Finite Group Databasebeta

← #29665

Subgroups — #29665

Order 256.

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

normal subgroups only

This group has 582 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
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4#2857C211yesmaximal, characteristic, complemented
5#2026C212yesmaximal, complemented
6#2015C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#2000C212yesmaximal
9#1803C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15#529—24no—
16#529—24no—
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2212yes—
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C2×#117—21no—
28C22×C42C2211yescharacteristic
29#462—24no—
30#461—24no—
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38#391C2212yes—
39#389—21no—
40#389—21no—
41#384C2211yescharacteristic
42#380—21no—
43#380—21no—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8C2312yes—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C2×Q16—24no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×SD16—24no—
59C2×SD16—42no—
60C2×SD16—42no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65#126—28no—
66#125—22no—
67#125—22no—
68#124—22no—
69#124—22no—
70#123—28no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—42no—
74C22⋊Q8—42no—
75#121—22no—
76#121—22no—
77C4×Q8—22no—
78C4×Q8D414yes—
79C4×D4—28no—
80C4×D4—22no—
81C4×D4D414yes—
82#117—28no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42D411yescharacteristic
105C2×C42D411yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C4⋊C8—22no—
109C4⋊C8—22no—
110#103—24no—
111#103—24no—
112#103—42no—
113#102—42no—
114C22⋊C8—42no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C2×Q8—21no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8C2×D412yes—
131C2×D4—24no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4C2×D412yes—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166Q16—48no—
167SD16—48no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4C2×D412yes—
188C4⋊C4C2×D412yes—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—44no—
195C42—22no—
196C42—22no—
197C42—24no—
198C42—21no—
199C42—21no—
200C42C2×D412yes—
201C42C2×D412yes—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23#13712yes—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—42no—
222Q8—42no—
223Q8—42no—
224Q8—42no—
225D4—22no—
226D4—22no—
227D4—48no—
228D4—41no—
229D4—41no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4#12012yes—
264C2×C4#12012yes—
265C2×C4#13512yes—
266C2×C4#13712yes—
267C2×C4C22×D411yescharacteristic
268C2×C4C22×D411yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C42−1+411yescharacteristic
273C8—42no—
274C8—42no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22C2×#12011yescharacteristic
290C22#53511yescharacteristic
291C22#53711yescharacteristic
292C22C2×#13511yescharacteristic
293C22C2×#13711yescharacteristic
294C22C2×#13711yescharacteristic
295C22#57511yescharacteristic
296C4—22no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—42no—
301C4—41no—
302C4—42no—
303C4—82no—
304C4—82no—
305C4#45214yes—
306C4#53511yescharacteristic
307C4#53511yescharacteristic
308C4#53711yescharacteristic
309C4#53711yescharacteristic
310C2—42no—
311C2—82no—
312C2#261511yescharacteristic
313C2C2×#45211yescharacteristic
314C2#300711yescharacteristic
315C2#321211yescharacteristic
316C2#321811yescharacteristic
317C2#324811yescharacteristic
318C2#325211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. C2×#462C2semidirect
3. C2×#461C2semidirect
4. #2857C2semidirect
5. #2026C2semidirect
6. #2015C2semidirect
7. #2001C2semidirect
9. #1803C2semidirect
10. #1802C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4622
C2×#4612
#28572
#20262
#20152
#20012
#20002
#18032
#18022
#17972