Finite Group Databasebeta

← #22876

Subgroups — #22876

Order 256.

565 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#452C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic
4#2362C211yesmaximal, characteristic, complemented
5#2339C211yesmaximal, characteristic
6#1969C211yesmaximal, characteristic, complemented
7#1933C211yesmaximal, characteristic, complemented
8#1927C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1878C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#452C418yescomplemented
26C4×SD16—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29#416—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#387—21no—
48#386—21no—
49#386—21no—
50#381—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4—21no—
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55C2×#95—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41C2×C414yes—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16C2×C414yescomplemented
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16C2×C414yes—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22⋊Q8—22no—
90C22⋊Q8C2×C414yescomplemented
91C4×Q8—22no—
92C4×Q8—41no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—41no—
96C4×D4—41no—
97#117—22no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4D411yescharacteristic
113C2×C22⋊C4D411yescharacteristic
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—41no—
120#106—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—41no—
125#103C2×C414yescomplemented
126#102—22no—
127#102C2×C414yes—
128#102D414yes—
129C22⋊C8C2×C414yescomplemented
130C4×C8—22no—
131C4×C8—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
143C24—21no—
144C24C2×D411yescharacteristic
145#41—28no—
146#41—44no—
147#41—44no—
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
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×Q8C22×C412yes—
160C2×Q8C22×C412yes—
161C2×D4—24no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4C22×C412yes—
168C2×D4C2×D412yes—
169C2×D4#4112yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C22×C412yes—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4#4112yes—
197C22×C4C2411yescharacteristic
198Q16—44no—
199SD16—28no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—44no—
210C2×C8C22×C412yes—
211C2×C8C22×C412yes—
212C2×C8C2×D412yes—
213C4⋊C4—24no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4C22×C412yes—
225C4⋊C4C2×D412yes—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—42no—
238C42—42no—
239C42—42no—
240C23—22no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23C4×D412yes—
254C23C22×D411yescharacteristic
255Q8—22no—
256Q8—22no—
257Q8—22no—
258Q8—22no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262Q8—41no—
263D4—22no—
264D4—22no—
265D4—22no—
266D4—48no—
267D4C4×D414yes—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—48no—
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—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4C4×D412yes—
309C2×C4#13512yes—
310C2×C4#13512yes—
311C2×C4C23×C411yescharacteristic
312C2×C4C22×D411yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C8—22no—
318C8—22no—
319C8—22no—
320C8—41no—
321C22—22no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—42no—
335C22C2×C4×D411yescharacteristic
336C22D4211yescharacteristic
337C22#54611yescharacteristic
338C22C2×#13511yescharacteristic
339C22C2×#13511yescharacteristic
340C22#57711yescharacteristic
341C22#57811yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—42no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4C2×C4×D411yescharacteristic
355C4#51811yescharacteristic
356C4#54611yescharacteristic
357C4#55111yescharacteristic
358C2—42no—
359C2—44no—
360C2#235811yescharacteristic
361C2C4×#13511yescharacteristic
362C2#294711yescharacteristic
363C2#328211yescharacteristic
364C2#328311yescharacteristic
365C2#329611yescharacteristic
366C2#330211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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×C4×SD16C2semidirect
4. #2362C2semidirect
6. #1969C2semidirect
7. #1933C2semidirect
8. #1927C2semidirect
9. #1892C2semidirect
10. #1878C2semidirect
11. #1877C2semidirect
12. #1862C2semidirect
13. #1854C2semidirect
14. C4×#102C2semidirect
15. C8×C22⋊C4C2semidirect
25. #452C4semidirect
78. C2×Q16C2×C4semidirect
90. C22⋊Q8C2×C4semidirect
125. #103C2×C4semidirect
129. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×C4×SD162
C2×#4162
#23622
#23392
#19692
#19332
#19272
#18922
#18782
#18772
#18622
#18542
C4×#1022
C8×C22⋊C42