Finite Group Databasebeta

← #21540

Subgroups — #21540

Order 256.

543 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×C4×M16C211yesmaximal, characteristic, complemented
4C4×C22⋊Q8C211yesmaximal, characteristic, complemented
5C4×#121C211yesmaximal, characteristic
6#1968C211yesmaximal, characteristic, complemented
7#1942C211yesmaximal, characteristic, complemented
8#1934C211yesmaximal, characteristic, complemented
9#1933C211yesmaximal, characteristic, complemented
10#1895C211yesmaximal, characteristic
11#1894C211yesmaximal, characteristic, complemented
12#1809C211yesmaximal, characteristic, complemented
13#1776C211yesmaximal, characteristic, complemented
14#1763C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C22×C42C2211yescharacteristic
25#468C418yescomplemented
26#440—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C4×M16—24no—
37C2×#97C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#387—21no—
47#386—21no—
48#385—21no—
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4—21no—
55C4×C22⋊C4—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4D411yescharacteristic
67C23×C4D411yescharacteristic
68C23×C4C2311yescharacteristic
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16C2×C414yescomplemented
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16C2×C414yescomplemented
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22⋊Q8—22no—
83C22⋊Q8C2×C414yescomplemented
84#121—22no—
85#121C2×C414yes—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—44no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42D411yescharacteristic
114C2×C42D411yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—22no—
118#107C2×C414yescomplemented
119#106—22no—
120#103—22no—
121#103—22no—
122#103C2×C414yescomplemented
123#102—22no—
124#102—22no—
125#102C2×C414yescomplemented
126#97—22no—
127#97D414yescomplemented
128C4×C8—22no—
129C4×C8—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8C22×C412yes—
146C2×D4—24no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4C22×C412yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C22×C412yes—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4#4112yes—
182C22×C4C2411yescharacteristic
183SD16—28no—
184M16—28no—
185M16—28no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8C22×C412yes—
193C2×C8C22×C412yes—
194C2×C8C2×D412yes—
195C2×C8C2×D412yes—
196C2×C8#4112yes—
197C2×C8#4112yes—
198C4⋊C4—24no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4C22×C412yes—
213C4⋊C4C22×C412yes—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—22no—
222C42—22no—
223C42—21no—
224C42—21no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42C2×D412yes—
229C42#4112yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C4×D412yes—
244C23C22×D411yescharacteristic
245Q8—22no—
246Q8—22no—
247Q8—41no—
248Q8—41no—
249D4—22no—
250D4—22no—
251D4—48no—
252D4—41no—
253D4—41no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—84no—
292C2×C4C4×D412yes—
293C2×C4#12112yes—
294C2×C4#12112yes—
295C2×C4AGL(1,ℤ/8)12yes—
296C2×C4#13712yes—
297C2×C4C23×C411yescharacteristic
298C2×C4C22×D411yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C8—22no—
304C8—22no—
305C8—22no—
306C8C4×D414yescomplemented
307C22—22no—
308C22—22no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×C4×D411yescharacteristic
322C22C2×#12111yescharacteristic
323C22#53211yescharacteristic
324C22C2×AGL(1,ℤ/8)11yescharacteristic
325C22C2×#13711yescharacteristic
326C22#57511yescharacteristic
327C22#57511yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4C2×C4×D411yescharacteristic
341C4C4×#4111yescharacteristic
342C4#52511yescharacteristic
343C4#52911yescharacteristic
344C2—42no—
345C2—82no—
346C2C4×#12111yescharacteristic
347C2C4×AGL(1,ℤ/8)11yescharacteristic
348C2C4×#13711yescharacteristic
349C2#305511yescharacteristic
350C2#305611yescharacteristic
351C2#318211yescharacteristic
352C2#318311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#440C2semidirect
3. C2×C4×M16C2semidirect
4. C4×C22⋊Q8C2semidirect
6. #1968C2semidirect
7. #1942C2semidirect
8. #1934C2semidirect
9. #1933C2semidirect
11. #1894C2semidirect
12. #1809C2semidirect
13. #1776C2semidirect
14. #1763C2semidirect
15. #1762C2semidirect
25. #468C4semidirect
71. C2×SD16C2×C4semidirect
77. C2×M16C2×C4semidirect
83. C22⋊Q8C2×C4semidirect
118. #107C2×C4semidirect
122. #103C2×C4semidirect
125. #102C2×C4semidirect
127. #97D4semidirect
306. C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4402
C2×C4×M162
C4×C22⋊Q82
C4×#1212
#19682
#19422
#19342
#19332
#18952
#18942
#18092
#17762
#17632
#17622