Finite Group Databasebeta

← #44449

Subgroups — #44449

Order 256.

1011 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic
2C2×#544C211yesmaximal, characteristic, complemented
3#2729C212yesmaximal, complemented
4#2720C212yesmaximal, complemented
5#2713C212yesmaximal, complemented
6#2713C212yesmaximal, complemented
7#2564C211yesmaximal, characteristic
8#2549C212yesmaximal, complemented
9#2506C212yesmaximal, complemented
10#2502C211yesmaximal, characteristic, complemented
11#2492C212yesmaximal, complemented
12#2473C212yesmaximal, complemented
13#2450C212yesmaximal
14#2446C212yesmaximal, complemented
15#2438C212yesmaximal, complemented
16C2×#400C211yesmaximal, characteristic, complemented
17#2368C211yesmaximal, characteristic, complemented
18#2359C212yesmaximal
19#2322C211yesmaximal, characteristic
20#554—28no—
21#544—28no—
22C2×C4⋊Q8C2212yes—
23C2×C4⋊Q8C2212yes—
24C2×C4⋊Q8C2211yescharacteristic
25C2×#125C2212yes—
26C2×#125C2212yes—
27C2×#125C2212yes—
28C2×#123C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×C22⋊Q8C2212yes—
31C2×C22⋊Q8C2212yes—
32C2×C22⋊Q8C2212yes—
33C2×C22⋊Q8C2211yescharacteristic
34C2×C22⋊Q8C2211yescharacteristic
35C2×C4×Q8C2211yescharacteristic
36C2×C4×D4C2211yescharacteristic
37C2×C4×D4C2212yes—
38C2×#117C2212yes—
39C22×C4⋊C4C2211yescharacteristic
40C22×C4⋊C4C2211yescharacteristic
41C22×C4⋊C4C2211yescharacteristic
42#400C2211yescharacteristic
43#400C2212yes—
44#400C2212yes—
45#400C2212yes—
46#400C2212yes—
47#400C2212yes—
48#400C2212yes—
49#399C2211yescharacteristic
50#398C2212yes—
51#398C2212yes—
52#398C2212yes—
53#398C2212yes—
54#398C2212yes—
55#398C2212yes—
56#398C2212yes—
57#396C2212yes—
58#395C2211yescharacteristic
59#395C2212yes—
60#395C2212yes—
61#395C2212yes—
62#393C2211yescharacteristic
63#393C2212yes—
64#391C2212yes—
65#391C2212yes—
66#391C2212yes—
67#391C2212yes—
68#391C2212yes—
69#391C2212yes—
70#389C2212yes—
71#389C2212yes—
72#389C2212yes—
73#389C2212yes—
74#389C2212yes—
75#388C2212yes—
76#388C2212yes—
77#388C2212yes—
78#387C2212yes—
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2212yes—
84#387C2212yes—
85#387C2212yes—
86#387C2212yes—
87#386C2212yes—
88#386C2211yescharacteristic
89#385C2212yes—
90#385C2212yes—
91#385C2211yescharacteristic
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#385C2212yes—
97#384C2212yes—
98#381C2211yescharacteristic
99#380C2212yes—
100#380C2212yes—
101C4×C4⋊C4C2212yes—
102C4×C4⋊C4C2212yes—
103C4×C22⋊C4C2211yescharacteristic
104C4×C22⋊C4C2212yes—
105C4×C22⋊C4C2212yes—
106C2×#95C2211yescharacteristic
107C2×#95C2212yes—
108C2×#95C2212yes—
109C22×Q8C2312yes—
110C22×Q8C2311yescharacteristic
111C22×Q8C2312yes—
112C22×Q8C2311yescharacteristic
113C22×D4C2311yescharacteristic
114C23×C4C2311yescharacteristic
115C23×C4C2312yes—
116C23×C4C2312yes—
117C23×C4C2311yescharacteristic
118C23×C4C2311yescharacteristic
119C23×C4C2312yes—
120C23×C4C2311yescharacteristic
121C4⋊Q8—28no—
122C4⋊Q8—28no—
123C4⋊Q8—24no—
124#125—28no—
125#125—28no—
126#125—28no—
127#123Q818yescomplemented
128C22⋊Q8—24no—
129C22⋊Q8—24no—
130C22⋊Q8—28no—
131C22⋊Q8—28no—
132C22⋊Q8—28no—
133C22⋊Q8D418yes—
134C4×Q8—24no—
135C4×D4—24no—
136C4×D4—28no—
137#117—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2311yescharacteristic
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⋊C4C2311yescharacteristic
154C2×C4⋊C4C2312yes—
155C2×C4⋊C4C2312yes—
156C2×C4⋊C4C2312yes—
157C2×C4⋊C4C2312yes—
158C2×C4⋊C4C2312yes—
159C2×C4⋊C4C2312yes—
160C2×C4⋊C4C2312yes—
161C2×C4⋊C4C2312yes—
162C2×C4⋊C4C2312yes—
163C2×C4⋊C4C2312yes—
164C2×C4⋊C4C2312yes—
165C2×C4⋊C4C2312yes—
166C2×C4⋊C4C2312yes—
