Finite Group Databasebeta

← #15635

Subgroups — #15635

Order 256.

1077 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1077 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394C212yesmaximal, complemented
2C2×#392C211yesmaximal, characteristic
3C2×#386C212yesmaximal, complemented
4C2×#385C211yesmaximal, characteristic, complemented
5#1442C212yesmaximal, complemented
6#1440C212yesmaximal, complemented
7#1438C212yesmaximal, complemented
8#1437C212yesmaximal, complemented
9#1437C211yesmaximal, characteristic, complemented
10C23×D4C2212yes—
11C23×D4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2212yes—
14C22×C22⋊C4C2211yescharacteristic
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C42C2211yescharacteristic
23#394—28no—
24#392—24no—
25#386—28no—
26#385—24no—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2211yescharacteristic
35C25C2312yes—
36C25C2311yescharacteristic
37C22×D4—24no—
38C22×D4—24no—
39C22×D4—24no—
40C22×D4—24no—
41C22×D4—24no—
42C22×D4—22no—
43C22×D4—24no—
44C22×D4—42no—
45C22×D4—42no—
46C22×D4—41no—
47C22×D4—41no—
48C22×D4—42no—
49C23×C4—22no—
50C23×C4—24no—
51C23×C4—41no—
52C23×C4—41no—
53C23×C4—42no—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2312yes—
60C23×C4C2312yes—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—42no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4D414yes—
86C2×C22⋊C4D418yescomplemented
87C2×C42—41no—
88C2×C42—42no—
89#95—24no—
90#95—24no—
91#95—24no—
92#95—22no—
93#95—24no—
94#95—24no—
95#95—42no—
96#95—41no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—21no—
101C24—21no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24C2×D414yes—
119C24C2×D412yes—
120C24#4114yes—
121C24C2411yescharacteristic
122C2×D4—216no—
123C2×D4—48no—
124C2×D4—48no—
125C2×D4—48no—
126C2×D4—44no—
127C2×D4—48no—
128C2×D4—44no—
129C2×D4—48no—
130C2×D4—48no—
131C2×D4—48no—
132C2×D4—44no—
133C2×D4—48no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
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—44no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4C2×D412yes—
179C22×C4C2×D414yes—
180C22×C4C2×D414yes—
181C22×C4C2×D412yes—
182C22×C4#4114yes—
183C22×C4#4114yes—
184C4⋊C4—48no—
185C22⋊C4—28no—
186C22⋊C4—44no—
187C22⋊C4—48no—
188C22⋊C4—48no—
189C22⋊C4—48no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C42—44no—
194C23—22no—
195C23—24no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—24no—
202C23—24no—
203C23—24no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23#12414yes—
250C23C22×D411yescharacteristic
251C23C22×D412yes—
252C23C2×#4112yes—
253C23C2×#4112yes—
254C23C2×#4112yes—
255C23H(5,𝔽2)11yescharacteristic
256C23H(5,𝔽2)12yes—
257C23H(5,𝔽2)11yescharacteristic
258C23H(5,𝔽2)12yes—
259D4—432no—
260D4—416no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
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—24no—
276C2×C4—24no—
277C2×C4—416no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C22—24no—
308C22—28no—
309C22—44no—
310C22—42no—
311C22—44no—
312C22—42no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22#52511yescharacteristic
332C22#52512yes—
333C22C2×#12411yescharacteristic
334C22#53212yes—
335C22#53411yescharacteristic
336C22#53412yes—
337C22#53812yes—
338C22#54012yes—
339C22#54012yes—
340C22D4211yescharacteristic
341C22D4212yes—
342C22#54612yes—
343C22#54612yes—
344C22#54612yes—
345C22#55112yes—
346C22#55312yes—
347C22#55312yes—
348C22#55312yes—
349C22#55911yescharacteristic
350C22#56011yescharacteristic
351C22H(3,𝔽4)11yescharacteristic
352C4—44no—
353C4—42no—
354C4—44no—
355C4—42no—
356C4—44no—
357C4—44no—
358C4—82no—
359C2—44no—
360C2—84no—
361C2#246412yes—
362C2#246412yes—
363C2#247212yes—
364C2#259111yescharacteristic
365C2#260212yes—
366C2#267911yescharacteristic
367C2#268812yes—
368C2#270911yescharacteristic
369C2#275312yes—
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. C2×#394C2semidirect
3. C2×#386C2semidirect
4. C2×#385C2semidirect
5. #1442C2semidirect
6. #1440C2semidirect
7. #1438C2semidirect
8. #1437C2semidirect
9. #1437C2semidirect
86. C2×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3942
C2×#3922
C2×#3862
C2×#3852
#14422
#14402
#14382
#14372