Finite Group Databasebeta

← #42748

Subgroups — #42748

Order 256.

674 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2836C212yesmaximal
2#2798C212yesmaximal
3#2783C212yesmaximal
4#2766C212yesmaximal, complemented
5#2762C212yesmaximal, complemented
6#2650C212yesmaximal
7#2647C211yesmaximal, characteristic, complemented
8#2645C212yesmaximal, complemented
9#2570C211yesmaximal, characteristic, complemented
10#2550C211yesmaximal, characteristic, complemented
11#2528C212yesmaximal, complemented
12#2509C211yesmaximal, characteristic, complemented
13#2509C211yesmaximal, characteristic, complemented
14#2508C211yesmaximal, characteristic, complemented
15#2495C211yesmaximal, characteristic
16#2494C211yesmaximal, characteristic
17#2494C211yesmaximal, characteristic
18#2441C211yesmaximal, characteristic
19#2439C211yesmaximal, characteristic
20#2401C211yesmaximal, characteristic
21#2370C211yesmaximal, characteristic, complemented
22#2346C211yesmaximal, characteristic
23#2331C211yesmaximal, characteristic, complemented
24C2×C4⋊Q8C2211yescharacteristic
25C2×#125C2211yescharacteristic
26C2×#117C2211yescharacteristic
27#401C2212yes—
28#401C2212yes—
29#401C2211yescharacteristic
30#400C2212yes—
31#400C2212yes—
32#400C2212yes—
33#400C2211yescharacteristic
34#400C2212yes—
35#400C2212yes—
36#400C2212yes—
37#400C2212yes—
38#400C2212yes—
39#400C2212yes—
40#400C2211yescharacteristic
41#400C2211yescharacteristic
42#399C2212yes—
43#398C2212yes—
44#398C2212yes—
45#398C2212yes—
46#398C2212yes—
47#398C2212yes—
48#397C2212yes—
49#397C2212yes—
50#397C2212yes—
51#397C2211yescharacteristic
52#397C2211yescharacteristic
53#397C2211yescharacteristic
54#396C2211yescharacteristic
55#396C2212yes—
56#396C2212yes—
57#395C2212yes—
58#393C2212yes—
59#391C2212yes—
60#391C2212yes—
61#391C2212yes—
62#391C2212yes—
63#389C2211yescharacteristic
64#389C2211yescharacteristic
65#389C2212yes—
66#389C2211yescharacteristic
67#389C2211yescharacteristic
68#389C2211yescharacteristic
69#389C2211yescharacteristic
70#389C2212yes—
71#389C2212yes—
72#389C2211yescharacteristic
73#389C2211yescharacteristic
74#389C2211yescharacteristic
75#389C2211yescharacteristic
76#389C2211yescharacteristic
77#389C2211yescharacteristic
78#389C2212yes—
79#389C2212yes—
80#388C2212yes—
81#388C2212yes—
82#388C2212yes—
83#388C2212yes—
84#388C2212yes—
85#387C2212yes—
86#387C2212yes—
87#387C2212yes—
88#387C2212yes—
89#387C2212yes—
90#387C2212yes—
91#387C2212yes—
92#387C2212yes—
93#387C2212yes—
94#387C2212yes—
95#387C2212yes—
96#387C2212yes—
97#387C2212yes—
98#384C2211yescharacteristic
99#383C2212yes—
100#382C2211yescharacteristic
101#382C2211yescharacteristic
102#382C2211yescharacteristic
103#382C2212yes—
104#382C2211yescharacteristic
105#382C2212yes—
106#380C2211yescharacteristic
107#380C2211yescharacteristic
108#380C2211yescharacteristic
109C4×C4⋊C4C2211yescharacteristic
110C4×C4⋊C4C2211yescharacteristic
111C4×C4⋊C4C2211yescharacteristic
112C4×C4⋊C4C2211yescharacteristic
113C4×C4⋊C4C2211yescharacteristic
114C4×C4⋊C4C2212yes—
115C4×C4⋊C4C2211yescharacteristic
116C4×C4⋊C4C2211yescharacteristic
117C4×C4⋊C4C2211yescharacteristic
118C4×C22⋊C4C2211yescharacteristic
119C4×C22⋊C4C2211yescharacteristic
120C4×C22⋊C4C2211yescharacteristic
121#376C2211yescharacteristic
122#376C2212yes—
123C22×Q8C2312yes—
124C23×C4C2311yescharacteristic
125C4⋊Q8—24no—
126#125—24no—
127#117—24no—
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
155C2×C4⋊C4C2312yes—
156C2×C4⋊C4C2312yes—
157C2×C4⋊C4C2312yes—
158C2×C4⋊C4C2311yescharacteristic
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2311yescharacteristic
161C2×C22⋊C4C2311yescharacteristic
162C2×C22⋊C4C2311yescharacteristic
163C2×C22⋊C4C2311yescharacteristic
164C2×C22⋊C4C2311yescharacteristic
165C2×C22⋊C4C2311yescharacteristic
