Finite Group Databasebeta

← #23575

Subgroups — #23575

Order 256.

630 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529C211yesmaximal, characteristic, complemented
2C2×#442C211yesmaximal, characteristic, complemented
3C2×#440C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2025C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#1999C211yesmaximal, characteristic
13#1763C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15#1749C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18#529—24no—
19#529—24no—
20C2×#126—21no—
21C2×#125C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×#123—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117—21no—
33C22×C42C2211yescharacteristic
34#442—24no—
35#440—24no—
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×#97C2211yescharacteristic
49C2×#97C2211yescharacteristic
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#388—21no—
53#386—21no—
54#380C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C22⋊C4—21no—
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4D411yescharacteristic
71C23×C4D411yescharacteristic
72C23×C4C2311yescharacteristic
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#126—28no—
84#125—22no—
85#125—22no—
86#124—22no—
87#124—22no—
88#123—28no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95C4×Q8—22no—
96C4×Q8D414yescomplemented
97C4×D4—28no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—44no—
102C4×D4D414yescomplemented
103#117—28no—
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×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42D411yescharacteristic
129C2×C42D411yescharacteristic
130C2×C42C2311yescharacteristic
131C2×C42C2311yescharacteristic
132#107—22no—
133#106—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144#97—22no—
145#97—22no—
146#95—21no—
147#95—21no—
148#95—21no—
149C24—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8C2×D412yes—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4C2×D412yes—
175C2×D4#4112yes—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2411yescharacteristic
206SD16—28no—
207D8—28no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8#4112yes—
219C2×C8#4112yes—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4C2×D412yes—
233C4⋊C4C2×D412yes—
234C4⋊C4#4112yes—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C42—22no—
246C42—22no—
247C42—24no—
248C42—21no—
249C42—21no—
250C42—42no—
251C42C2×D412yes—
252C42C2×D412yes—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23AGL(1,ℤ/8)12yes—
274C23C22×D411yescharacteristic
275Q8—22no—
276Q8—22no—
277Q8—41no—
278Q8—41no—
279D4—22no—
280D4—22no—
281D4—22no—
282D4—22no—
283D4—48no—
284D4—48no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—48no—
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—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—84no—
328C2×C4#12012yes—
329C2×C4#12012yes—
330C2×C4C22×D411yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4C2×#4111yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C8—22no—
337C8—22no—
338C8—41no—
339C8—41no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22C2×#12011yescharacteristic
362C22#52511yescharacteristic
363C22#53211yescharacteristic
364C22C2×AGL(1,ℤ/8)11yescharacteristic
365C22#57511yescharacteristic
366C22#57511yescharacteristic
367C22#57511yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—41no—
372C4—42no—
373C4—41no—
374C4—42no—
375C4—41no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4#52511yescharacteristic
380C4#52511yescharacteristic
381C4#53211yescharacteristic
382C4#53211yescharacteristic
383C2—42no—
384C2—82no—
385C2—82no—
386C2#238811yescharacteristic
387C2#300511yescharacteristic
388C2#300611yescharacteristic
389C2#311111yescharacteristic
390C2#311411yescharacteristic
391C2#318011yescharacteristic
392C2#318311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
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
1. C2×#529C2semidirect
2. C2×#442C2semidirect
3. C2×#440C2semidirect
4. C4×#121C2semidirect
5. #2059C2semidirect
6. #2042C2semidirect
7. #2035C2semidirect
8. #2033C2semidirect
9. #2025C2semidirect
10. #2012C2semidirect
11. #2000C2semidirect
13. #1763C2semidirect
14. #1762C2semidirect
15. #1749C2semidirect
96. C4×Q8D4semidirect
102. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4422
C2×#4402
C4×#1212
#20592
#20422
#20352
#20332
#20252
#20122
#20002
#19992
#17632
#17622
#17492