Finite Group Databasebeta

← #16680

Subgroups — #16680

Order 256.

1091 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic, complemented
2C2×#393C211yesmaximal, characteristic, complemented
3C2×#392C211yesmaximal, characteristic
4C2×#388C211yesmaximal, characteristic, complemented
5#1446C211yesmaximal, characteristic, complemented
6#1440C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C212yesmaximal, complemented
9#1438C212yesmaximal, complemented
10#1438C211yesmaximal, characteristic, complemented
11#1434C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2212yes—
14C23×D4C2211yescharacteristic
15C2×#124—24no—
16C2×#124—24no—
17C2×#124—24no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2212yes—
28C22×C42C2211yescharacteristic
29#393—24no—
30#392—24no—
31#388—24no—
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2312yes—
42C25C2311yescharacteristic
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—42no—
46C22×Q8—42no—
47C22×Q8D414yescomplemented
48C22×D4—24no—
49C22×D4—24no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—24no—
53C22×D4—42no—
54C22×D4—42no—
55C22×D4—42no—
56C22×D4—42no—
57C22×D4—42no—
58C22×D4—42no—
59C22×D4D414yes—
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2312yes—
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69#124—232no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—42no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—42no—
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×C42—22no—
99C2×C42—22no—
100C2×C42D414yescomplemented
101#95—22no—
102#95—22no—
103#95—22no—
104#95—24no—
105#95—22no—
106#95—24no—
107#95—42no—
108#95—42no—
109#95—42no—
110C24—21no—
111C24—21no—
112C24—22no—
113C24—22no—
114C24—21no—
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—22no—
128C24—22no—
129C24—22no—
130C24—22no—
131C24—21no—
132C24—22no—
133C24—21no—
134C24C2×D414yes—
135C24#4112yes—
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—44no—
149C2×D4—48no—
150C2×D4—48no—
151C2×D4—48no—
152C2×D4—48no—
153C2×D4—48no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
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—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4C2×D412yes—
200C22×C4C2×D414yes—
201C22×C4C2×D412yes—
202C22×C4C2×D412yes—
203C22×C4C2×D412yes—
204C22×C4C2×D412yes—
205C22×C4#4114yes—
206C22×C4#4112yes—
207C4⋊C4—44no—
208C22⋊C4—28no—
209C22⋊C4—216no—
210C22⋊C4—216no—
211C22⋊C4—48no—
212C22⋊C4—48no—
213C22⋊C4—44no—
214C22⋊C4—48no—
215C22⋊C4—48no—
216C22⋊C4—44no—
217C42—28no—
218C23—24no—
219C23—22no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—22no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—22no—
229C23—24no—
230C23—41no—
231C23—42no—
232C23—41no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—41no—
259C23—42no—
260C23—41no—
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—
275C23—42no—
276C23C22×D411yescharacteristic
277C23C22×D411yescharacteristic
278C23C22×D411yescharacteristic
279C23C2×#4111yescharacteristic
280C23C2×#4111yescharacteristic
281C23C2×#4112yes—
282C23H(5,𝔽2)12yes—
283C23H(5,𝔽2)12yes—
284C23H(5,𝔽2)12yes—
285C23H(5,𝔽2)11yescharacteristic
286C232−1+411yescharacteristic
287Q8—416no—
288D4—416no—
289D4—432no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
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—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4#12014yes—
343C22—42no—
344C22—44no—
345C22—42no—
346C22—44no—
347C22—42no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22—81no—
359C22—82no—
360C22—82no—
361C22—81no—
362C22—82no—
363C22—81no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22C2×#12011yescharacteristic
370C22#52511yescharacteristic
371C22#53211yescharacteristic
372C22#53412yes—
373C22#53412yes—
374C22#53511yescharacteristic
375C22#53511yescharacteristic
376C22#53511yescharacteristic
377C22#53711yescharacteristic
378C22#53812yes—
379C22#53812yes—
380C22#53812yes—
381C22#54212yes—
382C22#54611yescharacteristic
383C22#54612yes—
384C22#54711yescharacteristic
385C22#54811yescharacteristic
386C22#55011yescharacteristic
387C22#55012yes—
388C22#55112yes—
389C22#55512yes—
390C22H(3,𝔽4)12yes—
391C22#56212yes—
392C4—24no—
393C4—44no—
394C4—42no—
395C4—42no—
396C4—44no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—82no—
401C2—82no—
402C2—84no—
403C2#240711yescharacteristic
404C2#243111yescharacteristic
405C2#261011yescharacteristic
406C2#261311yescharacteristic
407C2#263111yescharacteristic
408C2#266211yescharacteristic
409C2#268412yes—
410C2#268612yes—
411C2#273012yes—
412C2#273312yes—
413C2#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×#393C2semidirect
4. C2×#388C2semidirect
5. #1446C2semidirect
6. #1440C2semidirect
7. #1439C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
10. #1438C2semidirect
11. #1434C2semidirect
47. C22×Q8D4semidirect
100. 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×#3932
C2×#3922
C2×#3882
#14462
#14402
#14392
#14382
#14342