Finite Group Databasebeta

← #30567

Subgroups — #30567

Order 256.

497 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3#2535C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2085C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1807C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#121—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#481—24no—
27#466—24no—
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42#397—21no—
43#396—21no—
44#394—21no—
45#392—21no—
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#386—21no—
49#386—21no—
50#386—21no—
51#382C2211yescharacteristic
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—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—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—
91C4×D4—41no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—41no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4D411yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#107—41no—
121#106—22no—
122#106—22no—
123#106—41no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128#102—41no—
129#102D414yescomplemented
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
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—41no—
157C2×D4—42no—
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—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4C2×D412yes—
170C22×C4—21no—
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—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199D8—44no—
200C2×C8—21no—
201C2×C8—21no—
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—44no—
211C2×C8C2×D412yes—
212C2×C8#4112yes—
213C4⋊C4—21no—
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—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4C2×D412yes—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#13512yes—
269C23AGL(1,ℤ/8)12yes—
270C23C22×D411yescharacteristic
271D4—22no—
272D4—22no—
273D4—44no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—84no—
281D4—84no—
282D4—84no—
283D4—84no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
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—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—84no—
323C2×C4—84no—
324C2×C4C22×D411yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C2×C42−1+411yescharacteristic
330C8—22no—
331C8—22no—
332C8—41no—
333C8—41no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#53411yescharacteristic
363C22D4211yescharacteristic
364C22#54711yescharacteristic
365C22C2×#13511yescharacteristic
366C22C2×AGL(1,ℤ/8)11yescharacteristic
367C22#57611yescharacteristic
368C22#57711yescharacteristic
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—42no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4#53711yescharacteristic
380C4#54011yescharacteristic
381C4#54611yescharacteristic
382C4#56211yescharacteristic
383C2—42no—
384C2—42no—
385C2—82no—
386C2—161no—
387C2#267111yescharacteristic
388C2#318811yescharacteristic
389C2#319811yescharacteristic
390C2#326411yescharacteristic
391C2#328211yescharacteristic
392C2#333911yescharacteristic
393C2#334511yescharacteristic
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×#466C2semidirect
3. #2535C2semidirect
4. #2416C2semidirect
5. #2092C2semidirect
6. #2085C2semidirect
7. #2034C2semidirect
8. #2034C2semidirect
9. #2033C2semidirect
10. #1999C2semidirect
11. #1879C2semidirect
12. #1869C2semidirect
13. #1864C2semidirect
14. #1852C2semidirect
15. #1807C2semidirect
129. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4662
#25352
#24162
#20922
#20852
#20342
#20332
#19992
#18792
#18692
#18642
#18522
#18072