Finite Group Databasebeta

← #16504

Subgroups — #16504

Order 256.

1133 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic, complemented
2C2×#394C212yesmaximal, complemented
3C2×#393C211yesmaximal, characteristic, complemented
4C2×#392C211yesmaximal, characteristic
5C2×#388C211yesmaximal, characteristic, complemented
6#1446C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C212yesmaximal, complemented
10#1438C211yesmaximal, characteristic, complemented
11C4⋊#95C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C23×D4C2212yes—
15C2×#124—28no—
16C2×#124—24no—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2212yes—
28C22×C42C2211yescharacteristic
29#394—28no—
30#393—24no—
31#392—24no—
32#388—24no—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2312yes—
43C22×Q8—24no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×D4—24no—
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—24no—
54C22×D4—22no—
55C22×D4—42no—
56C22×D4—42no—
57C22×D4—42no—
58C22×D4—42no—
59C22×D4—42no—
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2312yes—
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69#124—232no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4D414yes—
99C2×C22⋊C4D418yescomplemented
100C2×C42—24no—
101C2×C42D414yescomplemented
102#95—22no—
103#95—22no—
104#95—22no—
105#95—24no—
106#95—22no—
107#95—24no—
108#95—42no—
109#95—42no—
110C24—21no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—21no—
128C24—21no—
129C24—22no—
130C24—21no—
131C24—21no—
132C24—22no—
133C24—22no—
134C24C2×D412yes—
135C24C2×D414yes—
136C24C2411yescharacteristic
137C2×Q8—28no—
138C2×Q8—48no—
139C2×Q8—48no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×D4—28no—
143C2×D4—48no—
144C2×D4—48no—
145C2×D4—48no—
146C2×D4—48no—
147C2×D4—44no—
148C2×D4—48no—
149C2×D4—48no—
150C2×D4—48no—
151C2×D4—48no—
152C2×D4—48no—
153C2×D4—44no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4C2×D412yes—
197C22×C4C2×D414yes—
198C22×C4C2×D412yes—
199C22×C4C2×D412yes—
200C22×C4C2×D414yes—
201C22×C4C2×D412yes—
202C22×C4C2×D412yes—
203C22×C4#4114yes—
204C22×C4#4112yes—
205C4⋊C4—48no—
206C4⋊C4—44no—
207C22⋊C4—216no—
208C22⋊C4—216no—
209C22⋊C4—28no—
210C22⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—44no—
213C22⋊C4—48no—
214C22⋊C4—48no—
215C22⋊C4—44no—
216C42—28no—
217C23—24no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23—22no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23C22×D411yescharacteristic
276C23C22×D412yes—
277C23C22×D411yescharacteristic
278C23C2×#4111yescharacteristic
279C23C2×#4112yes—
280C23H(5,𝔽2)11yescharacteristic
281C23H(5,𝔽2)12yes—
282C23H(5,𝔽2)11yescharacteristic
283C23H(5,𝔽2)12yes—
284C232−1+411yescharacteristic
285C232−1+411yescharacteristic
286Q8—416no—
287D4—416no—
288D4—432no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4#12014yes—
339C2×C4#12114yes—
340C22—44no—
341C22—42no—
342C22—44no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—81no—
353C22—82no—
354C22—81no—
355C22—82no—
356C22—82no—
357C22—81no—
358C22—82no—
359C22—82no—
360C22—81no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22C2×#12011yescharacteristic
367C22C2×#12111yescharacteristic
368C22#53211yescharacteristic
369C22#53511yescharacteristic
370C22#53511yescharacteristic
371C22#53512yes—
372C22#53511yescharacteristic
373C22#53611yescharacteristic
374C22#53711yescharacteristic
375C22#53711yescharacteristic
376C22#53712yes—
377C22#53812yes—
378C22#54211yescharacteristic
379C22#54211yescharacteristic
380C22#54212yes—
381C22D4212yes—
382C22#54612yes—
383C22#54612yes—
384C22#54612yes—
385C22#54712yes—
386C22#55912yes—
387C22H(3,𝔽4)12yes—
388C22#56212yes—
389C4—24no—
390C4—24no—
391C4—42no—
392C4—44no—
393C4—44no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—82no—
398C2—84no—
399C2—82no—
400C2#240811yescharacteristic
401C2#243011yescharacteristic
402C2#254411yescharacteristic
403C2#261511yescharacteristic
404C2#262011yescharacteristic
405C2#266211yescharacteristic
406C2#267012yes—
407C2#267212yes—
408C2#271012yes—
409C2#271212yes—
410C2#281911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC2×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#124C2semidirect
2. C2×#394C2semidirect
3. C2×#393C2semidirect
5. C2×#388C2semidirect
6. #1446C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
9. #1438C2semidirect
10. #1438C2semidirect
11. C4⋊#95C2semidirect
99. C2×C22⋊C4D4semidirect
101. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3942
C2×#3932
C2×#3922
C2×#3882
#14462
#14392
#14382
C4⋊#952