Finite Group Databasebeta

← #23714

Subgroups — #23714

Order 256.

438 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#440C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2324C211yesmaximal, characteristic
4#2323C211yesmaximal, characteristic, complemented
5#2087C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#2055C211yesmaximal, characteristic, complemented
10#2036C211yesmaximal, characteristic, complemented
11#2034C211yesmaximal, characteristic
12#2000C211yesmaximal, characteristic
13#1776C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#440—24no—
23#433—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#391C2211yescharacteristic
40#390—21no—
41#390C2211yescharacteristic
42#389—21no—
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#388—21no—
48#387—21no—
49#386—21no—
50#385—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75#121—22no—
76#121—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×D4—22no—
80C4×D4—22no—
81#117—22no—
82#117—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111#106—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122#97D414yescomplemented
123C4×C8—22no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8#4112yes—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—81no—
147C2×D4—81no—
148C2×D4#4112yes—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—28no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8C2×D412yes—
183C2×C8C2×D412yes—
184C4⋊C4—24no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4#4112yes—
201C4⋊C4#4112yes—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—81no—
215C42—81no—
216C42C2×D412yes—
217C42#4112yes—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C22×D411yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—41no—
236Q8—41no—
237D4—22no—
238D4—22no—
239D4—41no—
240D4—41no—
241D4—84no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—84no—
281C2×C4#12112yes—
282C2×C4#12112yes—
283C2×C4C22×D411yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C8—22no—
290C8—22no—
291C8—41no—
292C8—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—42no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22C2×#12111yescharacteristic
308C22#52511yescharacteristic
309C22#52511yescharacteristic
310C22#57611yescharacteristic
311C22#57711yescharacteristic
312C22#57711yescharacteristic
313C22#57811yescharacteristic
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—41no—
319C4—42no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4#52511yescharacteristic
327C4#52911yescharacteristic
328C4#52911yescharacteristic
329C4#53211yescharacteristic
330C2—81no—
331C2—82no—
332C2#239211yescharacteristic
333C2#306211yescharacteristic
334C2#306311yescharacteristic
335C2#311911yescharacteristic
336C2#312111yescharacteristic
337C2#312311yescharacteristic
338C2#312411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#440C2semidirect
2. C2×#433C2semidirect
4. #2323C2semidirect
5. #2087C2semidirect
6. #2075C2semidirect
7. #2067C2semidirect
8. #2066C2semidirect
9. #2055C2semidirect
10. #2036C2semidirect
13. #1776C2semidirect
14. C4×#103C2semidirect
15. C4×#102C2semidirect
122. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
C2×#4332
#23242
#23232
#20872
#20752
#20672
#20662
#20552
#20362
#20342
#20002
#17762
C4×#1032
C4×#1022