Finite Group Databasebeta

← #24400

Subgroups — #24400

Order 256.

661 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#137C211yesmaximal, characteristic
2C2×#473C212yesmaximal, complemented
3#3007C211yesmaximal, characteristic
4C2×#451C211yesmaximal, characteristic, complemented
5#2880C211yesmaximal, characteristic
6#2052C212yesmaximal, complemented
7#2020C212yesmaximal, complemented
8#1908C212yesmaximal, complemented
9#1901C211yesmaximal, characteristic, complemented
10#1855C211yesmaximal, characteristic, complemented
11#1784C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C2×#137—21no—
15C2×#137—21no—
16C2×#137—28no—
17C2×#137C2212yes—
18C2×#137C2211yescharacteristic
19C2×#137C2212yes—
20C2×#137C2211yescharacteristic
21C22×Q16—21no—
22C22×Q16C2212yescomplemented
23C22×SD16—21no—
24C2×#131C2212yes—
25C22×M16C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27#473—28no—
28#452—21no—
29#452—21no—
30#451—21no—
31#451—21no—
32#451—21no—
33#451—22no—
34#450—21no—
35C2×#108C2212yescomplemented
36#417C2211yescharacteristic
37#417C2211yescharacteristic
38C2×#103C2212yescomplemented
39C2×#101C2212yescomplemented
40C2×#101C2212yescomplemented
41C2×#101C2212yescomplemented
42#408—22no—
43#408C2212yes—
44#407—21no—
45#407C2211yescharacteristic
46#407C2211yescharacteristic
47C2×C22⋊C8—21no—
48C2×C22⋊C8C2211yescharacteristic
49#343C2212yescomplemented
50#341C2212yescomplemented
51C2×#41—21no—
52C2×#41—21no—
53C2×#41—24no—
54C2×#41—42no—
55C2×#41D411yescharacteristic
56C2×#41D411yescharacteristic
57C2×#41C2311yescharacteristic
58C2×#41C2311yescharacteristic
59C22×Q8—21no—
60C22×Q8—24no—
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8—21no—
64C22×Q8—21no—
65C22×Q8—42no—
66C22×Q8D411yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×Q8C2312yes—
69C22×D4—21no—
70C22×D4D411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74#137—28no—
75#137—28no—
76#137—24no—
77#137—24no—
78#137—24no—
79#137—24no—
80#137—42no—
81#137D418yes—
82C2×Q16—28no—
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—24no—
86C2×Q16—21no—
87C2×Q16—21no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—42no—
91C2×SD16—28no—
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16—21no—
95C2×SD16—21no—
96#131—24no—
97#131—42no—
98C2×M16—21no—
99C2×M16—24no—
100C2×M16—22no—
101C2×M16C2312yes—
102C2×M16C2311yescharacteristic
103C2×M16C2311yescharacteristic
104C2×M16C2312yes—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8C2312yes—
108C22⋊Q8—21no—
109C22⋊Q8—21no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115#108—24no—
116#103—21no—
117#103—21no—
118#103—22no—
119#103—22no—
120#102—21no—
121#102—21no—
122#101—24no—
123#101—24no—
124#101—42no—
125C22⋊C8—21no—
126C22⋊C8—21no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8D411yescharacteristic
132C22⋊C8D411yescharacteristic
133C22⋊C8D412yes—
134C24—21no—
135C24#4111yescharacteristic
136#41—22no—
137#41—22no—
138#41—22no—
139#41—22no—
140#41—22no—
141#41—44no—
142#41—41no—
143#41—41no—
144#41—44no—
145#41C2×D414yes—
146C2×Q8—21no—
147C2×Q8—22no—
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—44no—
160C2×Q8—41no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8C2×D414yes—
168C2×Q8C2×D411yescharacteristic
169C2×Q8C2×D411yescharacteristic
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—44no—
178C2×D4C2×D411yescharacteristic
179C2×D4C2×D411yescharacteristic
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4#4111yescharacteristic
199C22×C4C2411yescharacteristic
200Q16—28no—
201Q16—28no—
202Q16—44no—
203Q16—44no—
204Q16—42no—
205Q16—42no—
206SD16—28no—
207SD16—28no—
208SD16—42no—
209SD16—42no—
210M16—24no—
211M16—22no—
212M16—22no—
213M16—22no—
214M16—22no—
215M16—22no—
216M16—44no—
217M16C2×D414yes—
218C2×C8—24no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—44no—
226C2×C8C2×D412yes—
227C2×C8C2×D412yes—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—81no—
242C23#12111yescharacteristic
243C23#12111yescharacteristic
244C23C2×#4111yescharacteristic
245Q8—21no—
246Q8—24no—
247Q8—21no—
248Q8—21no—
249Q8—24no—
250Q8—42no—
251Q8—44no—
252Q8—44no—
253Q8—42no—
254Q8—42no—
255Q8—41no—
256Q8—41no—
257Q8—42no—
258Q8#12012yes—
259D4—24no—
260D4—21no—
261D4—21no—
262D4—21no—
263D4—42no—
264D4—42no—
265D4—42no—
266D4—42no—
267D4#12012yes—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4#12111yescharacteristic
292C2×C4#12111yescharacteristic
293C2×C4C22×D411yescharacteristic
294C2×C4C22×D412yes—
295C2×C4C22×D411yescharacteristic
296C2×C4H(5,𝔽2)12yes—
297C8—24no—
298C8—22no—
299C8—22no—
300C8—41no—
301C8—41no—
302C8—42no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—44no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—84no—
318C22#47312yescomplemented
319C22C2×#12111yescharacteristic
320C22#54612yes—
321C4—21no—
322C4—21no—
323C4—22no—
324C4—41no—
325C4—44no—
326C4—81no—
327C4—81no—
328C4C2×#12011yescharacteristic
329C4#53411yescharacteristic
330C4D4212yes—
331C2—22no—
332C2—22no—
333C2—41no—
334C2—42no—
335C2—81no—
336C2#241011yescharacteristic
337C2C2×#47311yescharacteristic
338C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#473C2semidirect
4. C2×#451C2semidirect
6. #2052C2semidirect
7. #2020C2semidirect
8. #1908C2semidirect
9. #1901C2semidirect
10. #1855C2semidirect
11. #1784C2semidirect
22. C22×Q16C22semidirect
35. C2×#108C22semidirect
38. C2×#103C22semidirect
39. C2×#101C22semidirect
40. C2×#101C22semidirect
41. C2×#101C22semidirect
49. #343C22semidirect
50. #341C22semidirect
318. C22#473semidirect

Maximal subgroups

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

subgroupindex
C22×#1372
C2×#4732
#30072
C2×#4512
#28802
#20522
#20202
#19082
#19012
#18552
#17842