Finite Group Databasebeta

← #22505

Subgroups — #22505

Order 256.

631 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic
2D4×M16C211yesmaximal, characteristic, complemented
3C42×D4C211yesmaximal, characteristic, complemented
4#1994C211yesmaximal, characteristic, complemented
5#1986C211yesmaximal, characteristic, complemented
6#1896C212yesmaximal, complemented
7#1871C212yesmaximal, complemented
8#1826C212yesmaximal, complemented
9#1806C211yesmaximal, characteristic, complemented
10C4×#104C211yesmaximal, characteristic, complemented
11C2×#339C212yesmaximal, complemented
12C22×M16C2212yes—
13#556C411yescharacteristic, complemented
14#555C411yescharacteristic, complemented
15#554C411yescharacteristic, complemented
16#553C412yescomplemented
17#552—21no—
18#552C412yescomplemented
19#551—22no—
20#547C411yescharacteristic, complemented
21#546—21no—
22#533C2211yescharacteristic
23#529—22no—
24#529—21no—
25#525C2212yes—
26C4×#41—21no—
27C4×#41C2211yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2211yescharacteristic
30C2×#117C2212yes—
31C22×C42C2212yes—
32#436—21no—
33#435—22no—
34C8×D4—21no—
35#423C2211yescharacteristic
36C2×#104C2211yescharacteristic
37C2×#104C2211yescharacteristic
38C2×#104C2212yes—
39#407C2212yes—
40C4×M16C2211yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2211yescharacteristic
43C4×C22⋊C4—22no—
44C4×C22⋊C4C2212yes—
45C43C2211yescharacteristic
46#339C2212yes—
47#339C2211yescharacteristic
48#339C2212yes—
49#339C2212yes—
50#339C2212yes—
51#339C2212yes—
52#339C2211yescharacteristic
53C2×#41—21no—
54C2×#41C2311yescharacteristic
55C22×D4D411yescharacteristic
56C23×C4—22no—
57C23×C4C2312yes—
58C2×M16—22no—
59C2×M16—24no—
60C2×M16—42no—
61C2×M16C2312yes—
62C2×M16C2311yescharacteristic
63C2×M16C2311yescharacteristic
64C22×C8—22no—
65C4⋊Q8C2×C411yescharacteristic, complemented
66C4⋊Q8C2×C411yescharacteristic, complemented
67#126—22no—
68#126—22no—
69#126—22no—
70#125—22no—
71#125—21no—
72#125C2×C412yescomplemented
73#124—21no—
74#124—22no—
75#123—22no—
76#123—22no—
77#123—22no—
78#123—22no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—22no—
83C22⋊Q8C2×C412yescomplemented
84C22⋊Q8C2×C412yescomplemented
85#121—22no—
86#121—21no—
87#121C2×C412yescomplemented
88#121C2×C412yescomplemented
89#120—22no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8C2×C411yescharacteristic, complemented
94C4×Q8C2×C411yescharacteristic, complemented
95C4×D4—28no—
96C4×D4—21no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4C2×C411yescharacteristic
103C4×D4C2×C412yes—
104C4×D4C2×C411yescharacteristic
105C4×D4C2×C411yescharacteristic, complemented
106C4×D4C2×C411yescharacteristic, complemented
107C4×D4D411yescharacteristic
108C4×D4D412yes—
109C4×D4D411yescharacteristic
110#117—22no—
111#117—22no—
112#117—21no—
113#117—22no—
114#117—21no—
115#117C2×C412yescomplemented
116#117C2×C412yescomplemented
117#117C2311yescharacteristic
118#117C2312yes—
119C2×C4⋊C4—21no—
120C2×C4⋊C4C2×C412yescomplemented
121C2×C4⋊C4C2×C412yescomplemented
122C2×C4⋊C4D411yescharacteristic
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4D412yes—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—24no—
130C2×C42—21no—
131C2×C42—22no—
132C2×C42—42no—
133C2×C42C2311yescharacteristic
134C2×C42C2312yes—
135C2×C42C2311yescharacteristic
136C2×C42C2311yescharacteristic
137C4⋊C8—21no—
138C4⋊C8—21no—
139#104—22no—
140#104—22no—
141#104—42no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144#97—21no—
145C4×C8—21no—
146#95—21no—
147#95—22no—
148C24C2×D412yes—
149#41—21no—
150#41—21no—
151#41—21no—
152#41—21no—
153#41—44no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8C22×C411yescharacteristic
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4C22⋊C412yes—
162C2×D4C22⋊C411yescharacteristic
