Finite Group Databasebeta

← #24504

Subgroups — #24504

Order 256.

529 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic
2C2×#431C211yesmaximal, characteristic, complemented
3#2854C211yesmaximal, characteristic, complemented
4#2297C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#1970C211yesmaximal, characteristic, complemented
10#1968C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic, complemented
12#1933C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#123—21no—
20C2×#121C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#449—24no—
24#431—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#388—21no—
50#388—21no—
51#386—21no—
52#386—21no—
53#386—21no—
54#380C2211yescharacteristic
55C4×C22⋊C4—21no—
56C4×C22⋊C4—21no—
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4D411yescharacteristic
74C23×C4D411yescharacteristic
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—41no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#123—44no—
87#121—22no—
88#121—22no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
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×C42—22no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42D411yescharacteristic
113C2×C42D411yescharacteristic
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#107—41no—
117#106—41no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120#103—22no—
121#103—22no—
122#103—41no—
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127#102—41no—
128C22⋊C8—22no—
129C22⋊C8D414yes—
130C4×C8—22no—
131C4×C8—22no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142#41—44no—
143#41—44no—
144#41—44no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8#4112yes—
152C2×D4—24no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4#4112yes—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D412yes—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4#4112yes—
196C22×C4C2411yescharacteristic
197SD16—44no—
198D8—44no—
199C2×C8—21no—
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×C8C2×D412yes—
210C2×C8C2×D412yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
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—42no—
222C4⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—22no—
235C42—22no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—42no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#12112yes—
262C23C22×D411yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—41no—
266Q8—41no—
267D4—22no—
268D4—22no—
269D4—48no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—84no—
277C2×C4—28no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—48no—
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—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4#12112yes—
314C2×C4#13512yes—
315C2×C4#13512yes—
316C2×C4AGL(1,ℤ/8)12yes—
317C2×C4C22×D411yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C8—22no—
324C8—22no—
325C8—41no—
326C8—41no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22C2×#12111yescharacteristic
349C22#52511yescharacteristic
350C22#53511yescharacteristic
351C22C2×#13511yescharacteristic
352C22C2×#13511yescharacteristic
353C22C2×AGL(1,ℤ/8)11yescharacteristic
354C22#57511yescharacteristic
355C4—22no—
356C4—22no—
357C4—22no—
358C4—22no—
359C4—42no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4#54611yescharacteristic
367C4#54611yescharacteristic
368C4#55111yescharacteristic
369C4#55111yescharacteristic
370C2—42no—
371C2—82no—
372C2—161no—
373C2#241111yescharacteristic
374C2#305011yescharacteristic
375C2#305511yescharacteristic
376C2#310211yescharacteristic
377C2#310611yescharacteristic
378C2#320711yescharacteristic
379C2#321211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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×#431C2semidirect
3. #2854C2semidirect
4. #2297C2semidirect
5. #2068C2semidirect
6. #2065C2semidirect
7. #2055C2semidirect
8. #2054C2semidirect
9. #1970C2semidirect
10. #1968C2semidirect
11. #1934C2semidirect
12. #1933C2semidirect
13. #1875C2semidirect
14. #1854C2semidirect
15. #1788C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
C2×#4312
#28542
#22972
#20682
#20652
#20552
#20542
#19702
#19682
#19342
#19332
#18752
#18542
#17882