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C42C2312yes—
170C2×C42C2311yescharacteristic
171C2×C42C2311yescharacteristic
172C2×C42C2311yescharacteristic
173C2×C42C2311yescharacteristic
174C2×C42C2311yescharacteristic
175C2×C42C2311yescharacteristic
176C2×C42C2312yes—
177C2×C42C2312yes—
178C2×C42C2312yes—
179C2×C42C2312yes—
180C2×C42C2312yes—
181#95C2311yescharacteristic
182#95C2312yes—
183#95C2312yes—
184#95C2311yescharacteristic
185#95C2311yescharacteristic
186#95C2312yes—
187#95C2311yescharacteristic
188#95C2311yescharacteristic
189#95C2311yescharacteristic
190#95C2311yescharacteristic
191#95C2311yescharacteristic
192#95C2311yescharacteristic
193#95C2311yescharacteristic
194#95C2311yescharacteristic
195#95C2311yescharacteristic
196#95C2312yes—
197#95C2312yes—
198#95C2311yescharacteristic
199#95C2311yescharacteristic
200#95C2312yes—
201#95C2311yescharacteristic
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2311yescharacteristic
206#95C2312yes—
207#95C2312yes—
208#95C2312yes—
209#95C2312yes—
210#95C2312yes—
211#95C2312yes—
212#95C2312yes—
213#95C2312yes—
214#95C2312yes—
215#95C2312yes—
216#95C2312yes—
217#95C2312yes—
218#95C2312yes—
219#95C2312yes—
220#95C2312yes—
221#95C2312yes—
222#95C2312yes—
223#95C2312yes—
224#95C2312yes—
225C24C2411yescharacteristic
226C2×Q8—24no—
227C2×Q8—42no—
228C2×Q8—42no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4C2411yescharacteristic
233C22×C4C2411yescharacteristic
234C22×C4C2411yescharacteristic
235C22×C4C2411yescharacteristic
236C22×C4C2411yescharacteristic
237C22×C4C2411yescharacteristic
238C22×C4C2412yes—
239C22×C4C2411yescharacteristic
240C22×C4C2411yescharacteristic
241C22×C4C2411yescharacteristic
242C22×C4C2411yescharacteristic
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2411yescharacteristic
248C22×C4C2412yes—
249C22×C4C2411yescharacteristic
250C22×C4C2411yescharacteristic
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2411yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
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—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4#4114yes—
285C22⋊C4—24no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C42—24no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23C2511yescharacteristic
315Q8—84no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
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—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—44no—
379C2×C4C2×#4114yes—
380C2×C4C2×#4112yes—
381C2×C42−1+412yes—
382C2×C42−1+412yes—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22C22×#4111yescharacteristic
391C22C2×2−1+411yescharacteristic
392C22C2×2−1+411yescharacteristic
393C22#58511yescharacteristic
394C22#58511yescharacteristic
395C22#58511yescharacteristic
396C22#58511yescharacteristic
397C4—24no—
398C4—41no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—41no—
404C4—41no—
405C4—42no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4#54114yes—
419C2—81no—
420C2C2×#54111yescharacteristic
421C2#352111yescharacteristic
422C2#352411yescharacteristic
423C2#356511yescharacteristic
424C2#356511yescharacteristic
425C2#356511yescharacteristic
426C2#356511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC24C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2766C2semidirect
5. #2762C2semidirect
7. #2647C2semidirect
8. #2645C2semidirect
9. #2570C2semidirect
10. #2550C2semidirect
11. #2528C2semidirect
12. #2509C2semidirect
13. #2509C2semidirect
14. #2508C2semidirect
21. #2370C2semidirect
23. #2331C2semidirect

Maximal subgroups

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

subgroupindex
#28362
#27982
#27832
#27662
#27622
#26502
#26472
#26452
#25702
#25502
#25282
#25092
#25082
#24952
#24942
#24412
#24392
#24012
#23702
#23462
#23312