Finite Group Databasebeta

← #42167

Subgroups — #42167

Order 256.

1023 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552C212yesmaximal, complemented
2#2823C211yesmaximal, characteristic, complemented
3#2822C211yesmaximal, characteristic, complemented
4#2726C212yesmaximal, complemented
5#2725C212yesmaximal, complemented
6#2704C212yesmaximal, complemented
7#2702C212yesmaximal, complemented
8#2637C212yesmaximal, complemented
9#2629C212yesmaximal, complemented
10#2618C211yesmaximal, characteristic
11#2609C211yesmaximal, characteristic, complemented
12#2582C211yesmaximal, characteristic, complemented
13#2450C212yesmaximal, complemented
14#2438C212yesmaximal, complemented
15#2425C212yesmaximal
16#2413C212yesmaximal, complemented
17#2413C212yesmaximal, complemented
18C2×#398C211yesmaximal, characteristic, complemented
19#2317C211yesmaximal, characteristic, complemented
20#552—216no—
21C2×#126C2212yes—
22C2×#126C2212yes—
23C2×#125C2212yes—
24C2×#125C2212yes—
25C2×#125C2212yes—
26C2×#123C2212yes—
27C2×#123C2212yes—
28C2×#123C2212yes—
29C2×C22⋊Q8C2212yes—
30C2×C22⋊Q8C2212yes—
31C2×#121C2211yescharacteristic
32C2×C4×Q8C2212yes—
33C2×C4×D4C2212yes—
34C2×#117C2212yes—
35C2×#117C2212yes—
36C2×#117C2211yescharacteristic
37C2×#117C2211yescharacteristic
38C22×C4⋊C4C2212yes—
39C22×C4⋊C4C2211yescharacteristic
40C22×C4⋊C4C2211yescharacteristic
41#400C2212yes—
42#400C2212yes—
43#398C2211yescharacteristic
44#398C2211yescharacteristic
45#398C2212yes—
46#398C2212yes—
47#398C2212yes—
48#398C2212yes—
49#398C2211yescharacteristic
50#398C2212yes—
51#398C2211yescharacteristic
52#398C2212yes—
53#398C2212yes—
54#398C2212yes—
55#398C2212yes—
56#398C2212yes—
57#398C2211yescharacteristic
58#398C2211yescharacteristic
59#397C2212yes—
60#396C2212yes—
61#396C2212yes—
62#396C2212yes—
63#396C2212yes—
64#396C2212yes—
65#396C2212yes—
66#396C2212yes—
67#396C2212yes—
68#395C2212yes—
69#394C2211yescharacteristic
70#394C2212yes—
71#394C2211yescharacteristic
72#393C2212yes—
73#391C2212yes—
74#391C2212yes—
75#389C2212yes—
76#389C2212yes—
77#389C2212yes—
78#389C2212yes—
79#389C2212yes—
80#389C2212yes—
81#389C2212yes—
82#389C2212yes—
83#388C2212yes—
84#388C2212yes—
85#388C2212yes—
86#388C2212yes—
87#388C2212yes—
88#387C2212yes—
89#387C2212yes—
90#387C2212yes—
91#386C2212yes—
92#386C2212yes—
93#386C2212yes—
94#385C2212yes—
95#385C2212yes—
96#385C2212yes—
97#385C2212yes—
98#384C2211yescharacteristic
99#384C2211yescharacteristic
100#382C2211yescharacteristic
101#382C2211yescharacteristic
102#381C2212yes—
103#380C2211yescharacteristic
104#380C2212yes—
105#380C2212yes—
106C2×#95C2212yes—
107C2×#95C2211yescharacteristic
108C22×Q8C2312yes—
109C22×D4C2312yes—
110C22×D4C2311yescharacteristic
111C23×C4C2312yes—
112C23×C4C2312yes—
113C23×C4C2312yes—
114C23×C4C2311yescharacteristic
115C23×C4C2311yescharacteristic
116C23×C4C2311yescharacteristic
117#126—28no—
118#126—28no—
119#125—28no—
120#125—28no—
121#125—28no—
122#123—28no—
123#123—28no—
124#123—44no—
125C22⋊Q8—28no—
126C22⋊Q8—28no—
127#121—42no—
128C4×Q8—28no—
129C4×D4—28no—
130#117—28no—
131#117—28no—
132#117—42no—
133#117D418yescomplemented
134C2×C4⋊C4—28no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
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⋊C4C2312yes—
152C2×C4⋊C4C2311yescharacteristic
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2311yescharacteristic
155C2×C4⋊C4C2312yes—
156C2×C4⋊C4C2311yescharacteristic
157C2×C4⋊C4C2311yescharacteristic
158C2×C4⋊C4C2311yescharacteristic
