Finite Group Databasebeta

← #22497

Subgroups — #22497

Order 256.

587 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic
3#2352C211yesmaximal, characteristic
4#1993C211yesmaximal, characteristic, complemented
5#1988C211yesmaximal, characteristic, complemented
6#1894C211yesmaximal, characteristic, complemented
7#1892C211yesmaximal, characteristic, complemented
8#1873C211yesmaximal, characteristic, complemented
9#1872C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1823C211yesmaximal, characteristic, complemented
12#1805C211yesmaximal, characteristic, complemented
13#1763C211yesmaximal, characteristic, complemented
14#1736C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18#554—24no—
19#554C418yescomplemented
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#435—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#388—21no—
43#387—21no—
44#386—21no—
45#382C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4D411yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—41no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8C2×C414yescomplemented
82#125—28no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—28no—
86C22⋊Q8—28no—
87C22⋊Q8C2×C414yes—
88C22⋊Q8C2×C414yescomplemented
89C4×Q8—22no—
90C4×Q8C2×C414yescomplemented
91C4×D4—28no—
92C4×D4C2×C414yes—
93C4×D4D414yes—
94#117—22no—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C2×C414yes—
108C2×C4⋊C4C2×C414yescomplemented
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C2311yescharacteristic
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—41no—
132#103—41no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135#97—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C2×Q8—24no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—81no—
150C2×Q8—81no—
151C2×Q8C22×C412yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4C22⋊C414yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4C22×C412yes—
181C22×C4C22×C412yes—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D412yes—
187C22×C4C2×D412yes—
188C22×C4C2411yescharacteristic
189M16—44no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—44no—
202C2×C8#4112yes—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—21no—
210C4⋊C4—21no—
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⋊C4C22⋊C414yes—
221C4⋊C4C22×C412yes—
222C4⋊C4C22×C412yes—
223C4⋊C4C22×C412yes—
224C4⋊C4#4112yes—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—81no—
230C22⋊C4C22⋊C414yes—
231C22⋊C4C22⋊C414yes—
232C42—24no—
233C42—42no—
234C42—81no—
235C42C22×C412yes—
236C42C2×D412yes—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23C2×C22⋊C412yes—
248C23C2×C22⋊C412yes—
249C23#13712yes—
250C23#13712yes—
251C23C22×D411yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—41no—
255Q8—41no—
256Q8—84no—
257D4—28no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
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—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—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4C2×C22⋊C412yes—
302C2×C4C2×C22⋊C412yes—
303C2×C4C2×C22⋊C412yes—
304C2×C4C2×C22⋊C412yes—
305C2×C4C23×C411yescharacteristic
306C2×C4C22×D411yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C42−1+411yescharacteristic
311C8—22no—
312C8—41no—
313C8—41no—
314C8—41no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22#41914yes—
329C22C22×C22⋊C411yescharacteristic
330C22#54611yescharacteristic
331C22#54711yescharacteristic
332C22C2×#13711yescharacteristic
333C22C2×#13711yescharacteristic
334C22#57611yescharacteristic
335C22#57711yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4C4×#4111yescharacteristic
349C4#52011yescharacteristic
350C4#54611yescharacteristic
351C4#54711yescharacteristic
352C2—24no—
353C2—42no—
354C2#233911yescharacteristic
355C2C2×#41911yescharacteristic
356C2#289711yescharacteristic
357C2#331411yescharacteristic
358C2#331611yescharacteristic
359C2#334411yescharacteristic
360C2#334611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1993C2semidirect
5. #1988C2semidirect
6. #1894C2semidirect
7. #1892C2semidirect
8. #1873C2semidirect
9. #1872C2semidirect
10. #1824C2semidirect
11. #1823C2semidirect
12. #1805C2semidirect
13. #1763C2semidirect
14. #1736C2semidirect
15. C2×#340C2semidirect
19. #554C4semidirect
81. C4⋊Q8C2×C4semidirect
88. C22⋊Q8C2×C4semidirect
90. C4×Q8C2×C4semidirect
108. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4352
#23522
#19932
#19882
#18942
#18922
#18732
#18722
#18242
#18232
#18052
#17632
#17362
C2×#3402