Finite Group Databasebeta

← #22855

Subgroups — #22855

Order 256.

585 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#448C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C22×#103C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic
5Q8×C22⋊C4C211yesmaximal, characteristic, complemented
6#1973C211yesmaximal, characteristic, complemented
7#1935C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic
10#1880C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1852C211yesmaximal, characteristic, complemented
14#1763C211yesmaximal, characteristic, complemented
15#1752C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#448C418yescomplemented
27#440—24no—
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#103—24no—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#391—21no—
43#391—21no—
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#386—21no—
50#385—21no—
51#380—21no—
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
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×C414yescomplemented
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16C2×C414yescomplemented
82C2×SD16C2×C414yescomplemented
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#121—22no—
91#121C2×C414yes—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—44no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121C2×C42C2311yescharacteristic
122#107—22no—
123#107—41no—
124#103—22no—
125#103—22no—
126#103—22no—
127#103C2×C414yescomplemented
128#103C2×C414yescomplemented
129#103D414yescomplemented
130#102—22no—
131#102—41no—
132C22⋊C8C2×C414yescomplemented
133#97—22no—
134#97—41no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24C2×D411yescharacteristic
146C2×Q8—24no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—28no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8C22×C412yes—
162C2×Q8C22×C412yes—
163C2×Q8C2×D412yes—
164C2×Q8#4112yes—
165C2×D4—24no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4C22×C412yes—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4C22×C412yes—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4#4112yes—
201C22×C4C2411yescharacteristic
202SD16—28no—
203SD16—44no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—44no—
214C2×C8C22×C412yes—
215C2×C8C22×C412yes—
216C2×C8C2×D412yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4C22×C412yes—
230C4⋊C4C2×D412yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—24no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—81no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—42no—
243C42—42no—
244C42—42no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—22no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C4×D412yes—
257C23AGL(1,ℤ/8)12yes—
258C23#13712yes—
259C23C22×D411yescharacteristic
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—22no—
264Q8—22no—
265Q8—48no—
266Q8—41no—
267Q8—41no—
268Q8C4×D414yescomplemented
269D4—22no—
270D4—22no—
271D4—48no—
272D4—41no—
273D4—41no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—48no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4C4×D412yes—
317C2×C4C23×C411yescharacteristic
318C2×C4C22×D411yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C8—22no—
324C8—22no—
325C8—22no—
326C8—41no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#44014yescomplemented
341C22C2×C4×D411yescharacteristic
342C22D4211yescharacteristic
343C22#54611yescharacteristic
344C22C2×AGL(1,ℤ/8)11yescharacteristic
345C22C2×#13711yescharacteristic
346C22#57711yescharacteristic
347C22#57811yescharacteristic
348C4—22no—
349C4—24no—
350C4—42no—
351C4—42no—
352C4—41no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4C2×C4×D411yescharacteristic
361C4#51811yescharacteristic
362C4#54611yescharacteristic
363C4#55111yescharacteristic
364C2—24no—
365C2—82no—
366C2#235811yescharacteristic
367C2C2×#44011yescharacteristic
368C2#294711yescharacteristic
369C2#326511yescharacteristic
370C2#327011yescharacteristic
371C2#330811yescharacteristic
372C2#332011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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×#440C2semidirect
3. C22×#103C2semidirect
5. Q8×C22⋊C4C2semidirect
6. #1973C2semidirect
7. #1935C2semidirect
8. #1928C2semidirect
10. #1880C2semidirect
11. #1877C2semidirect
12. #1863C2semidirect
13. #1852C2semidirect
14. #1763C2semidirect
15. #1752C2semidirect
26. #448C4semidirect
68. C22×Q8C2×C4semidirect
81. C2×SD16C2×C4semidirect
82. C2×SD16C2×C4semidirect
127. #103C2×C4semidirect
128. #103C2×C4semidirect
129. #103D4semidirect
132. C22⋊C8C2×C4semidirect
268. Q8C4×D4semidirect
340. C22#440semidirect

Maximal subgroups

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

subgroupindex
C2×#4482
C2×#4402
C22×#1032
#23612
Q8×C22⋊C42
#19732
#19352
#19282
#18932
#18802
#18772
#18632
#18522
#17632
#17522