Finite Group Databasebeta

← #25543

Subgroups — #25543

Order 256.

614 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×#554C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic
5#2058C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1880C211yesmaximal, characteristic, complemented
10#1874C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18#554—24no—
19#554—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35#481—24no—
36#465—24no—
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#106C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49#396—21no—
50#394—21no—
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#386—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86#125—28no—
87C22⋊Q8—28no—
88C22⋊Q8—28no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8D414yescomplemented
93#121—22no—
94#121—22no—
95#121—22no—
96#121—41no—
97#121—44no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×D4—28no—
101C4×D4—22no—
102C4×D4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4D411yescharacteristic
117C2×C4⋊C4D414yescomplemented
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4D411yescharacteristic
132C2×C22⋊C4D411yescharacteristic
133C2×C42—21no—
134C2×C42C2311yescharacteristic
135#106—22no—
136#106—22no—
137#106—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141#102—22no—
142#102—22no—
143#102—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147#95—21no—
148#95—21no—
149C24—21no—
150C24C2×D411yescharacteristic
151C24C2×D411yescharacteristic
152C2×Q8—24no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—81no—
159C2×Q8—81no—
160C2×Q8C2×D412yes—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—81no—
175C2×D4SD1614yescomplemented
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—24no—
186C22×C4—24no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D412yes—
206C22×C4C2×D412yes—
207C22×C4C2×D411yescharacteristic
208C22×C4C2411yescharacteristic
209SD16—44no—
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—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—44no—
222C2×C8#4112yes—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4C2×D412yes—
241C4⋊C4C2×D412yes—
242C4⋊C4C2×D412yes—
243C4⋊C4#4112yes—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—81no—
253C42—24no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23#12012yes—
274C23C2×SD1612yes—
275C23C2×SD1612yes—
276C23#13712yes—
277C23C22×D411yescharacteristic
278Q8—22no—
279Q8—22no—
280Q8—41no—
281Q8—41no—
282Q8—84no—
283D4—28no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—84no—
289D4—84no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
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—41no—
328C2×C4—41no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4#12012yes—
333C2×C4C2×SD1612yes—
334C2×C4C22×D411yescharacteristic
335C2×C4C22×D411yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C42−1+411yescharacteristic
340C8—22no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#44814yes—
365C22C2×#12011yescharacteristic
366C22#54611yescharacteristic
367C22#54711yescharacteristic
368C22C22×SD1611yescharacteristic
369C22C2×#13711yescharacteristic
370C22#57611yescharacteristic
371C22#57711yescharacteristic
372C4—22no—
373C4—22no—
374C4—41no—
375C4—42no—
376C4—41no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—81no—
382C4—81no—
383C4#53211yescharacteristic
384C4#53711yescharacteristic
385C4#54611yescharacteristic
386C4#54711yescharacteristic
387C2—24no—
388C2—42no—
389C2—161no—
390C2#243011yescharacteristic
391C2C2×#44811yescharacteristic
392C2#301211yescharacteristic
393C2#329511yescharacteristic
394C2#331611yescharacteristic
395C2#333511yescharacteristic
396C2#334611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2416C2
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×#554C2semidirect
2. C2×#481C2semidirect
3. C2×#465C2semidirect
5. #2058C2semidirect
6. #2035C2semidirect
7. #2023C2semidirect
8. #2000C2semidirect
9. #1880C2semidirect
10. #1874C2semidirect
11. #1872C2semidirect
12. #1869C2semidirect
13. #1865C2semidirect
14. #1853C2semidirect
15. C2×#340C2semidirect
92. C22⋊Q8D4semidirect
117. C2×C4⋊C4D4semidirect
175. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4812
C2×#4652
#24162
#20582
#20352
#20232
#20002
#18802
#18742
#18722
#18692
#18652
#18532
C2×#3402