Finite Group Databasebeta

← #23927

Subgroups — #23927

Order 256.

880 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 151 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×#537C211yesmaximal, characteristic
2C2×#536C211yesmaximal, characteristic, complemented
3C2×#452C212yesmaximal, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5C2×#448C211yesmaximal, characteristic, complemented
6#2001C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1873C211yesmaximal, characteristic, complemented
10#1872C212yesmaximal, complemented
11#1780C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C23×Q8C2211yescharacteristic
15C22×Q16C2212yes—
16C22×SD16C2212yes—
17#537—24no—
18#537—24no—
19#536—28no—
20C2×C4⋊Q8—21no—
21C2×#124—21no—
22C2×#123—21no—
23C2×#123—21no—
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8C2212yes—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2212yes—
29C2×#121—21no—
30C2×#121—21no—
31C2×#121—21no—
32C2×#121C2211yescharacteristic
33C2×#121C2211yescharacteristic
34C2×#117C2211yescharacteristic
35C22×C4⋊C4C2211yescharacteristic
36#452—28no—
37#451—24no—
38#448—24no—
39C2×#103C2212yes—
40C2×#103C2212yes—
41C2×#103C2212yes—
42C2×#102C2212yes—
43C2×C22⋊C8C2212yes—
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2212yes—
47#340C2212yes—
48#340C2212yes—
49C2×#41—22no—
50C2×#41—22no—
51C2×#41—24no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—24no—
55C2×#41—42no—
56C2×#41—42no—
57C22×Q8—24no—
58C22×Q8—21no—
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—42no—
62C22×Q8D411yescharacteristic
63C22×Q8C2312yes—
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C22×D4D411yescharacteristic
70C22×D4D411yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×Q16—24no—
79C2×Q16—24no—
80C2×Q16—42no—
81C2×SD16—24no—
82C2×SD16—24no—
83C2×SD16—42no—
84C22×C8C2312yes—
85C22×C8C2312yes—
86C4⋊Q8—28no—
87#124—28no—
88#123—28no—
89#123—28no—
90C22⋊Q8—28no—
91C22⋊Q8—24no—
92C22⋊Q8—22no—
93C22⋊Q8—24no—
94C22⋊Q8—22no—
95C22⋊Q8—24no—
96C22⋊Q8—22no—
97C22⋊Q8D414yes—
98C22⋊Q8D418yescomplemented
99#121—28no—
100#121—28no—
101#121—28no—
102#121—22no—
103#121—22no—
104#121—22no—
105#121D414yes—
106#117—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2312yes—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125#103—24no—
126#103—24no—
127#103—24no—
128#102—24no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—24no—
132C22⋊C8—42no—
133C24—21no—
134C24C2×D411yescharacteristic
135C24C2×D411yescharacteristic
136C24C2×D411yescharacteristic
137#41—48no—
138#41—48no—
139#41—48no—
140#41—48no—
141C2×Q8—24no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×Q8—48no—
154C2×Q8—48no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8C2×D412yes—
158C2×Q8C2×D414yes—
159C2×D4—24no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4C2×D412yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—82no—
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D412yes—
205C22×C4C2×D412yes—
206C22×C4C2×D414yes—
207C22×C4C2411yescharacteristic
208Q16—48no—
209SD16—48no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—28no—
219C4⋊C4—24no—
220C4⋊C4—28no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—22no—
224C4⋊C4C2×D414yes—
225C4⋊C4C2×D412yes—
226C4⋊C4C2×D412yes—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—28no—
231C22⋊C4—24no—
232C22⋊C4—28no—
233C22⋊C4—24no—
234C42—24no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23#12012yes—
254C23#12012yes—
255C23C22×D411yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—24no—
259Q8—44no—
260Q8—48no—
261Q8—42no—
262Q8—42no—
263Q8—42no—
264D4—24no—
265D4—44no—
266D4—44no—
267D4—44no—
268D4—48no—
269D4—42no—
270D4—84no—
271D4—84no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—84no—
301C2×C4#12012yes—
302C2×C4#12012yes—
303C2×C4C22×D412yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C22×D411yescharacteristic
306C2×C4H(5,𝔽2)12yes—
307C8—42no—
308C8—42no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—24no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—82no—
329C22C2×#12011yescharacteristic
330C22C2×#12011yescharacteristic
331C22#53411yescharacteristic
332C22#57712yes—
333C22#57812yes—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—42no—
338C4—44no—
339C4—42no—
340C4—42no—
341C4—44no—
342C4D4212yes—
343C4D4212yes—
344C2—42no—
345C2—42no—
346C2—81no—
347C2—82no—
348C2#240311yescharacteristic
349C2#300911yescharacteristic
350C2#301111yescharacteristic
351C2#301312yes—
352C2#320112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#536C2semidirect
3. C2×#452C2semidirect
4. C2×#451C2semidirect
5. C2×#448C2semidirect
6. #2001C2semidirect
7. #2000C2semidirect
8. #1875C2semidirect
9. #1873C2semidirect
10. #1872C2semidirect
11. #1780C2semidirect
98. C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#5362
C2×#4522
C2×#4512
C2×#4482
#20012
#20002
#18752
#18732
#18722
#17802