Finite Group Databasebeta

← #53492

Subgroups — #53492

Order 256.

860 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#2836C212yesmaximal, complemented
2#2823C211yesmaximal, characteristic, complemented
3#2792C211yesmaximal, characteristic, complemented
4#2729C212yesmaximal, complemented
5#2710C212yesmaximal, complemented
6#2700C212yesmaximal, complemented
7#2692C212yesmaximal, complemented
8#2691C212yesmaximal, complemented
9#2683C212yesmaximal, complemented
10#2681C212yesmaximal, complemented
11#2680C211yesmaximal, characteristic, complemented
12#2649C212yesmaximal, complemented
13#2638C211yesmaximal, characteristic, complemented
14#2582C211yesmaximal, characteristic, complemented
15#2549C212yesmaximal, complemented
16#2543C211yesmaximal, characteristic
17#2542C211yesmaximal, characteristic, complemented
18#2469C212yesmaximal, complemented
19#2426C212yesmaximal, complemented
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126C2212yes—
22C2×#124C2212yes—
23C2×#123C2212yes—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2212yes—
26C2×C22⋊Q8C2212yes—
27C2×C22⋊Q8C2212yes—
28C2×C22⋊Q8C2212yes—
29C2×C22⋊Q8C2212yes—
30C2×#121C2211yescharacteristic
31C2×#121C2212yes—
32C2×C4×Q8C2212yes—
33C2×C4×D4C2211yescharacteristic
34#400C2212yes—
35#400C2212yes—
36#399C2212yes—
37#399C2212yes—
38#398C2212yes—
39#398C2211yescharacteristic
40#398C2212yes—
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2211yescharacteristic
45#397C2211yescharacteristic
46#397C2212yes—
47#397C2212yes—
48#397C2211yescharacteristic
49#397C2212yes—
50#396C2212yes—
51#396C2212yes—
52#396C2212yes—
53#396C2212yes—
54#396C2212yes—
55#396C2212yes—
56#395C2211yescharacteristic
57#395C2212yes—
58#395C2212yes—
59#395C2212yes—
60#394C2211yescharacteristic
61#394C2212yes—
62#394C2212yes—
63#394C2212yes—
64#394C2212yes—
65#394C2212yes—
66#393C2211yescharacteristic
67#393C2212yes—
68#393C2212yes—
69#392C2211yescharacteristic
70#391C2212yes—
71#391C2212yes—
72#391C2212yes—
73#390C2212yes—
74#390C2211yescharacteristic
75#390C2212yes—
76#390C2211yescharacteristic
77#389C2212yes—
78#389C2212yes—
79#389C2212yes—
80#389C2212yes—
81#389C2212yes—
82#388C2212yes—
83#388C2212yes—
84#388C2212yes—
85#388C2212yes—
86#388C2212yes—
87#388C2212yes—
88#387C2212yes—
89#387C2212yes—
90#387C2212yes—
91#387C2212yes—
92#386C2212yes—
93#386C2212yes—
94#386C2212yes—
95#386C2212yes—
96#385C2212yes—
97#385C2212yes—
98#385C2212yes—
99#385C2212yes—
100#384C2211yescharacteristic
101#384C2211yescharacteristic
102#381C2211yescharacteristic
103#380C2212yes—
104C4×C4⋊C4C2211yescharacteristic
105C4×C22⋊C4C2212yes—
106C2×#95C2211yescharacteristic
107C22×Q8C2312yes—
108C22×Q8C2312yes—
109C22×Q8C2311yescharacteristic
110C22×D4C2311yescharacteristic
111C22×D4C2312yes—
112C22×D4C2312yes—
113C22×D4C2311yescharacteristic
114C23×C4C2311yescharacteristic
115C23×C4C2311yescharacteristic
116C23×C4C2312yes—
117C23×C4C2312yes—
118C4⋊Q8—42no—
119#126—44no—
120#124—44no—
121#123—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C22⋊Q8—42no—
128#121—42no—
129#121—44no—
130C4×Q8—44no—
131C4×D4—42no—
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⋊C4C2311yescharacteristic
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2311yescharacteristic
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2311yescharacteristic
155C2×C4⋊C4C2311yescharacteristic
156C2×C4⋊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⋊C4C2311yescharacteristic
166C2×C22⋊C4C2311yescharacteristic
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2312yes—
178C2×C22⋊C4C2312yes—
179C2×C42C2311yescharacteristic
180C2×C42C2311yescharacteristic
181C2×C42C2311yescharacteristic
182C2×C42C2312yes—
183C2×C42C2312yes—
184C2×C42C2312yes—
185#95C2312yes—
186#95C2311yescharacteristic
187#95C2311yescharacteristic
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2311yescharacteristic
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208C24C2411yescharacteristic
209C24C2412yes—
210C24C2411yescharacteristic
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C2411yescharacteristic
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2411yescharacteristic
250C22×C4C2411yescharacteristic
251C22×C4C2412yes—
252C22×C4C2411yescharacteristic
253C22×C4C2412yes—
254C22×C4C2411yescharacteristic
255C22×C4C2412yes—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—42no—
307C42—42no—
308C42—42no—
309C23—22no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23—21no—
322C23—22no—
323C23—22no—
324C23—22no—
325C232−1+412yes—
326C23C2511yescharacteristic
327Q8—48no—
328Q8—84no—
329Q8—82no—
330D4—44no—
331D4—84no—
332D4—84no—
333D4—82no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
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—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4H(5,𝔽2)12yes—
379C2×C4H(5,𝔽2)12yes—
380C2×C4H(5,𝔽2)14yes—
381C2×C42−1+414yes—
382C22—22no—
383C22—24no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—42no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22C2×H(5,𝔽2)11yescharacteristic
400C22C2×H(5,𝔽2)11yescharacteristic
401C22C2×H(5,𝔽2)12yes—
402C22C2×2−1+412yes—
403C22C2×2−1+411yescharacteristic
404C4—24no—
405C4—22no—
406C4—24no—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—42no—
420C2—42no—
421C2—81no—
422C2—82no—
423C2#353612yes—
424C2#353611yescharacteristic
425C2#354311yescharacteristic
426C2#355011yescharacteristic
427C2#355212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radical#2543C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2836C2semidirect
2. #2823C2semidirect
3. #2792C2semidirect
4. #2729C2semidirect
5. #2710C2semidirect
6. #2700C2semidirect
7. #2692C2semidirect
8. #2691C2semidirect
9. #2683C2semidirect
10. #2681C2semidirect
11. #2680C2semidirect
12. #2649C2semidirect
13. #2638C2semidirect
14. #2582C2semidirect
15. #2549C2semidirect
17. #2542C2semidirect
18. #2469C2semidirect
19. #2426C2semidirect

Maximal subgroups

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

subgroupindex
#28362
#28232
#27922
#27292
#27102
#27002
#26922
#26912
#26832
#26812
#26802
#26492
#26382
#25822
#25492
#25432
#25422
#24692
#24262