Finite Group Databasebeta

← #22859

Subgroups — #22859

Order 256.

543 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#451C211yesmaximal, characteristic
2C2×#441C211yesmaximal, characteristic, complemented
3C22×#103C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic
5Q8×C22⋊C4C211yesmaximal, characteristic
6#1971C211yesmaximal, characteristic, complemented
7#1935C211yesmaximal, characteristic, complemented
8#1927C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic
10#1878C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1865C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic
14#1763C211yesmaximal, characteristic, complemented
15#1752C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#451C418yescomplemented
26#441—24no—
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#103—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#391—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#389—21no—
47#389C2211yescharacteristic
48#387—21no—
49#385—21no—
50#380—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2×C414yes—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16C2×C414yescomplemented
80C2×Q16C2×C414yescomplemented
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22⋊Q8—22no—
89C22⋊Q8C2×C414yes—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#106—22no—
120#106—41no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103C2×C414yescomplemented
127#103C2×C414yescomplemented
128#103D414yes—
129C22⋊C8C2×C414yes—
130#97—22no—
131#97—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95D411yescharacteristic
142#95D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×Q8—24no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—28no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8C22×C412yes—
165C2×Q8C22×C412yes—
166C2×Q8C22×C412yes—
167C2×Q8C2×D412yes—
168C2×Q8#4112yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4C22×C412yes—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4#4112yes—
194C22×C4C2411yescharacteristic
195Q16—28no—
196Q16—44no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—44no—
207C2×C8C22×C412yes—
208C2×C8C22×C412yes—
209C2×C8C2×D412yes—
210C4⋊C4—24no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4C22×C412yes—
225C4⋊C4C2×D412yes—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—42no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—42no—
235C42—42no—
236C42—42no—
237C42—81no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23C4×D412yes—
243C23#13712yes—
244C23#13712yes—
245C23C22×D411yescharacteristic
246Q8—22no—
247Q8—22no—
248Q8—22no—
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—22no—
253Q8—48no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8C4×D414yes—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—48no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—84no—
303C2×C4C4×D412yes—
304C2×C4C23×C411yescharacteristic
305C2×C4C22×D411yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C8—22no—
311C8—22no—
312C8—22no—
313C8—41no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22#44114yescomplemented
321C22C2×C4×D411yescharacteristic
322C22D4211yescharacteristic
323C22#54611yescharacteristic
324C22C2×#13711yescharacteristic
325C22C2×#13711yescharacteristic
326C22#57711yescharacteristic
327C22#57811yescharacteristic
328C4—22no—
329C4—24no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4C2×C4×D411yescharacteristic
342C4#51811yescharacteristic
343C4#54611yescharacteristic
344C4#55111yescharacteristic
345C2—24no—
346C2#235811yescharacteristic
347C2C2×#44111yescharacteristic
348C2#294711yescharacteristic
349C2#326611yescharacteristic
350C2#327011yescharacteristic
351C2#331211yescharacteristic
352C2#331611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#441C2semidirect
6. #1971C2semidirect
7. #1935C2semidirect
8. #1927C2semidirect
10. #1878C2semidirect
11. #1878C2semidirect
12. #1865C2semidirect
14. #1763C2semidirect
25. #451C4semidirect
79. C2×Q16C2×C4semidirect
80. C2×Q16C2×C4semidirect
126. #103C2×C4semidirect
127. #103C2×C4semidirect
320. C22#441semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
C2×#4412
C22×#1032
#23602
Q8×C22⋊C42
#19712
#19352
#19272
#18922
#18782
#18652
#18532
#17632
#17522