Finite Group Databasebeta

← #24514

Subgroups — #24514

Order 256.

448 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#451C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic
3#2625C211yesmaximal, characteristic
4#2310C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2066C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#1971C211yesmaximal, characteristic, complemented
10#1969C211yesmaximal, characteristic, complemented
11#1935C211yesmaximal, characteristic, complemented
12#1933C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic
14#1854C211yesmaximal, characteristic
15#1787C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#451—24no—
23#433—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#400—21no—
41#398—21no—
42#397—21no—
43#397—21no—
44#391—21no—
45#391—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#381—21no—
55#381—21no—
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85#117—22no—
86#117—22no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—41no—
108#106—41no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#103—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119C22⋊C8—22no—
120C22⋊C8D414yes—
121#97—22no—
122C4×C8—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24C2×D411yescharacteristic
139C2×Q8—24no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8#4112yes—
163C2×Q8#4112yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D412yes—
185C22×C4#4112yes—
186C22×C4C2411yescharacteristic
187Q16—44no—
188Q16—44no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8C2×D412yes—
200C2×C8C2×D412yes—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—81no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—42no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23#12112yes—
233C23C22×D411yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—22no—
238Q8—48no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—48no—
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—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4#12112yes—
291C2×C4C22×D411yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C8—22no—
298C8—22no—
299C8—41no—
300C8—41no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22C2×#12111yescharacteristic
309C22#52511yescharacteristic
310C22#53511yescharacteristic
311C22#57711yescharacteristic
312C22#57811yescharacteristic
313C22#57811yescharacteristic
314C22#57811yescharacteristic
315C4—22no—
316C4—42no—
317C4—42no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4#54611yescharacteristic
329C4#54611yescharacteristic
330C4#55111yescharacteristic
331C4#55111yescharacteristic
332C2—42no—
333C2#241111yescharacteristic
334C2#305911yescharacteristic
335C2#306311yescharacteristic
336C2#312111yescharacteristic
337C2#312711yescharacteristic
338C2#322311yescharacteristic
339C2#322511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
5. #2068C2semidirect
6. #2066C2semidirect
7. #2056C2semidirect
8. #2056C2semidirect
9. #1971C2semidirect
10. #1969C2semidirect
11. #1935C2semidirect
12. #1933C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
C2×#4332
#26252
#23102
#20682
#20662
#20562
#19712
#19692
#19352
#19332
#18732
#18542
#17872