Finite Group Databasebeta

← #18789

Subgroups — #18789

Order 256.

458 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 131 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
1Q8×#41C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic
4C2×#477C211yesmaximal, characteristic, complemented
5#1637C212yesmaximal, complemented
6#1636C212yesmaximal, complemented
7#1633C212yesmaximal, complemented
8#1632C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic, complemented
10#1487C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic
12Q82C2211yescharacteristic
13#557—21no—
14#557C2211yescharacteristic
15#554—22no—
16#554—21no—
17#554—21no—
18#554C2211yescharacteristic
19D4×Q8—21no—
20D4×Q8C2211yescharacteristic
21#533—22no—
22#533—21no—
23#533—21no—
24#533C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C4×#41—21no—
27C4×#41C2211yescharacteristic
28C2×C4×Q8—21no—
29C2×C4×Q8C2211yescharacteristic
30#494C2212yes—
31#487C2212yes—
32#477—21no—
33#477—21no—
34#477C2212yes—
35#477C2212yes—
36#477C2212yes—
37#476C2212yes—
38#475—21no—
39#474—21no—
40#462—22no—
41#451—22no—
42#431C2211yescharacteristic
43#428—21no—
44C2×#107—21no—
45#423C2211yescharacteristic
46C2×C4⋊C8C2211yescharacteristic
47#417—21no—
48C2×#103—21no—
49#333C2212yes—
50#332C2212yes—
51Q8⋊C8C2212yes—
52Q8⋊C8C2212yes—
53Q8⋊C8C2212yes—
54D4⋊C8C2212yes—
55C2×#41D411yescharacteristic
56C22×Q8—21no—
57C22×Q8—21no—
58C22×Q8D411yescharacteristic
59C2×Q16—24no—
60C2×M16—21no—
61C22×C8—21no—
62C4⋊Q8—22no—
63C4⋊Q8—21no—
64C4⋊Q8—22no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8C2312yes—
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2311yescharacteristic
74#125—22no—
75#125—22no—
76#125—22no—
77#125—21no—
78#125—21no—
79#125—21no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—21no—
86C22⋊Q8—22no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—22no—
90C22⋊Q8D412yes—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8D411yescharacteristic
102C4×Q8D411yescharacteristic
103C4×Q8D411yescharacteristic
104C4×Q8D412yes—
105C4×Q8C2311yescharacteristic
106C4×Q8C2311yescharacteristic
107C4×Q8C2311yescharacteristic
108C4×D4—22no—
109C4×D4—22no—
110C4×D4—21no—
111C4×D4—21no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4D411yescharacteristic
115C4×D4C2311yescharacteristic
116#117—22no—
117#117—21no—
118#117—21no—
119#117D411yescharacteristic
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4D411yescharacteristic
126C2×C42—22no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130#107—22no—
131#107—22no—
132#107—22no—
133#106—22no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8C2312yes—
138C4⋊C8C2312yes—
139#103—22no—
140#103—22no—
141#103—22no—
142#103—24no—
143#102—22no—
144C22⋊C8—21no—
145C4×C8C2312yes—
146#41—22no—
147#41Q1614yescomplemented
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—41no—
158C2×Q8—82no—
159C2×Q8C2×D411yescharacteristic
160C2×Q8C2×D411yescharacteristic
161C2×Q8C2×D411yescharacteristic
162C2×Q8C2×D412yes—
163C2×D4—21no—
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176Q16—84no—
177M16—42no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—42no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—22no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4C2×D412yes—
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C42—22no—
218C42—22no—
219C42—21no—
220C42—22no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42C2×D411yescharacteristic
228C42C2×D411yescharacteristic
229C42C2×D411yescharacteristic
230C42C2411yescharacteristic
231C23—21no—
232C23#12011yescharacteristic
233Q8—21no—
234Q8—21no—
235Q8—21no—
236Q8—21no—
237Q8—42no—
238Q8—42no—
239Q8—44no—
240Q8—44no—
241Q8—41no—
242Q8—41no—
243Q8—82no—
244Q8—82no—
245Q8C2×Q1612yes—
246Q8#13712yes—
247D4—24no—
248D4—21no—
249D4C2×Q1612yes—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4#12011yescharacteristic
273C2×C4#12011yescharacteristic
274C2×C4#12011yescharacteristic
275C2×C4C2×Q1612yes—
276C2×C4#13712yes—
277C2×C4C22×D411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4C22×D411yescharacteristic
280C8—42no—
281C8—42no—
282C8—42no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22#45112yes—
289C22C2×#12011yescharacteristic
290C4—22no—
291C4—21no—
292C4—22no—
293C4—21no—
294C4—21no—
295C4—41no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—42no—
300C4—41no—
301C4—41no—
302C4—82no—
303C4#45112yes—
304C4C22×Q1611yescharacteristic
305C4C2×#13711yescharacteristic
306C4C2×#13711yescharacteristic
307C4#57511yescharacteristic
308C2—22no—
309C2—42no—
310C2C2×#45111yescharacteristic
311C2#300711yescharacteristic
312C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3209C2semidirect
4. C2×#477C2semidirect
5. #1637C2semidirect
6. #1636C2semidirect
7. #1633C2semidirect
8. #1632C2semidirect
9. #1550C2semidirect
10. #1487C2semidirect
147. #41Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×#412
#32092
#30792
C2×#4772
#16372
#16362
#16332
#16322
#15502
#14872
#14772