163C2×D4C22⋊C411yescharacteristic
164C2×D4C22×C411yescharacteristic
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—28no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4C22×C412yes—
181C22×C4C22×C412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D411yescharacteristic
187C22×C4C2411yescharacteristic
188M16—21no—
189M16—21no—
190M16—22no—
191M16—22no—
192M16—21no—
193M16—44no—
194M16#4112yescomplemented
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—22no—
198C2×C8—44no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—21no—
210C4⋊C4C22⋊C411yescharacteristic
211C4⋊C4C22⋊C411yescharacteristic
212C4⋊C4C22⋊C412yes—
213C4⋊C4C22×C411yescharacteristic
214C4⋊C4C22×C411yescharacteristic
215C4⋊C4C22×C412yes—
216C4⋊C4C22×C412yes—
217C22⋊C4—28no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4C22⋊C412yes—
229C22⋊C4C22⋊C412yes—
230C22⋊C4C22⋊C412yes—
231C22⋊C4C22⋊C412yes—
232C42—24no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42—21no—
240C42—21no—
241C42—22no—
242C42—22no—
243C42—21no—
244C42—44no—
245C42—42no—
246C42C22×C411yescharacteristic
247C42C22×C411yescharacteristic
248C42C2×D411yescharacteristic
249C42C2×D411yescharacteristic
250C42#4112yescomplemented
251C23—22no—
252C23—22no—
253C23—24no—
254C23—21no—
255C23—42no—
256C23C2×C22⋊C412yes—
257C23C2×C22⋊C412yes—
258C23C22×D411yescharacteristic
259Q8—21no—
260Q8—21no—
261Q8—42no—
262Q8—81no—
263D4—24no—
264D4—21no—
265D4—21no—
266D4—22no—
267D4—41no—
268D4—41no—
269D4—44no—
270D4—82no—
271D4#10414yescomplemented
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—416no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4C2×C22⋊C412yes—
306C2×C4C2×C22⋊C411yescharacteristic
307C2×C4C2×C22⋊C411yescharacteristic
308C2×C4C2×C22⋊C411yescharacteristic
309C2×C4C2×C22⋊C412yes—
310C2×C4C2×C22⋊C411yescharacteristic
311C2×C4C23×C411yescharacteristic
312C2×C4C22×D411yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C42−1+411yescharacteristic
317C8—22no—
318C8—42no—
319C8—41no—
320C22—22no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—44no—
328C22—44no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C2×#10414yes—
333C22C22×C22⋊C411yescharacteristic
334C22#54611yescharacteristic
335C22#54711yescharacteristic
336C4—24no—
337C4—22no—
338C4—22no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—42no—
343C4—44no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—41no—
348C4—42no—
349C4C2×#10412yes—
350C4C4×#4111yescharacteristic
351C4#52011yescharacteristic
352C4#54611yescharacteristic
353C4#54711yescharacteristic
354C2—21no—
355C2—21no—
356C2—24no—
357C2—42no—
358C2—81no—
359C2#233911yescharacteristic
360C2C22×#10411yescharacteristic
361C2#290111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radical#3475C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×M16C2semidirect
3. C42×D4C2semidirect
4. #1994C2semidirect
5. #1986C2semidirect
6. #1896C2semidirect
7. #1871C2semidirect
8. #1826C2semidirect
9. #1806C2semidirect
10. C4×#104C2semidirect
11. C2×#339C2semidirect
13. #556C4semidirect
14. #555C4semidirect
15. #554C4semidirect
16. #553C4semidirect
18. #552C4semidirect
20. #547C4semidirect
65. C4⋊Q8C2×C4semidirect
66. C4⋊Q8C2×C4semidirect
72. #125C2×C4semidirect
83. C22⋊Q8C2×C4semidirect
84. C22⋊Q8C2×C4semidirect
87. #121C2×C4semidirect
88. #121C2×C4semidirect
93. C4×Q8C2×C4semidirect
94. C4×Q8C2×C4semidirect
105. C4×D4C2×C4semidirect
106. C4×D4C2×C4semidirect
115. #117C2×C4semidirect
116. #117C2×C4semidirect
120. C2×C4⋊C4C2×C4semidirect
121. C2×C4⋊C4C2×C4semidirect
194. M16#41semidirect
250. C42#41semidirect
271. D4#104semidirect

Maximal subgroups

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

subgroupindex
#34752
D4×M162
C42×D42
#19942
#19862
#18962
#18712
#18262
#18062
C4×#1042
C2×#3392