Finite Group Databasebeta

← #25549

Subgroups — #25549

Order 256.

695 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#469C211yesmaximal, characteristic, complemented
4#2414C211yesmaximal, characteristic, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1879C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19#547—24no—
20#547—24no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×#123—21no—
23C2×#123—21no—
24C2×C22⋊Q8—21no—
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#481—24no—
35#469—24no—
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#394—21no—
49#392—21no—
50#390—21no—
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—28no—
64C2×#41—44no—
65C2×#41—44no—
66C22×Q8—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4D411yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—41no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91#123—28no—
92#123—28no—
93C22⋊Q8—28no—
94#121—22no—
95#121—22no—
96#121—22no—
97#121—22no—
98#121—22no—
99#121—22no—
100#121—41no—
101#121D414yescomplemented
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106#117—22no—
107#117—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4D411yescharacteristic
116C2×C4⋊C4D414yescomplemented
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C22⋊C4—21no—
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⋊C4—21no—
132C2×C22⋊C4D411yescharacteristic
133C2×C22⋊C4D411yescharacteristic
134C2×C42—21no—
135C2×C42C2311yescharacteristic
136#106—22no—
137#106—22no—
138#106—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—41no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148#95—21no—
149#95—21no—
150C24—21no—
151C24—21no—
152C24C2×D411yescharacteristic
153C24C2×D411yescharacteristic
154#41—432no—
155C2×Q8—24no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4C2×D412yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D412yes—
214C22×C4C2×D412yes—
215C22×C4C2×D411yescharacteristic
216C22×C4C2411yescharacteristic
217D8—44no—
218M16—44no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8#4112yescomplemented
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—81no—
244C4⋊C4C2×D412yes—
245C4⋊C4C2×D412yes—
246C4⋊C4C2×D412yes—
247C4⋊C4#4112yescomplemented
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—22no—
251C22⋊C4—24no—
252C22⋊C4—22no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C42—21no—
260C42—21no—
261C42—42no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#12012yes—
289C23AGL(1,ℤ/8)12yes—
290C23C22×D411yescharacteristic
291Q8—48no—
292D4—28no—
293D4—22no—
294D4—22no—
295D4—48no—
296D4—48no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—84no—
304D4—84no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—48no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4#12012yes—
345C2×C4AGL(1,ℤ/8)12yes—
346C2×C4C22×D411yescharacteristic
347C2×C4C22×D411yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C2×C42−1+411yescharacteristic
352C8—22no—
353C8—41no—
354C8—41no—
355C8—41no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C2×#12011yescharacteristic
385C22#54611yescharacteristic
386C22#54711yescharacteristic
387C22C2×AGL(1,ℤ/8)11yescharacteristic
388C22C2×AGL(1,ℤ/8)11yescharacteristic
389C22#57711yescharacteristic
390C22#57711yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—41no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4#53211yescharacteristic
402C4#53711yescharacteristic
403C4#54611yescharacteristic
404C4#54711yescharacteristic
405C2—42no—
406C2—42no—
407C2—82no—
408C2—161no—
409C2#243011yescharacteristic
410C2#300311yescharacteristic
411C2#301011yescharacteristic
412C2#330611yescharacteristic
413C2#333211yescharacteristic
414C2#334511yescharacteristic
415C2#335411yescharacteristic
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×#547C2semidirect
2. C2×#481C2semidirect
3. C2×#469C2semidirect
4. #2414C2semidirect
5. #2058C2semidirect
6. #2034C2semidirect
7. #2023C2semidirect
8. #1999C2semidirect
9. #1879C2semidirect
10. #1875C2semidirect
11. #1874C2semidirect
12. #1870C2semidirect
13. #1864C2semidirect
14. #1853C2semidirect
15. #1736C2semidirect
101. #121D4semidirect
116. C2×C4⋊C4D4semidirect
230. C2×C8#41semidirect
247. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4812
C2×#4692
#24142
#20582
#20342
#20232
#19992
#18792
#18752
#18742
#18702
#18642
#18532
#17362