Finite Group Databasebeta

← #27690

Subgroups — #27690

Order 256.

366 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2323C211yesmaximal, characteristic
2#2317C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2097C211yesmaximal, characteristic, complemented
6#2095C211yesmaximal, characteristic, complemented
7#1949C211yesmaximal, characteristic, complemented
8#1947C211yesmaximal, characteristic, complemented
9#1873C211yesmaximal, characteristic
10#1872C211yesmaximal, characteristic
11#1854C211yesmaximal, characteristic
12#1852C211yesmaximal, characteristic
13#1805C211yesmaximal, characteristic, complemented
14#1803C211yesmaximal, characteristic, complemented
15#1798C211yesmaximal, characteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#391C2211yescharacteristic
34#389—21no—
35#389—21no—
36#389—21no—
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#387—21no—
41#385—21no—
42#385—21no—
43#384C2211yescharacteristic
44#384C2211yescharacteristic
45#382C2211yescharacteristic
46#382C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22⋊Q8—22no—
67C22⋊Q8Q814yes—
68C4×Q8—22no—
69C4×Q8—22no—
70#117—22no—
71#117—22no—
72#117—41no—
73#117—41no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—41no—
104#107—41no—
105#107—41no—
106#106—41no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109#103—41no—
110#103—41no—
111#103—41no—
112#103—41no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124C24C2×D411yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C2×Q812yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—81no—
145C22×C4—81no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×Q812yes—
152C22×C4#4112yes—
153C22×C4C2411yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4C2×Q812yes—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—42no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42—42no—
199C42—81no—
200C42#4112yes—
201C42#4112yes—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23C22⋊Q812yes—
209C23C22×D411yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
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—84no—
255C2×C4C22⋊Q812yes—
256C2×C4#12312yes—
257C2×C4#12312yes—
258C2×C4C22×Q811yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22C2×C22⋊Q811yescharacteristic
276C22C2×#12311yescharacteristic
277C22#53511yescharacteristic
278C22#57611yescharacteristic
279C22#57711yescharacteristic
280C22#57811yescharacteristic
281C22#57811yescharacteristic
282C4—22no—
283C4—22no—
284C4—22no—
285C4—42no—
286C4—41no—
287C4—41no—
288C4—42no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4#55311yescharacteristic
296C4#55311yescharacteristic
297C4#55411yescharacteristic
298C4#55411yescharacteristic
299C2—42no—
300C2#248711yescharacteristic
301C2#308211yescharacteristic
302C2#308311yescharacteristic
303C2#309711yescharacteristic
304C2#309811yescharacteristic
305C2#322311yescharacteristic
306C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2099C2semidirect
5. #2097C2semidirect
6. #2095C2semidirect
7. #1949C2semidirect
8. #1947C2semidirect
13. #1805C2semidirect
14. #1803C2semidirect

Maximal subgroups

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

subgroupindex
#23232
#23172
#21012
#20992
#20972
#20952
#19492
#19472
#18732
#18722
#18542
#18522
#18052
#18032
#17982