Finite Group Databasebeta

← #24847

Subgroups — #24847

Order 256.

426 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474C211yesmaximal, characteristic, complemented
2C2×#424C211yesmaximal, characteristic, complemented
3#2628C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#1987C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic
12#1893C211yesmaximal, characteristic
13#1809C211yesmaximal, characteristic, complemented
14#1808C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#474—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27#424—24no—
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36#400—21no—
37#399—21no—
38#395—21no—
39#393—21no—
40#390—21no—
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#386—21no—
47#385—21no—
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C2×M16—22no—
67C2×M16—22no—
68C2×M16—41no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78#121—22no—
79#121Q814yescomplemented
80C4×D4—22no—
81C4×D4—22no—
82#117—22no—
83#117—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
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—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111#107—41no—
112#106—41no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8D414yescomplemented
117#102—22no—
118#102—22no—
119#102—41no—
120#102—41no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130C24—21no—
131C24C2×D411yescharacteristic
132C2×Q8—42no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×D4—24no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—81no—
142C2×D4—81no—
143C2×D4C2×Q812yes—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×Q812yes—
164C22×C4#4112yes—
165C22×C4C2411yescharacteristic
166M16—44no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—44no—
178C2×C8C2×D412yes—
179C2×C8C2×D412yes—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4C2×Q812yes—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—24no—
203C22⋊C4—42no—
204C22⋊C4—81no—
205C22⋊C4—81no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C42—21no—
210C42—21no—
211C42—81no—
212C42C2×D412yes—
213C42#4112yes—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23C22⋊Q812yes—
228C23C22×D411yescharacteristic
229Q8—84no—
230D4—22no—
231D4—22no—
232D4—41no—
233D4—41no—
234D4—84no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—84no—
273C2×C4#12112yes—
274C2×C4#12112yes—
275C2×C4C22⋊Q812yes—
276C2×C4C22×D411yescharacteristic
277C2×C4C22×Q811yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C2×C4H(5,𝔽2)11yescharacteristic
281C2×C42−1+411yescharacteristic
282C8—22no—
283C8—22no—
284C8—41no—
285C8—41no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22C2×#12111yescharacteristic
301C22C2×C22⋊Q811yescharacteristic
302C22#53711yescharacteristic
303C22#57611yescharacteristic
304C22#57611yescharacteristic
305C22#57711yescharacteristic
306C22#57811yescharacteristic
307C4—22no—
308C4—22no—
309C4—22no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4#54611yescharacteristic
320C4D4×Q811yescharacteristic
321C4#55211yescharacteristic
322C4#55411yescharacteristic
323C2—42no—
324C2—82no—
325C2#241711yescharacteristic
326C2#305811yescharacteristic
327C2#306311yescharacteristic
328C2#308111yescharacteristic
329C2#308311yescharacteristic
330C2#325711yescharacteristic
331C2#326211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
1. C2×#474C2semidirect
2. C2×#424C2semidirect
3. #2628C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #1987C2semidirect
10. #1949C2semidirect
13. #1809C2semidirect
14. #1808C2semidirect
15. #1804C2semidirect
79. #121Q8semidirect
116. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4242
#26282
#23242
#20872
#20862
#20672
#20652
#19872
#19492
#18952
#18932
#18092
#18082
#18042