Finite Group Databasebeta

← #22879

Subgroups — #22879

Order 256.

553 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#452C211yesmaximal, characteristic
2C2×C4×Q16C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic, complemented
5#2341C211yesmaximal, characteristic
6#1973C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1925C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1852C211yesmaximal, characteristic, complemented
14#1763C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#452C418yescomplemented
26C4×Q16—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29#417—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#391C2211yescharacteristic
42#390—21no—
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
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
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×C4C2311yescharacteristic
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16C2×C414yescomplemented
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16C2×C414yescomplemented
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—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⋊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×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123#107—22no—
124#107—41no—
125#103—22no—
126#103—22no—
127#103—41no—
128#103C2×C414yescomplemented
129#103D414yes—
130#102—22no—
131#102—22no—
132#102C2×C414yescomplemented
133C22⋊C8C2×C414yes—
134#97—41no—
135C4×C8—22no—
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×Q8C22×C412yes—
158C2×Q8C22×C412yes—
159C2×Q8C2×D412yes—
160C2×Q8#4112yes—
161C2×D4—24no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4C22×C412yes—
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—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—
196SD16—44no—
197M16—44no—
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—21no—
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—41no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—81no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C4×D412yes—
256C23C22×D411yescharacteristic
257Q8—22no—
258Q8—22no—
259Q8—22no—
260Q8—22no—
261Q8—22no—
262Q8—41no—
263Q8—41no—
264Q8C4×D414yes—
265D4—22no—
266D4—22no—
267D4—48no—
268D4—41no—
269D4—41no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—48no—
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—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—84no—
310C2×C4C4×D412yes—
311C2×C4#13712yes—
312C2×C4C23×C411yescharacteristic
313C2×C4C22×D411yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C2×C4H(5,𝔽2)11yescharacteristic
318C8—22no—
319C8—22no—
320C8—22no—
321C8—41no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×C4×D411yescharacteristic
337C22D4211yescharacteristic
338C22#54611yescharacteristic
339C22C2×#13711yescharacteristic
340C22#57511yescharacteristic
341C22#57611yescharacteristic
342C22#57811yescharacteristic
343C4—22no—
344C4—24no—
345C4—22no—
346C4—41no—
347C4—42no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4C2×C4×D411yescharacteristic
356C4#51811yescharacteristic
357C4#54611yescharacteristic
358C4#55111yescharacteristic
359C2—42no—
360C2—82no—
361C2#235811yescharacteristic
362C2C4×#13711yescharacteristic
363C2#295111yescharacteristic
364C2#326811yescharacteristic
365C2#327011yescharacteristic
366C2#332711yescharacteristic
367C2#333311yescharacteristic
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×Q16C2semidirect
4. #2360C2semidirect
6. #1973C2semidirect
7. #1934C2semidirect
8. #1925C2semidirect
9. #1894C2semidirect
10. #1880C2semidirect
11. #1878C2semidirect
12. #1863C2semidirect
13. #1852C2semidirect
14. #1763C2semidirect
15. #1753C2semidirect
25. #452C4semidirect
77. C2×Q16C2×C4semidirect
80. C2×SD16C2×C4semidirect
128. #103C2×C4semidirect
132. #102C2×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×Q162
C2×#4172
#23602
#23412
#19732
#19342
#19252
#18942
#18802
#18782
#18632
#18522
#17632
#17532