Finite Group Databasebeta

← #26488

Subgroups — #26488

Order 256.

435 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×C4×D8C211yesmaximal, characteristic
3#2454C211yesmaximal, characteristic, complemented
4#2452C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1972C211yesmaximal, characteristic
10#1972C211yesmaximal, characteristic
11#1943C211yesmaximal, characteristic, complemented
12#1941C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#501—24no—
23C4×D8—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#399—21no—
40#396—21no—
41#396—21no—
42#393—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#385—21no—
52#385—21no—
53#380—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C2×D8—22no—
67C2×D8—22no—
68C2×D8Q814yescomplemented
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—22no—
74C4⋊Q8—41no—
75#125—22no—
76#125—41no—
77C22⋊Q8—44no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—41no—
81C4×D4—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
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⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#107D414yes—
112#106—22no—
113#106—22no—
114#106—22no—
115C4⋊C8—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—41no—
121#102—41no—
122#97—22no—
123#97—22no—
124C4×C8—22no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
132C24—21no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—81no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C2×Q812yes—
147C2×D4C2×Q812yes—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167D8—28no—
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×C8C2×D412yes—
177C2×C8C2×Q812yes—
178C2×C8#4112yes—
179C2×C8#4112yes—
180C4⋊C4—22no—
181C4⋊C4—21no—
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—41no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4C2×D412yes—
200C4⋊C4C2×D412yes—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C23—22no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23C22×D411yescharacteristic
232Q8—84no—
233D4—22no—
234D4—22no—
235D4—22no—
236D4—22no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
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—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4AGL(1,ℤ/8)12yes—
279C2×C4C22×D411yescharacteristic
280C2×C4C22×Q811yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C2×C42−1+411yescharacteristic
285C8—22no—
286C8—41no—
287C8—41no—
288C8C22⋊Q814yescomplemented
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#53211yescharacteristic
304C22D4211yescharacteristic
305C22D4×Q811yescharacteristic
306C22C2×AGL(1,ℤ/8)11yescharacteristic
307C22#57511yescharacteristic
308C22#57611yescharacteristic
309C22#57711yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4C2×C22⋊Q811yescharacteristic
323C4#54211yescharacteristic
324C4#54811yescharacteristic
325C4#55411yescharacteristic
326C2—82no—
327C2—82no—
328C2#245211yescharacteristic
329C2#317111yescharacteristic
330C2#317911yescharacteristic
331C2#326411yescharacteristic
332C2#327111yescharacteristic
333C2#338411yescharacteristic
334C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
3. #2454C2semidirect
4. #2452C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2025C2semidirect
8. #2023C2semidirect
11. #1943C2semidirect
12. #1941C2semidirect
13. #1864C2semidirect
14. #1862C2semidirect
15. #1771C2semidirect
68. C2×D8Q8semidirect
288. C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×C4×D82
#24542
#24522
#20672
#20652
#20252
#20232
#19722
#19432
#19412
#18642
#18622
#17712