristic, complemented
16#2434C211yesmaximal, characteristic, complemented
17#2432C211yesmaximal, characteristic, complemented
18#2431C211yesmaximal, characteristic
19#2298C211yesmaximal, characteristic, complemented
20C22×#41C2211yescharacteristic
21C23×Q8C2211yescharacteristic
22#542—28no—
23#536—28no—
24C2×C4⋊Q8C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C2×C4⋊Q8C2211yescharacteristic
27C2×C4⋊Q8C2211yescharacteristic
28C2×C4⋊Q8C2211yescharacteristic
29C2×C4⋊Q8C2211yescharacteristic
30C2×#127C2211yescharacteristic
31C2×#126C2212yes—
32C2×#124C2211yescharacteristic
33C2×#124C2211yescharacteristic
34C2×#124C2211yescharacteristic
35C2×#124C2212yes—
36C2×#124C2211yescharacteristic
37C2×#124C2211yescharacteristic
38C2×#124C2211yescharacteristic
39C2×#124C2211yescharacteristic
40C2×#123C2212yes—
41C2×#123C2212yes—
42C2×#123C2212yes—
43C2×C22⋊Q8C2212yes—
44C2×C22⋊Q8C2212yes—
45C2×C22⋊Q8C2211yescharacteristic
46C2×C22⋊Q8C2212yes—
47C2×#121C2211yescharacteristic
48C2×C4×Q8C2211yescharacteristic
49C2×C4×D4C2211yescharacteristic
50C2×#117C2211yescharacteristic
51#399C2214yes—
52#399C2214yes—
53#398C2214yes—
54#397C2214yes—
55#397C2214yes—
56#396C2214yes—
57#395C2214yes—
58#394C2214yes—
59#394C2214yes—
60#393C2214yes—
61#393C2214yes—
62#392C2214yes—
63#391C2211yescharacteristic
64#391C2211yescharacteristic
65#391C2211yescharacteristic
66#391C2211yescharacteristic
67#391C2214yes—
68#391C2211yescharacteristic
69#391C2211yescharacteristic
70#390C2214yes—
71#390C2214yes—
72#390C2211yescharacteristic
73#390C2211yescharacteristic
74#389C2214yes—
75#388C2212yes—
76#388C2212yes—
77#388C2214yes—
78#387C2212yes—
79#387C2212yes—
80#386C2214yes—
81#386C2214yes—
82#386C2214yes—
83#385C2214yes—
84#385C2214yes—
85#382C2214yes—
86#380C2214yes—
87C4×C4⋊C4C2211yescharacteristic
88C4×C4⋊C4C2211yescharacteristic
89C4×C22⋊C4C2211yescharacteristic
90C4×C22⋊C4C2211yescharacteristic
91#376C2211yescharacteristic
92C43C2211yescharacteristic
93C2×#41—24no—
94C2×#41—42no—
95C2×#41—42no—
96C22×Q8—24no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8C2311yescharacteristic
100C22×Q8C2311yescharacteristic
101C22×Q8C2311yescharacteristic
102C22×Q8C2311yescharacteristic
103C22×Q8C2312yes—
104C22×Q8C2312yes—
105C22×D4C2311yescharacteristic
106C22×D4C2311yescharacteristic
107C22×D4C2312yes—
108C22×D4C2312yes—
109C22×D4C2314yes—
110C23×C4C2311yescharacteristic
111C23×C4C2312yes—
112C23×C4C2312yes—
113C23×C4C2312yes—
114C4⋊Q8—24no—
115C4⋊Q8—24no—
116C4⋊Q8—24no—
117C4⋊Q8—42no—
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120#127—42no—
121#126—28no—
122#124—28no—
123#124—24no—
124#124—24no—
125#124—24no—
126#124—42no—
127#124—42no—
128#124—42no—
129#124—42no—
130#123—28no—
131#123—28no—
132#123—28no—
133C22⋊Q8—24no—
134C22⋊Q8—28no—
135C22⋊Q8—28no—
136C22⋊Q8—28no—
137#121—24no—
138C4×Q8—24no—
139C4×D4—24no—
140#117D418yescomplemented
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2314yes—
152C2×C4⋊C4C2314yes—
153C2×C4⋊C4C2314yes—
154C2×C4⋊C4C2312yes—
155C2×C4⋊C4C2311yescharacteristic
156C2×C4⋊C4C2311yescharacteristic
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⋊C4C2314yes—
168C2×C22⋊C4C2314yes—
169C2×C22⋊C4C2314yes—
170C2×C22⋊C4C2314yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2314yes—
173C2×C22⋊C4C2314yes—
174C2×C22⋊C4C2311yescharacteristic
175C2×C22⋊C4C2311yescharacteristic
176C2×C42C2312yes—
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
181C2×C42C2312yes—
182C2×C42C2312yes—
183C2×C42C2311yescharacteristic
184C2×C42C2311yescharacteristic
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2314yes—
192#95C2314yes—
193#95C2314yes—
194#95C2314yes—
195#95C2314yes—
196#95C2314yes—
197C24C2412yes—
198C24C2412yes—
199C24C2411yescharacteristic
200#41—416no—
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—416no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—28no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4C2×D414yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2411yescharacteristic
258C22×C4C2412yes—
259C22×C4C2411yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4C2×D414yes—
275C4⋊C4C2×D414yes—
276C22⋊C4—24no—
277C22⋊C4—24no—
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—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4C2×D414yes—
294C22⋊C4C2×D414yes—
295C42—24no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—24no—
301C42—42no—
302C42C2×D414yes—
303C42C2×D414yes—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23C22×D412yes—
319C23C2511yescharacteristic
320Q8—48no—
321Q8—44no—
322Q8—44no—
323Q8—44no—
324Q8—44no—
325Q8—84no—
326D4—48no—
327D4—44no—
328D4—44no—
329D4—88no—
330D4—84no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—28no—
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—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—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—84no—
368C2×C4C22×D412yes—
369C2×C4C22×D412yes—
370C2×C4C22×D412yes—
371C2×C4C22×D412yes—
372C2×C4C22×D412yes—
373C2×C4C22×D412yes—
374C2×C4H(5,𝔽2)12yes—
375C2×C4H(5,𝔽2)12yes—
376C2×C42−1+414yes—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22C23×D411yescharacteristic
393C22C2×H(5,𝔽2)11yescharacteristic
394C22C2×H(5,𝔽2)11yescharacteristic
395C22C2×2−1+412yes—
396C22#58512yes—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4#53514yes—
413C2—42no—
414C2—82no—
415C2—82no—
416C2C2×#53511yescharacteristic
417C2#349911yescharacteristic
418C2#350011yescharacteristic
419C2#355412yes—
420C2#356112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2431C2
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×#542C2semidirect
2. C2×#536C2semidirect
3. #2862C2semidirect
4. #2858C2semidirect
5. #2723C2semidirect
6. #2710C2semidirect
7. #2706C2semidirect
8. #2692C2semidirect
9. #2658C2semidirect
10. #2640C2semidirect
11. #2547C2semidirect
12. #2546C2semidirect
13. #2545C2semidirect
14. #2544C2semidirect
15. #2436C2semidirect
16. #2434C2semidirect
17. #2432C2semidirect
19. #2298C2semidirect
140. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#5362
#28622
#28582
#27232
#27102
#27062
#26922
#26582
#26402
#25472
#25462
#25452
#25442
#24362
#24342
#24322
#24312
#22982