159C2×C4⋊C4C2311yescharacteristic
160C2×C4⋊C4C2311yescharacteristic
161C2×C4⋊C4C2311yescharacteristic
162C2×C4⋊C4C2312yes—
163C2×C4⋊C4C2312yes—
164C2×C4⋊C4C2312yes—
165C2×C4⋊C4C2312yes—
166C2×C4⋊C4C2312yes—
167C2×C4⋊C4C2312yes—
168C2×C4⋊C4C2312yes—
169C2×C4⋊C4C2312yes—
170C2×C4⋊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×C22⋊C4C2312yes—
180C2×C22⋊C4C2312yes—
181C2×C22⋊C4C2312yes—
182C2×C22⋊C4C2312yes—
183C2×C22⋊C4C2312yes—
184C2×C22⋊C4C2312yes—
185C2×C22⋊C4C2312yes—
186C2×C42C2312yes—
187C2×C42C2312yes—
188C2×C42C2312yes—
189C2×C42C2312yes—
190C2×C42C2311yescharacteristic
191C2×C42C2311yescharacteristic
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2311yescharacteristic
199#95C2312yes—
200#95C2312yes—
201#95C2311yescharacteristic
202#95C2311yescharacteristic
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208#95C2312yes—
209#95C2312yes—
210#95C2312yes—
211#95C2312yes—
212#95C2312yes—
213#95C2312yes—
214#95C2312yes—
215C24C2412yes—
216C24C2411yescharacteristic
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—42no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—41no—
224C2×D4—42no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C2×D414yes—
241C22×C4C2412yes—
242C22×C4C2411yescharacteristic
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2411yescharacteristic
246C22×C4C2411yescharacteristic
247C22×C4C2411yescharacteristic
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2412yes—
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
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—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4C2×D418yes—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4C2×D418yes—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—42no—
311C42C2×D414yes—
312C42C2×D414yes—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23C22×D412yes—
323C23H(5,𝔽2)12yes—
324C232−1+412yes—
325C23C2511yescharacteristic
326Q8—48no—
327D4—48no—
328D4—82no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
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—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
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—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—82no—
377C2×C4C22×D414yes—
378C2×C4C22×D414yes—
379C2×C4C22×D412yes—
380C2×C4C22×D412yes—
381C2×C42−1+414yes—
382C22—24no—
383C22—22no—
384C22—22no—
385C22—24no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22#53714yes—
394C22C23×D411yescharacteristic
395C22C2×H(5,𝔽2)11yescharacteristic
396C22C2×2−1+411yescharacteristic
397C22C2×2−1+412yes—
398C22#58512yes—
399C4—22no—
400C4—22no—
401C4—24no—
402C4—24no—
403C4—24no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—42no—
416C2—24no—
417C2—82no—
418C2C2×#53711yescharacteristic
419C2#348912yes—
420C2#353311yescharacteristic
421C2#354411yescharacteristic
422C2#356312yes—
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
1. C2×#552C2semidirect
2. #2823C2semidirect
3. #2822C2semidirect
4. #2726C2semidirect
5. #2725C2semidirect
6. #2704C2semidirect
7. #2702C2semidirect
8. #2637C2semidirect
9. #2629C2semidirect
11. #2609C2semidirect
12. #2582C2semidirect
13. #2450C2semidirect
14. #2438C2semidirect
16. #2413C2semidirect
17. #2413C2semidirect
18. C2×#398C2semidirect
19. #2317C2semidirect
133. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
#28232
#28222
#27262
#27252
#27042
#27022
#26372
#26292
#26182
#26092
#25822
#24502
#24382
#24252
#24132
C2×#3982
#23172