Finite Group Databasebeta

← #22507

Subgroups — #22507

Order 256.

563 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic
2D4×M16C211yesmaximal, characteristic, complemented
3#2293C211yesmaximal, characteristic, complemented
4#1994C211yesmaximal, characteristic, complemented
5#1986C211yesmaximal, characteristic, complemented
6#1897C212yesmaximal, complemented
7#1871C212yesmaximal, complemented
8#1826C212yesmaximal, complemented
9#1806C211yesmaximal, characteristic, complemented
10C4×#104C211yesmaximal, characteristic, complemented
11#1733C212yesmaximal, 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×D4C2211yescharacteristic
29C2×#117C2212yes—
30C2×#117C2212yes—
31#436—21no—
32#435—22no—
33C8×D4—21no—
34#423C2211yescharacteristic
35C2×#104C2211yescharacteristic
36C2×#104C2211yescharacteristic
37C2×#104C2212yes—
38#407C2212yes—
39C4×M16C2211yescharacteristic
40#390—21no—
41#389—21no—
42#388—22no—
43C4×C4⋊C4C2211yescharacteristic
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×C4C2312yes—
57C2×M16—22no—
58C2×M16—24no—
59C2×M16—42no—
60C2×M16C2312yes—
61C2×M16C2311yescharacteristic
62C2×M16C2311yescharacteristic
63C22×C8—22no—
64C4⋊Q8C2×C411yescharacteristic
65C4⋊Q8C2×C411yescharacteristic
66#126—22no—
67#126—22no—
68#126—22no—
69#125—22no—
70#125—21no—
71#125C2×C412yes—
72#124—21no—
73#124—22no—
74#123—22no—
75#123—22no—
76#123—22no—
77#123—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—22no—
82C22⋊Q8C2×C412yes—
83C22⋊Q8C2×C412yes—
84#121—22no—
85#121—21no—
86#121C2×C412yes—
87#121C2×C412yes—
88#120—22no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8C2×C411yescharacteristic
93C4×Q8C2×C411yescharacteristic
94C4×D4—21no—
95C4×D4—22no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4C2×C411yescharacteristic
101C4×D4C2×C411yescharacteristic
102C4×D4C2×C411yescharacteristic
103C4×D4C2×C412yes—
104C4×D4C2×C411yescharacteristic
105C4×D4D411yescharacteristic
106C4×D4D411yescharacteristic
107C4×D4D412yes—
108#117—22no—
109#117—22no—
110#117—21no—
111#117—22no—
112#117—21no—
113#117—24no—
114#117—42no—
115#117C2×C412yes—
116#117C2×C412yes—
117#117C2311yescharacteristic
118#117C2312yes—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C2×C412yes—
122C2×C4⋊C4C2×C412yes—
123C2×C4⋊C4D411yescharacteristic
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4D412yes—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42C2312yes—
133C2×C42C2311yescharacteristic
134C2×C42C2311yescharacteristic
135C2×C42C2311yescharacteristic
136C4⋊C8—21no—
137C4⋊C8—21no—
138#104—22no—
139#104—22no—
140#104—42no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143#97—21no—
144C4×C8—21no—
145#95—22no—
146#95—21no—
147C24C2×D412yes—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152#41—44no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8C22×C411yescharacteristic
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4C22⋊C411yescharacteristic
161C2×D4C22⋊C411yescharacteristic
162C2×D4C22⋊C412yes—
163C2×D4C22×C411yescharacteristic
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4C22×C412yes—
177C22×C4C22×C412yes—
178C22×C4C2×D412yes—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4C2×D411yescharacteristic
183C22×C4C2411yescharacteristic
184M16—21no—
185M16—21no—
186M16—22no—
187M16—22no—
188M16—21no—
189M16—44no—
190M16#4112yescomplemented
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—22no—
194C2×C8—44no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—21no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4C22⋊C411yescharacteristic
208C4⋊C4C22⋊C411yescharacteristic
209C4⋊C4C22⋊C412yes—
210C4⋊C4C22×C411yescharacteristic
211C4⋊C4C22×C411yescharacteristic
212C4⋊C4C22×C412yes—
213C4⋊C4C22×C412yes—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—21no—
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4C22⋊C412yes—
227C22⋊C4C22⋊C412yes—
228C22⋊C4C22⋊C412yes—
229C22⋊C4C22⋊C412yes—
230C42—22no—
231C42—21no—
232C42—22no—
233C42—21no—
234C42—21no—
235C42—22no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—21no—
241C42—42no—
242C42—42no—
243C42C22×C411yescharacteristic
244C42C22×C411yescharacteristic
245C42C2×D411yescharacteristic
246C42C2×D411yescharacteristic
247C42#4112yescomplemented
248C23—22no—
249C23—22no—
250C23—21no—
251C23—42no—
252C23—42no—
253C23C2×C22⋊C412yes—
254C23C2×C22⋊C412yes—
255C23C22×D411yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—42no—
259Q8—81no—
260D4—21no—
261D4—21no—
262D4—22no—
263D4—22no—
264D4—42no—
265D4—44no—
266D4—41no—
267D4—41no—
268D4—82no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
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—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4C2×C22⋊C412yes—
302C2×C4C2×C22⋊C411yescharacteristic
303C2×C4C2×C22⋊C411yescharacteristic
304C2×C4C2×C22⋊C411yescharacteristic
305C2×C4C2×C22⋊C412yes—
306C2×C4C2×C22⋊C411yescharacteristic
307C2×C4C23×C411yescharacteristic
308C2×C4C22×D411yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C42−1+411yescharacteristic
313C8—22no—
314C8—42no—
315C8—41no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—82no—
328C22—82no—
329C22C22×C22⋊C411yescharacteristic
330C22#54611yescharacteristic
331C22#54711yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—42no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—42no—
344C4—82no—
345C4#42112yes—
346C4C4×#4111yescharacteristic
347C4#52011yescharacteristic
348C4#54611yescharacteristic
349C4#54711yescharacteristic
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2—81no—
355C2#233911yescharacteristic
356C2C2×#42111yescharacteristic
357C2#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. #2293C2semidirect
4. #1994C2semidirect
5. #1986C2semidirect
6. #1897C2semidirect
7. #1871C2semidirect
8. #1826C2semidirect
9. #1806C2semidirect
10. C4×#104C2semidirect
11. #1733C2semidirect
13. #556C4semidirect
14. #555C4semidirect
15. #554C4semidirect
16. #553C4semidirect
18. #552C4semidirect
20. #547C4semidirect
190. M16#41semidirect
247. C42#41semidirect

Maximal subgroups

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

subgroupindex
#34752
D4×M162
#22932
#19942
#19862
#18972
#18712
#18262
#18062
C4×#1042
#17332