Finite Group Databasebeta

← #30575

Subgroups — #30575

Order 256.

505 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3#2537C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic, complemented
5#2094C211yesmaximal, characteristic, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#121—21no—
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#481—24no—
28#469—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43#396—21no—
44#394—21no—
45#394—21no—
46#394—21no—
47#390C2211yescharacteristic
48#389C2211yescharacteristic
49#386—21no—
50#384C2211yescharacteristic
51#380—21no—
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4D411yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—41no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#121—22no—
83#121—22no—
84#121—22no—
85#121—41no—
86#121—41no—
87#121—41no—
88#121—41no—
89#121—41no—
90#121—44no—
91#121—44no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—41no—
95C4×D4—41no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4D411yescharacteristic
120C2×C22⋊C4D411yescharacteristic
121C2×C42C2311yescharacteristic
122#107—41no—
123#106—22no—
124#106—22no—
125#106—22no—
126#106—41no—
127#102—22no—
128#102—22no—
129#102—41no—
130#102—41no—
131#102—41no—
132#102D414yescomplemented
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140C24—21no—
141C24—21no—
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—21no—
147C2×D4—21no—
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—41no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4C2×D412yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2411yescharacteristic
200D8—44no—
201M16—44no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8C2×D412yes—
213C2×C8#4112yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4C2×D412yes—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23AGL(1,ℤ/8)12yes—
273C23C22×D411yescharacteristic
274D4—22no—
275D4—22no—
276D4—44no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—84no—
284D4—84no—
285D4—84no—
286D4—84no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
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—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—41no—
323C2×C4—41no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4AGL(1,ℤ/8)12yes—
327C2×C4C22×D411yescharacteristic
328C2×C4C2×#4111yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C2×C4H(5,𝔽2)11yescharacteristic
332C2×C42−1+411yescharacteristic
333C8—22no—
334C8—22no—
335C8—41no—
336C8—41no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—41no—
353C22—42no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22#53411yescharacteristic
366C22D4211yescharacteristic
367C22#54711yescharacteristic
368C22C2×AGL(1,ℤ/8)11yescharacteristic
369C22C2×AGL(1,ℤ/8)11yescharacteristic
370C22#57711yescharacteristic
371C22#57711yescharacteristic
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4#53711yescharacteristic
383C4#54011yescharacteristic
384C4#54611yescharacteristic
385C4#56211yescharacteristic
386C2—42no—
387C2—81no—
388C2—82no—
389C2—161no—
390C2#267111yescharacteristic
391C2#319211yescharacteristic
392C2#320411yescharacteristic
393C2#326511yescharacteristic
394C2#327211yescharacteristic
395C2#334511yescharacteristic
396C2#335411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×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
1. C2×#481C2semidirect
2. C2×#469C2semidirect
3. #2537C2semidirect
4. #2416C2semidirect
5. #2094C2semidirect
6. #2084C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #2033C2semidirect
10. #1999C2semidirect
11. #1879C2semidirect
12. #1870C2semidirect
13. #1862C2semidirect
14. #1853C2semidirect
15. #1809C2semidirect
132. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4692
#25372
#24162
#20942
#20842
#20342
#20332
#19992
#18792
#18702
#18622
#18532
#18092