Finite Group Databasebeta

← #15668

Subgroups — #15668

Order 256.

999 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C2×#393C211yesmaximal, characteristic, complemented
3C2×#393C211yesmaximal, characteristic, complemented
4C2×#391C211yesmaximal, characteristic, complemented
5#1445C211yesmaximal, characteristic, complemented
6#1442C212yesmaximal, complemented
7#1440C212yesmaximal, complemented
8#1438C212yesmaximal
9#1437C211yesmaximal, characteristic, complemented
10#1437C211yesmaximal, characteristic, complemented
11C22⋊#95C212yesmaximal, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C2×#123—24no—
16C2×#123—24no—
17C2×#123—24no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27C22×C42C2211yescharacteristic
28#393—24no—
29#393—24no—
30#391—24no—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2311yescharacteristic
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—24no—
47C22×Q8—42no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—24no—
51C22×D4—22no—
52C22×D4—42no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—24no—
57C23×C4—42no—
58C23×C4C2312yes—
59C23×C4C2312yes—
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2312yes—
65C23×C4C2312yes—
66C23×C4C2311yescharacteristic
67#123—232no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4D414yes—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4D414yes—
94C2×C22⋊C4D414yes—
95C2×C42—22no—
96C2×C42—42no—
97#95—24no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95—22no—
102#95—22no—
103#95—24no—
104#95—42no—
105#95—42no—
106#95—42no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24C2×D412yes—
123C24C2×D412yes—
124C24#4114yes—
125C24C2411yescharacteristic
126C2×Q8—28no—
127C2×Q8—48no—
128C2×Q8—48no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×D4—28no—
132C2×D4—48no—
133C2×D4—48no—
134C2×D4—44no—
135C2×D4—44no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—28no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
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—48no—
182C22×C4—48no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4#4114yes—
193C22×C4#4114yes—
194C4⋊C4—28no—
195C4⋊C4—28no—
196C22⋊C4—28no—
197C22⋊C4—28no—
198C22⋊C4—28no—
199C22⋊C4—48no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C42—44no—
204C23—22no—
205C23—24no—
206C23—22no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—24no—
218C23—24no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23#12414yes—
246C23C22×D411yescharacteristic
247C23C22×D411yescharacteristic
248C23C22×D411yescharacteristic
249C23C2×#4112yes—
250C23C2×#4112yes—
251C23C2×#4112yes—
252C23H(5,𝔽2)12yes—
253C23H(5,𝔽2)12yes—
254C232−1+412yes—
255Q8—416no—
256D4—416no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—416no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4#12014yes—
311C22—24no—
312C22—28no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22C2×#12011yescharacteristic
330C22#52511yescharacteristic
331C22C2×#12411yescharacteristic
332C22#52912yes—
333C22#53211yescharacteristic
334C22#53511yescharacteristic
335C22#53511yescharacteristic
336C22#53611yescharacteristic
337C22#53812yes—
338C22#53912yes—
339C22#54212yes—
340C22#54612yes—
341C22#54612yes—
342C22#54612yes—
343C22#54612yes—
344C22#54712yes—
345C22#54812yes—
346C22#55112yes—
347C22#55212yes—
348C22#55912yes—
349C22#56212yes—
350C4—24no—
351C4—44no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—44no—
357C4—82no—
358C4—82no—
359C2—44no—
360C2—82no—
361C2#240411yescharacteristic
362C2#243211yescharacteristic
363C2#243511yescharacteristic
364C2#245812yes—
365C2#248812yes—
366C2#255712yes—
367C2#268712yes—
368C2#269612yes—
369C2#271012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#393C2semidirect
3. C2×#393C2semidirect
4. C2×#391C2semidirect
5. #1445C2semidirect
6. #1442C2semidirect
7. #1440C2semidirect
9. #1437C2semidirect
10. #1437C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#3932
C2×#3912
#14452
#14422
#14402
#14382
#14372
C22⋊#952