Finite Group Databasebeta

← #23485

Subgroups — #23485

Order 256.

741 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4C2×#416C211yesmaximal, characteristic, complemented
5C2×#408C211yesmaximal, characteristic, complemented
6#2311C211yesmaximal, characteristic, complemented
7#2036C212yesmaximal, complemented
8#2035C212yesmaximal, complemented
9#2034C212yesmaximal, complemented
10#2033C212yesmaximal, complemented
11C2×#340C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C22×M16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16#537—24no—
17#537—24no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20C2×#121—21no—
21C2×#121—22no—
22C2×#121C2212yes—
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#428—24no—
27C2×#107C2212yes—
28C2×#106C2212yes—
29#417—24no—
30#416—24no—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2212yes—
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2212yes—
37#408—24no—
38C2×C22⋊C8C2212yes—
39C2×C22⋊C8C2212yes—
40#391—21no—
41#390—21no—
42#386—22no—
43#380C2212yes—
44#380C2212yes—
45#340C2212yes—
46#340C2212yes—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—22no—
55C2×#41—22no—
56C2×#41—22no—
57C2×#41—22no—
58C2×#41D414yes—
59C22×Q8D411yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4—22no—
63C22×D4D412yes—
64C22×D4D411yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C2312yes—
70C23×C4C2311yescharacteristic
71C2×M16—22no—
72C2×M16—22no—
73C2×M16D414yescomplemented
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2312yes—
80C22⋊Q8—28no—
81C22⋊Q8—24no—
82C22⋊Q8—24no—
83#121—216no—
84#121—28no—
85#121—24no—
86#121—24no—
87#117—22no—
88#117—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D414yes—
96C2×C4⋊C4C2312yes—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2312yes—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C42—21no—
106#107—24no—
107#106—24no—
108#103—22no—
109#103—22no—
110#103—24no—
111#102—22no—
112#102—22no—
113#102—24no—
114C22⋊C8—24no—
115C22⋊C8—24no—
116#95—22no—
117#95—22no—
118C24—21no—
119C24C2×D412yes—
120C24C2×D411yescharacteristic
121#41—28no—
122#41—28no—
123#41—44no—
124#41—44no—
125#41—44no—
126#41—44no—
127#41—44no—
128#41—44no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8C2×D412yes—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—28no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—82no—
154C2×D4—82no—
155C2×D4C2×D412yes—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4C2×D412yes—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2×D412yes—
181C22×C4#4112yes—
182C22×C4#4112yes—
183C22×C4#4112yes—
184C22×C4C2411yescharacteristic
185M16—28no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—44no—
195C2×C8C2×D414yes—
196C4⋊C4—28no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4C2×D414yes—
207C22⋊C4—28no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—42no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23#12012yes—
232C23#12112yes—
233C23#12312yes—
234C23C22×D411yescharacteristic
235Q8—22no—
236Q8—22no—
237Q8—41no—
238Q8—41no—
239Q8—44no—
240D4—22no—
241D4—22no—
242D4—48no—
243D4—48no—
244D4—41no—
245D4—41no—
246D4—44no—
247D4—88no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—48no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4#12012yes—
274C2×C4#12114yes—
275C2×C4#12114yes—
276C2×C4#12112yes—
277C2×C4#12314yes—
278C2×C4#12312yes—
279C2×C4#12414yes—
280C2×C4C22×D411yescharacteristic
281C2×C4C22×D411yescharacteristic
282C2×C4C22×D411yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C8—24no—
287C8—41no—
288C8—41no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#39414yes—
310C22C2×#12011yescharacteristic
311C22C2×#12111yescharacteristic
312C22C2×#12311yescharacteristic
313C22#57611yescharacteristic
314C22#57711yescharacteristic
315C22#57711yescharacteristic
316C22#57811yescharacteristic
317C4—42no—
318C4—42no—
319C4—44no—
320C4—81no—
321C4—81no—
322C4—82no—
323C4#39414yes—
324C4C2×#12111yescharacteristic
325C4C2×#12111yescharacteristic
326C4C2×#12311yescharacteristic
327C4C2×#12411yescharacteristic
328C2—24no—
329C2—82no—
330C2—82no—
331C2C2×#39411yescharacteristic
332C2#301211yescharacteristic
333C2#301311yescharacteristic
334C2#306211yescharacteristic
335C2#306311yescharacteristic
336C2#309811yescharacteristic
337C2#309911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#428C2semidirect
3. C2×#417C2semidirect
4. C2×#416C2semidirect
5. C2×#408C2semidirect
6. #2311C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2034C2semidirect
10. #2033C2semidirect
11. C2×#340C2semidirect
73. C2×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#4282
C2×#4172
C2×#4162
C2×#4082
#23112
#20362
#20352
#20342
#20332
C2×#3402