Finite Group Databasebeta

← #38484

Subgroups — #38484

Order 256.

900 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2854C211yesmaximal, characteristic
2#2809C211yesmaximal, characteristic, complemented
3#2808C211yesmaximal, characteristic, complemented
4#2805C211yesmaximal, characteristic, complemented
5#2798C212yesmaximal, complemented
6#2732C211yesmaximal, characteristic, complemented
7#2689C212yesmaximal, complemented
8#2650C212yesmaximal, complemented
9#2588C212yesmaximal, complemented
10#2569C212yesmaximal, complemented
11#2560C212yesmaximal, complemented
12#2498C212yesmaximal, complemented
13#2481C212yesmaximal, complemented
14#2444C212yesmaximal
15#2409C212yesmaximal, complemented
16#2359C212yesmaximal, complemented
17#2332C212yesmaximal, complemented
18#2296C211yesmaximal, characteristic, complemented
19C2×#391C211yesmaximal, characteristic, complemented
20C23×Q8C2211yescharacteristic
21C2×#126C2212yes—
22C2×#123C2212yes—
23C2×C22⋊Q8C2212yes—
24C2×C22⋊Q8C2212yes—
25C2×#121C2211yescharacteristic
26C2×C4×D4C2212yes—
27C2×#117C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C42C2211yescharacteristic
31#400C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#400C2212yes—
37#399C2212yes—
38#398C2212yes—
39#398C2212yes—
40#398C2212yes—
41#397C2212yes—
42#397C2212yes—
43#397C2212yes—
44#397C2212yes—
45#396C2212yes—
46#395C2212yes—
47#395C2212yes—
48#394C2212yes—
49#393C2212yes—
50#393C2212yes—
51#391C2212yes—
52#391C2211yescharacteristic
53#391C2212yes—
54#391C2212yes—
55#391C2211yescharacteristic
56#391C2212yes—
57#391C2211yescharacteristic
58#391C2211yescharacteristic
59#390C2211yescharacteristic
60#390C2211yescharacteristic
61#389C2211yescharacteristic
62#389C2211yescharacteristic
63#389C2212yes—
64#389C2211yescharacteristic
65#389C2211yescharacteristic
66#389C2211yescharacteristic
67#389C2211yescharacteristic
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#388C2212yes—
72#388C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#387C2212yes—
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#386C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#381C2212yes—
97#380C2211yescharacteristic
98#380C2212yes—
99#380C2212yes—
100C4×C4⋊C4C2212yes—
101C4×C4⋊C4C2212yes—
102C4×C4⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104C4×C22⋊C4C2212yes—
105C2×#95C2212yes—
106C2×#95C2212yes—
107C22×Q8—24no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8C2312yes—
111C22×Q8C2311yescharacteristic
112C22×Q8C2311yescharacteristic
113C22×D4C2312yes—
114C22×D4C2311yescharacteristic
115C23×C4C2311yescharacteristic
116C23×C4C2312yes—
117C23×C4C2311yescharacteristic
118C23×C4C2311yescharacteristic
119C23×C4C2312yes—
120C23×C4C2312yes—
121#126—44no—
122#123—28no—
123C22⋊Q8—28no—
124C22⋊Q8—44no—
125#121—42no—
126C4×D4—44no—
127#117—24no—
128#117—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
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⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2311yescharacteristic
152C2×C4⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
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⋊C4C2312yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C42—42no—
169C2×C42C2312yes—
170C2×C42C2312yes—
171C2×C42C2312yes—
172C2×C42C2312yes—
173C2×C42C2312yes—
174C2×C42C2311yescharacteristic
175C2×C42C2311yescharacteristic
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178#95C2312yes—
179#95C2312yes—
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
209C2×Q8—24no—
210C2×Q8—22no—
211C2×Q8—24no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—416no—
218C2×Q8#4114yes—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—22no—
223C2×D4—41no—
224C2×D4—41no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4#4114yes—
243C22×C4C2411yescharacteristic
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2411yescharacteristic
258C22×C4C2411yescharacteristic
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—22no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
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—
320C232−1+412yes—
321C23C2511yescharacteristic
322Q8—44no—
323Q8—48no—
324Q8—44no—
325D4—48no—
326D4—82no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
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—21no—
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—21no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4C2×#4112yes—
379C2×C4C2×#4112yes—
380C2×C4C2×#4112yes—
381C2×C4C2×#4114yes—
382C2×C4H(5,𝔽2)12yes—
383C2×C42−1+412yes—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22#53914yes—
398C22C22×#4111yescharacteristic
399C22C22×#4111yescharacteristic
400C22C2×H(5,𝔽2)11yescharacteristic
401C22C2×H(5,𝔽2)11yescharacteristic
402C22C2×2−1+411yescharacteristic
403C22C2×2−1+411yescharacteristic
404C22#58511yescharacteristic
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—24no—
410C4—22no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—42no—
422C2—24no—
423C2—82no—
424C2C2×#53911yescharacteristic
425C2#348011yescharacteristic
426C2#349011yescharacteristic
427C2#351111yescharacteristic
428C2#352311yescharacteristic
429C2#353511yescharacteristic
430C2#356111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2809C2semidirect
3. #2808C2semidirect
4. #2805C2semidirect
5. #2798C2semidirect
6. #2732C2semidirect
7. #2689C2semidirect
8. #2650C2semidirect
9. #2588C2semidirect
10. #2569C2semidirect
11. #2560C2semidirect
12. #2498C2semidirect
13. #2481C2semidirect
15. #2409C2semidirect
16. #2359C2semidirect
17. #2332C2semidirect
18. #2296C2semidirect
19. C2×#391C2semidirect

Maximal subgroups

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

subgroupindex
#28542
#28092
#28082
#28052
#27982
#27322
#26892
#26502
#25882
#25692
#25602
#24982
#24812
#24442
#24092
#23592
#23322
#22962
C2×#3912