Finite Group Databasebeta

← #25575

Subgroups — #25575

Order 256.

619 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×#553C211yesmaximal, characteristic, complemented
2C2×#488C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4#2540C211yesmaximal, characteristic, complemented
5#2000C211yesmaximal, characteristic, complemented
6#2000C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1954C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18#553—24no—
19#553—24no—
20C2×#126—21no—
21C2×#125C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×#123—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×C4×D4—21no—
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×#117C2211yescharacteristic
35C22×C4⋊C4C2211yescharacteristic
36#488—24no—
37#463—24no—
38C2×C4⋊C8C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C4×C8C2211yescharacteristic
52#399—21no—
53#392—21no—
54#390—21no—
55#390—21no—
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×SD16—22no—
76C2×SD16—41no—
77C2×SD16—41no—
78C2×D8—22no—
79C2×D8—41no—
80C2×D8—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#126—28no—
86#125—22no—
87#125D414yescomplemented
88#124—22no—
89#124—22no—
90#123—28no—
91C22⋊Q8—22no—
92C22⋊Q8—22no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95#121—28no—
96#121—22no—
97#121—22no—
98#121—22no—
99#121—22no—
100#121—22no—
101#121—22no—
102#121—41no—
103#121—41no—
104C4×D4—28no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4D414yescomplemented
109#117—22no—
110#117—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4D411yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4D411yescharacteristic
135C2×C22⋊C4D411yescharacteristic
136C2×C42—21no—
137C2×C42—21no—
138C2×C42C2311yescharacteristic
139C4⋊C8—22no—
140#103—22no—
141#103—22no—
142#103—22no—
143#102—22no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C4×C8—22no—
153#95—21no—
154C24—21no—
155C24—21no—
156C24C2×D411yescharacteristic
157C24C2×D411yescharacteristic
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×D4—24no—
165C2×D4—22no—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4C2×D412yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—24no—
210C22×C4—21no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2411yescharacteristic
220SD16—44no—
221D8—44no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4C2×D412yes—
246C4⋊C4C2×D412yes—
247C4⋊C4C2×D412yes—
248C4⋊C4#4112yes—
249C4⋊C4#4112yes—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—24no—
263C42—81no—
264C42C2×D412yes—
265C42C2×D412yes—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#13512yes—
294C23C22×D411yescharacteristic
295Q8—41no—
296Q8—41no—
297Q8—41no—
298Q8—41no—
299D4—22no—
300D4—22no—
301D4—44no—
302D4—48no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—84no—
310D4—84no—
311D4—84no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4#12012yes—
351C2×C4#12012yes—
352C2×C4C22×D411yescharacteristic
353C2×C4C22×D411yescharacteristic
354C2×C4C2×#4111yescharacteristic
355C2×C4C2×#4111yescharacteristic
356C2×C4H(5,𝔽2)11yescharacteristic
357C2×C4H(5,𝔽2)11yescharacteristic
358C8—41no—
359C8—41no—
360C8—41no—
361C8—41no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—41no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C2×#12011yescharacteristic
391C22#54611yescharacteristic
392C22#54611yescharacteristic
393C22C2×#13511yescharacteristic
394C22#57511yescharacteristic
395C22#57611yescharacteristic
396C22#57711yescharacteristic
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—81no—
407C4#53211yescharacteristic
408C4#53511yescharacteristic
409C4#55011yescharacteristic
410C4#55011yescharacteristic
411C2—42no—
412C2—81no—
413C2—82no—
414C2—161no—
415C2#243111yescharacteristic
416C2#300211yescharacteristic
417C2#301211yescharacteristic
418C2#329711yescharacteristic
419C2#330111yescharacteristic
420C2#332211yescharacteristic
421C2#332611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#553C2semidirect
2. C2×#488C2semidirect
3. C2×#463C2semidirect
4. #2540C2semidirect
5. #2000C2semidirect
6. #2000C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1989C2semidirect
10. #1954C2semidirect
11. #1875C2semidirect
12. #1874C2semidirect
13. #1873C2semidirect
14. #1872C2semidirect
15. C4×#102C2semidirect
87. #125D4semidirect
108. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4882
C2×#4632
#25402
#20002
#19992
#19892
#19542
#18752
#18742
#18732
#18722
C4×#1022