Finite Group Databasebeta

← #29662

Subgroups — #29662

Order 256.

652 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 137 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×#529C211yesmaximal, characteristic, complemented
2C2×#460C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic, complemented
4#2856C211yesmaximal, characteristic, complemented
5#2025C212yesmaximal, complemented
6#2012C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1999C212yesmaximal, complemented
9#1803C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15#529—24no—
16#529—24no—
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121—21no—
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2212yes—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30C22×C42C2211yescharacteristic
31#460—24no—
32#459—24no—
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2212yes—
36C2×#102C2212yes—
37C2×#102C2212yes—
38C2×#102C2212yes—
39C2×C22⋊C8C2212yes—
40#390—21no—
41#390—21no—
42#390C2212yes—
43#384C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8C2311yescharacteristic
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2312yes—
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×SD16—24no—
59C2×SD16—42no—
60C2×SD16—42no—
61C2×D8—24no—
62C2×D8—42no—
63C2×D8—42no—
64C22×C8C2312yes—
65C22×C8C2312yes—
66#127—22no—
67#127—22no—
68#126—28no—
69#125—22no—
70#125—22no—
71#124—22no—
72#124—22no—
73#123—28no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76#121—22no—
77#121—22no—
78#121—42no—
79#121—42no—
80#121—44no—
81#121—44no—
82C4×Q8—22no—
83C4×Q8D414yescomplemented
84C4×D4—28no—
85C4×D4—22no—
86C4×D4D414yescomplemented
87#117—28no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42D411yescharacteristic
109C2×C42D411yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112C4⋊C8—22no—
113C4⋊C8—22no—
114#103—42no—
115#102—24no—
116#102—24no—
117#102—42no—
118C22⋊C8—42no—
119C24—21no—
120C24—22no—
121C24C2×D411yescharacteristic
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8C2×D412yes—
128C2×D4—24no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4C2×D412yes—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177SD16—48no—
178D8—48no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4C2×D412yes—
196C4⋊C4C2×D412yes—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C42—22no—
206C42—22no—
207C42—24no—
208C42—21no—
209C42—21no—
210C42C2×D412yes—
211C42C2×D412yes—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23AGL(1,ℤ/8)12yes—
233C23C22×D411yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—41no—
237Q8—41no—
238D4—22no—
239D4—22no—
240D4—48no—
241D4—41no—
242D4—42no—
243D4—42no—
244D4—41no—
245D4—42no—
246D4—42no—
247D4—84no—
248D4—88no—
249D4—84no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—44no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4#12012yes—
281C2×C4#12012yes—
282C2×C4#13512yes—
283C2×C4AGL(1,ℤ/8)12yes—
284C2×C4C22×D411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C2×C4H(5,𝔽2)11yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C42−1+411yescharacteristic
290C8—42no—
291C8—42no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22C2×#12011yescharacteristic
314C22#53511yescharacteristic
315C22#53711yescharacteristic
316C22C2×#13511yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22C2×AGL(1,ℤ/8)11yescharacteristic
319C22#57511yescharacteristic
320C4—22no—
321C4—41no—
322C4—42no—
323C4—41no—
324C4—42no—
325C4—41no—
326C4—42no—
327C4—82no—
328C4#44914yes—
329C4#53511yescharacteristic
330C4#53511yescharacteristic
331C4#53711yescharacteristic
332C4#53711yescharacteristic
333C2—42no—
334C2—82no—
335C2—162no—
336C2#261511yescharacteristic
337C2C2×#44911yescharacteristic
338C2#300611yescharacteristic
339C2#321211yescharacteristic
340C2#321711yescharacteristic
341C2#324711yescharacteristic
342C2#325111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#529C2semidirect
2. C2×#460C2semidirect
3. C2×#459C2semidirect
4. #2856C2semidirect
5. #2025C2semidirect
6. #2012C2semidirect
7. #2000C2semidirect
8. #1999C2semidirect
9. #1803C2semidirect
10. #1802C2semidirect
11. #1797C2semidirect
83. C4×Q8D4semidirect
86. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4602
C2×#4592
#28562
#20252
#20122
#20002
#19992
#18032
#18022
#17972