Finite Group Databasebeta

← #51228

Subgroups — #51228

Order 256.

703 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2798C212yesmaximal, complemented
2#2762C212yesmaximal, complemented
3#2759C212yesmaximal, complemented
4#2751C212yesmaximal
5#2745C212yesmaximal, complemented
6#2742C212yesmaximal, complemented
7#2642C211yesmaximal, characteristic, complemented
8#2633C211yesmaximal, characteristic
9#2607C212yesmaximal, complemented
10#2605C211yesmaximal, characteristic, complemented
11#2563C211yesmaximal, characteristic
12#2530C212yesmaximal, complemented
13#2528C212yesmaximal, complemented
14#2527C212yesmaximal, complemented
15#2516C212yesmaximal
16#2493C212yesmaximal, complemented
17#2381C211yesmaximal, characteristic, complemented
18#2353C211yesmaximal, characteristic
19#2336C211yesmaximal, characteristic, complemented
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#401C2212yes—
24#401C2212yes—
25#400C2212yes—
26#400C2212yes—
27#400C2212yes—
28#400C2212yes—
29#400C2212yes—
30#400C2212yes—
31#400C2212yes—
32#400C2212yes—
33#399C2211yescharacteristic
34#399C2212yes—
35#399C2211yescharacteristic
36#398C2212yes—
37#397C2211yescharacteristic
38#397C2211yescharacteristic
39#397C2212yes—
40#397C2212yes—
41#397C2212yes—
42#395C2212yes—
43#395C2212yes—
44#394C2212yes—
45#393C2212yes—
46#393C2212yes—
47#393C2212yes—
48#393C2212yes—
49#391C2212yes—
50#391C2212yes—
51#391C2212yes—
52#391C2212yes—
53#391C2212yes—
54#391C2212yes—
55#391C2212yes—
56#390C2211yescharacteristic
57#390C2211yescharacteristic
58#389C2212yes—
59#389C2212yes—
60#388C2212yes—
61#388C2212yes—
62#388C2212yes—
63#388C2212yes—
64#388C2212yes—
65#388C2212yes—
66#388C2212yes—
67#387C2212yes—
68#387C2212yes—
69#387C2212yes—
70#387C2212yes—
71#387C2212yes—
72#387C2212yes—
73#387C2212yes—
74#387C2212yes—
75#387C2212yes—
76#387C2212yes—
77#387C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2212yes—
84#386C2212yes—
85#385C2212yes—
86#385C2212yes—
87#385C2212yes—
88#383C2211yescharacteristic
89#383C2211yescharacteristic
90#383C2212yes—
91#382C2211yescharacteristic
92#382C2211yescharacteristic
93#381C2212yes—
94#381C2211yescharacteristic
95#381C2212yes—
96#380C2212yes—
97#380C2212yes—
98#380C2211yescharacteristic
99C4×C4⋊C4C2211yescharacteristic
100C4×C4⋊C4C2212yes—
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2211yescharacteristic
104#376C2211yescharacteristic
105#376C2212yes—
106C2×#95C2211yescharacteristic
107C22×Q8C2312yes—
108C22×Q8C2312yes—
109C22×D4C2311yescharacteristic
110C23×C4C2311yescharacteristic
111C23×C4C2311yescharacteristic
112C23×C4C2312yes—
113C4⋊Q8—24no—
114#125—24no—
115C4×D4—24no—
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2312yes—
118C2×C4⋊C4C2312yes—
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C22⋊C4C2311yescharacteristic
136C2×C22⋊C4C2311yescharacteristic
137C2×C22⋊C4C2312yes—
138C2×C22⋊C4C2312yes—
139C2×C22⋊C4C2312yes—
140C2×C22⋊C4C2312yes—
141C2×C22⋊C4C2312yes—
142C2×C22⋊C4C2312yes—
143C2×C22⋊C4C2312yes—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C42C2312yes—
149C2×C42C2312yes—
150C2×C42C2312yes—
151C2×C42C2312yes—
152C2×C42C2312yes—
153C2×C42C2312yes—
154C2×C42C2311yescharacteristic
155C2×C42C2311yescharacteristic
156C2×C42C2311yescharacteristic
157#95C2311yescharacteristic
158#95C2311yescharacteristic
159#95C2312yes—
160#95C2311yescharacteristic
161#95C2312yes—
162#95C2312yes—
163#95C2312yes—
164#95C2312yes—
165#95C2312yes—
166#95C2311yescharacteristic
167#95C2312yes—
168#95C2312yes—
169#95C2312yes—
170#95C2312yes—
171#95C2312yes—
172#95C2312yes—
173#95C2312yes—
174#95C2312yes—
175#95C2312yes—
176#95C2312yes—
177#95C2312yes—
178#95C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2311yescharacteristic
188#95C2311yescharacteristic
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197C24C2411yescharacteristic
198C24C2411yescharacteristic
199C2×Q8—24no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4C2412yes—
216C22×C4C2412yes—
217C22×C4C2411yescharacteristic
218C22×C4C2412yes—
219C22×C4C2412yes—
220C22×C4C2412yes—
221C22×C4C2412yes—
222C22×C4C2412yes—
223C22×C4C2412yes—
224C22×C4C2412yes—
225C22×C4C2411yescharacteristic
226C22×C4C2412yes—
227C22×C4C2412yes—
228C22×C4C2412yes—
229C22×C4C2411yescharacteristic
230C22×C4C2411yescharacteristic
231C22×C4C2411yescharacteristic
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—24no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C22⋊C4—24no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—41no—
265C42—41no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42#4114yes—
273C23—21no—
274C23—22no—
275C23—21no—
276C23—22no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—22no—
281C23—21no—
282C232−1+412yes—
283C23C2511yescharacteristic
284Q8—84no—
285Q8—84no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4C2×#4112yes—
335C2×C4C2×#4112yes—
336C2×C4C2×#4112yes—
337C22—22no—
338C22—24no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—42no—
347C22C22×#4111yescharacteristic
348C22C2×2−1+411yescharacteristic
349C22#58511yescharacteristic
350C22#58512yes—
351C22#58512yes—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—42no—
369C2—42no—
370C2—81no—
371C2#349511yescharacteristic
372C2#351011yescharacteristic
373C2#351011yescharacteristic
374C2#356412yes—
375C2#356612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC22×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2798C2semidirect
2. #2762C2semidirect
3. #2759C2semidirect
5. #2745C2semidirect
6. #2742C2semidirect
7. #2642C2semidirect
9. #2607C2semidirect
10. #2605C2semidirect
12. #2530C2semidirect
13. #2528C2semidirect
14. #2527C2semidirect
16. #2493C2semidirect
17. #2381C2semidirect
19. #2336C2semidirect

Maximal subgroups

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

subgroupindex
#27982
#27622
#27592
#27512
#27452
#27422
#26422
#26332
#26072
#26052
#25632
#25302
#25282
#25272
#25162
#24932
#23812
#23532
#23362