Finite Group Databasebeta

← #24663

Subgroups — #24663

Order 256.

456 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2620C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2000C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic, complemented
10#1973C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1947C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic, complemented
14#1933C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#463—24no—
25#433—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#396—21no—
43#396—21no—
44#394—21no—
45#394—21no—
46#391C2211yescharacteristic
47#390—21no—
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#389C2211yescharacteristic
51#388—21no—
52#386—21no—
53#385—21no—
54#384C2211yescharacteristic
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—41no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#124—22no—
79#124—22no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82#121—22no—
83#121—22no—
84#121—41no—
85#121—41no—
86C4×D4—22no—
87C4×D4—22no—
88#117—22no—
89#117—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
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×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4D411yescharacteristic
113C2×C22⋊C4D411yescharacteristic
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—41no—
118#107—41no—
119C4⋊C8—22no—
120C4⋊C8D414yescomplemented
121#103—22no—
122#103—22no—
123#103—41no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128#102—41no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131#97—22no—
132C4×C8—22no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×D4—24no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—42no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4#4112yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2411yescharacteristic
186SD16—44no—
187D8—44no—
188C2×C8—21no—
189C2×C8—21no—
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×C8C2×D412yes—
199C2×C8C2×D412yes—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4#4112yes—
216C22⋊C4—24no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C42—21no—
230C42—21no—
231C42—81no—
232C42C2×D412yes—
233C42#4112yes—
234C23—21no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C22×D411yescharacteristic
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260D4—22no—
261D4—22no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—84no—
269D4—84no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
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—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—84no—
306C2×C4#12112yes—
307C2×C4#12112yes—
308C2×C4C22×D411yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C8—22no—
315C8—22no—
316C8—41no—
317C8—41no—
318C22—41no—
319C22—42no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22C2×#12111yescharacteristic
340C22#52511yescharacteristic
341C22#53511yescharacteristic
342C22#57611yescharacteristic
343C22#57611yescharacteristic
344C22#57711yescharacteristic
345C22#57811yescharacteristic
346C4—22no—
347C4—22no—
348C4—42no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4#54611yescharacteristic
358C4#55011yescharacteristic
359C4#55311yescharacteristic
360C4#55511yescharacteristic
361C2—81no—
362C2—82no—
363C2—161no—
364C2#241411yescharacteristic
365C2#306111yescharacteristic
366C2#306211yescharacteristic
367C2#311811yescharacteristic
368C2#312411yescharacteristic
369C2#322011yescharacteristic
370C2#322511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#463C2semidirect
2. C2×#433C2semidirect
3. #2620C2semidirect
4. #2324C2semidirect
5. #2036C2semidirect
6. #2033C2semidirect
7. #2000C2semidirect
9. #1983C2semidirect
10. #1973C2semidirect
11. #1970C2semidirect
12. #1947C2semidirect
13. #1936C2semidirect
14. #1933C2semidirect
15. #1802C2semidirect
120. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4332
#26202
#23242
#20362
#20332
#20002
#19992
#19832
#19732
#19702
#19472
#19362
#19332
#18022