Finite Group Databasebeta

← #33011

Subgroups — #33011

Order 256.

673 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic
4#2855C211yesmaximal, characteristic, complemented
5#1878C211yesmaximal, characteristic, complemented
6#1878C211yesmaximal, characteristic, complemented
7#1877C211yesmaximal, characteristic, complemented
8#1877C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1864C211yesmaximal, characteristic, complemented
11#1863C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1817C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C4×#41—24no—
21C4×#41—24no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117—21no—
29C2×#117C2211yescharacteristic
30C22×C42—21no—
31C22×C42C2211yescharacteristic
32#485—24no—
33#452—24no—
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#387—21no—
51#387—21no—
52#385—21no—
53#381—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41D414yes—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4D411yescharacteristic
77C23×C4D411yescharacteristic
78C23×C4C2311yescharacteristic
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×D4—28no—
96C4×D4—28no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101#117—28no—
102#117—22no—
103#117D414yes—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C42—28no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42D411yescharacteristic
126C2×C42D411yescharacteristic
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129#107—22no—
130#106—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#102—22no—
137#102—22no—
138#102—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24C2×D411yescharacteristic
150#41—28no—
151#41—28no—
152#41—28no—
153#41—28no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8C2×D412yes—
164C2×Q8#4112yes—
165C2×Q8#4112yes—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4C2×D412yes—
173C2×D4#4112yes—
174C2×D4#4112yes—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D412yes—
205C22×C4C2×D412yes—
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2411yescharacteristic
209Q16—44no—
210SD16—44no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4C2×D412yes—
237C4⋊C4C2×D412yes—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—24no—
244C42—24no—
245C42—22no—
246C42—24no—
247C42—22no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23#12012yes—
266C23C22×D411yescharacteristic
267Q8—22no—
268Q8—22no—
269Q8—22no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8#12314yes—
275D4—216no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4#12314yes—
280C2×C4—216no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4#12012yes—
323C2×C4#13512yes—
324C2×C4#13712yes—
325C2×C4C22×D411yescharacteristic
326C2×C4C22×D411yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4C2×#4111yescharacteristic
329C2×C4C2×#4111yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C8—41no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—22no—
336C22—24no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22C2×#12011yescharacteristic
350C22#52511yescharacteristic
351C22#52511yescharacteristic
352C22C2×#13511yescharacteristic
353C22C2×#13711yescharacteristic
354C22#57511yescharacteristic
355C22#57511yescharacteristic
356C4—22no—
357C4—24no—
358C4—22no—
359C4—22no—
360C4—41no—
361C4—42no—
362C4—41no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4C2×#12311yescharacteristic
369C4C2×#12311yescharacteristic
370C4#52511yescharacteristic
371C4#52511yescharacteristic
372C2—42no—
373C2—44no—
374C2#284911yescharacteristic
375C2#300211yescharacteristic
376C2#300711yescharacteristic
377C2#310211yescharacteristic
378C2#310711yescharacteristic
379C2#311111yescharacteristic
380C2#311611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×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×#485C2semidirect
4. #2855C2semidirect
5. #1878C2semidirect
6. #1878C2semidirect
7. #1877C2semidirect
8. #1877C2semidirect
9. #1865C2semidirect
10. #1864C2semidirect
11. #1863C2semidirect
12. #1862C2semidirect
13. #1825C2semidirect
14. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
C2×#4852
C2×#4522
#28552
#18782
#18772
#18652
#18642
#18632
#18622
#18252
#18172
#17882