Finite Group Databasebeta

← #22866

Subgroups — #22866

Order 256.

609 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic, complemented
5#2339C211yesmaximal, characteristic, complemented
6#1968C211yesmaximal, characteristic, complemented
7#1935C211yesmaximal, characteristic, complemented
8#1927C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#449C418yescomplemented
27#442—24no—
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30#417—24no—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#386—21no—
48#386—21no—
49#386—21no—
50#385—21no—
51#381—21no—
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55C2×#95—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41C2×C414yescomplemented
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16C2×C414yescomplemented
82C2×D8—22no—
83C2×D8—22no—
84C2×D8C2×C414yescomplemented
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—41no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#121—22no—
93#121C2×C414yescomplemented
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—44no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121C2×C42C2311yescharacteristic
122#106—22no—
123#106—41no—
124#103—22no—
125#103—22no—
126#103C2×C414yescomplemented
127#102—22no—
128#102—22no—
129#102—41no—
130#102C2×C414yescomplemented
131#102D414yescomplemented
132C22⋊C8C2×C414yescomplemented
133#97—22no—
134C4×C8—41no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24C2×D411yescharacteristic
147#41—28no—
148#41—44no—
149#41—44no—
150#41—44no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8C22×C412yes—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4C22×C412yes—
172C2×D4C22×C412yes—
173C2×D4C2×D412yes—
174C2×D4#4112yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4C22×C412yes—
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4#4112yes—
207C22×C4C2411yescharacteristic
208SD16—44no—
209D8—28no—
210M16—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×C8C22×C412yes—
221C2×C8C22×C412yes—
222C2×C8C2×D412yes—
223C4⋊C4—21no—
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⋊C4—42no—
233C4⋊C4C22×C412yes—
234C4⋊C4C2×D412yes—
235C22⋊C4—24no—
236C22⋊C4—22no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—42no—
249C23—22no—
250C23—22no—
251C23—21no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23C4×D412yes—
270C23C22×D411yescharacteristic
271Q8—22no—
272Q8—22no—
273Q8—41no—
274Q8—41no—
275D4—22no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4—22no—
280D4—48no—
281D4—48no—
282D4—41no—
283D4—41no—
284D4C4×D414yescomplemented
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
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—84no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4C4×D412yes—
326C2×C4AGL(1,ℤ/8)12yes—
327C2×C4C23×C411yescharacteristic
328C2×C4C22×D411yescharacteristic
329C2×C4C2×#4111yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4H(5,𝔽2)11yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C8—22no—
334C8—22no—
335C8—22no—
336C8—41no—
337C22—22no—
338C22—24no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22C2×C4×D411yescharacteristic
359C22D4211yescharacteristic
360C22#54611yescharacteristic
361C22C2×AGL(1,ℤ/8)11yescharacteristic
362C22#57511yescharacteristic
363C22#57711yescharacteristic
364C22#57711yescharacteristic
365C4—22no—
366C4—22no—
367C4—42no—
368C4—42no—
369C4—41no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4C2×C4×D411yescharacteristic
377C4#51811yescharacteristic
378C4#54611yescharacteristic
379C4#55111yescharacteristic
380C2—42no—
381C2—44no—
382C2—82no—
383C2#235811yescharacteristic
384C2C4×AGL(1,ℤ/8)11yescharacteristic
385C2#295211yescharacteristic
386C2#327211yescharacteristic
387C2#327511yescharacteristic
388C2#330611yescharacteristic
389C2#330711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#442C2semidirect
3. C2×#417C2semidirect
4. #2361C2semidirect
5. #2339C2semidirect
6. #1968C2semidirect
7. #1935C2semidirect
8. #1927C2semidirect
9. #1895C2semidirect
10. #1879C2semidirect
11. #1877C2semidirect
12. #1864C2semidirect
13. #1853C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect
26. #449C4semidirect
67. C2×#41C2×C4semidirect
81. C2×SD16C2×C4semidirect
84. C2×D8C2×C4semidirect
93. #121C2×C4semidirect
126. #103C2×C4semidirect
130. #102C2×C4semidirect
131. #102D4semidirect
132. C22⋊C8C2×C4semidirect
284. D4C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
C2×#4422
C2×#4172
#23612
#23392
#19682
#19352
#19272
#18952
#18792
#18772
#18642
#18532
C4×#1022
#17532