167C2×C4⋊C4C2311yescharacteristic
168C2×C4⋊C4C2312yes—
169C2×C4⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2311yescharacteristic
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2311yescharacteristic
174C2×C22⋊C4C2311yescharacteristic
175C2×C22⋊C4C2312yes—
176C2×C22⋊C4C2312yes—
177C2×C22⋊C4C2312yes—
178C2×C22⋊C4C2312yes—
179C2×C22⋊C4C2311yescharacteristic
180C2×C22⋊C4C2312yes—
181C2×C42C2312yes—
182C2×C42C2312yes—
183C2×C42C2312yes—
184C2×C42C2311yescharacteristic
185C2×C42C2312yes—
186C2×C42C2312yes—
187C2×C42C2312yes—
188#95C2312yes—
189#95C2311yescharacteristic
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2311yescharacteristic
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207#95C2312yes—
208#95C2312yes—
209#95C2312yes—
210#95C2312yes—
211#95C2312yes—
212#95C2312yes—
213#95C2312yes—
214#95C2312yes—
215#95C2312yes—
216C24C2411yescharacteristic
217C24C2411yescharacteristic
218C2×Q8—24no—
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8C2×D414yes—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4C2×Q814yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4C2×D414yes—
248C22×C4C2×Q814yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C22×C4C2412yes—
252C22×C4C2411yescharacteristic
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2412yes—
256C22×C4C2412yes—
257C22×C4C2411yescharacteristic
258C22×C4C2412yes—
259C22×C4C2411yescharacteristic
260C22×C4C2412yes—
261C22×C4C2411yescharacteristic
262C22×C4C2412yes—
263C22×C4C2412yes—
264C22×C4C2411yescharacteristic
265C22×C4C2412yes—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4C2×D414yes—
293C4⋊C4C2×D418yes—
294C4⋊C4C2×Q818yes—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—24no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4C2×D418yes—
304C22⋊C4C2×Q814yes—
305C22⋊C4C2×Q818yes—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—22no—
310C42—42no—
311C42—42no—
312C42—42no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—22no—
319C23C22×D412yes—
320C23C22×Q812yes—
321C23C22×Q812yes—
322C232−1+412yes—
323C23C2511yescharacteristic
324Q8—44no—
325Q8—82no—
326Q8—84no—
327Q8—84no—
328D4—28no—
329C2×C4—28no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
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—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4C22×D414yes—
377C2×C4C22×D414yes—
378C2×C4C22×D412yes—
379C2×C4C22×D412yes—
380C2×C4C22×Q814yes—
381C2×C4C22×Q814yes—
382C2×C4C22×Q812yes—
383C22—22no—
384C22—22no—
385C22—24no—
386C22—24no—
387C22—22no—
388C22—24no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22D4×Q814yes—
393C22C23×D411yescharacteristic
394C22C23×Q811yescharacteristic
395C22C2×2−1+411yescharacteristic
396C22#58512yes—
397C22#58512yes—
398C4—24no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—24no—
404C4—24no—
405C4—41no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—42no—
415C2—24no—
416C2—42no—
417C2C2×D4×Q811yescharacteristic
418C2#348511yescharacteristic
419C2#350011yescharacteristic
420C2#350412yes—
421C2#356412yes—
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
2. C2×#544C2semidirect
3. #2729C2semidirect
4. #2720C2semidirect
5. #2713C2semidirect
6. #2713C2semidirect
8. #2549C2semidirect
9. #2506C2semidirect
10. #2502C2semidirect
11. #2492C2semidirect
12. #2473C2semidirect
14. #2446C2semidirect
15. #2438C2semidirect
16. C2×#400C2semidirect
17. #2368C2semidirect
127. #123Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#5442
#27292
#27202
#27132
#25642
#25492
#25062
#25022
#24922
#24732
#24502
#24462
#24382
C2×#4002
#23682
#23592
#23222