Finite Group Databasebeta

← #31104

Subgroups — #31104

Order 256.

410 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#480C211yesmaximal, characteristic
2C2×#476C211yesmaximal, characteristic, complemented
3#2440C211yesmaximal, characteristic, complemented
4#2420C211yesmaximal, characteristic
5#2087C211yesmaximal, characteristic, complemented
6#2085C211yesmaximal, characteristic, complemented
7#2077C211yesmaximal, characteristic, complemented
8#2071C211yesmaximal, characteristic, complemented
9#2057C211yesmaximal, characteristic, complemented
10#2033C211yesmaximal, characteristic
11#1936C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1847C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C22⋊Q8—21no—
18C2×#121C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#480—24no—
22#476—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#398—21no—
38#397—21no—
39#395—21no—
40#394—21no—
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#386—21no—
46#385—21no—
47#384C2211yescharacteristic
48#382C2211yescharacteristic
49#381—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C22⋊Q8—44no—
74#121—22no—
75#121—41no—
76C4×D4—22no—
77C4×D4—41no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#107—41no—
107#106—22no—
108#106—22no—
109#106—41no—
110C4⋊C8—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114#102D414yes—
115C22⋊C8—22no—
116C22⋊C8—22no—
117#97—22no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C24C2×D411yescharacteristic
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—81no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—81no—
140C2×D4C2×D412yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
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—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×C4C2411yescharacteristic
166C2×C8—21no—
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×C8C2×D412yes—
177C2×C8#4112yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
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—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4C2×D412yes—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—81no—
205C22⋊C4—81no—
206C22⋊C4—81no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23AGL(1,ℤ/8)12yes—
225C23C22×D411yescharacteristic
226Q8—84no—
227D4—22no—
228D4—22no—
229D4—41no—
230D4—41no—
231D4—84no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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—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—84no—
272C2×C4—84no—
273C2×C4#13712yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4H(5,𝔽2)11yescharacteristic
277C2×C42−1+411yescharacteristic
278C2×C42−1+411yescharacteristic
279C2×C42−1+411yescharacteristic
280C8—22no—
281C8—22no—
282C8—41no—
283C8—41no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#53611yescharacteristic
299C22D4211yescharacteristic
300C22#54711yescharacteristic
301C22C2×AGL(1,ℤ/8)11yescharacteristic
302C22C2×#13711yescharacteristic
303C22#57611yescharacteristic
304C22#57711yescharacteristic
305C4—22no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—42no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4#53611yescharacteristic
318C4#53911yescharacteristic
319C4#54711yescharacteristic
320C4#56311yescharacteristic
321C2—42no—
322C2—82no—
323C2#268011yescharacteristic
324C2#323811yescharacteristic
325C2#324111yescharacteristic
326C2#326411yescharacteristic
327C2#326611yescharacteristic
328C2#334511yescharacteristic
329C2#335711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#476C2semidirect
3. #2440C2semidirect
5. #2087C2semidirect
6. #2085C2semidirect
7. #2077C2semidirect
8. #2071C2semidirect
9. #2057C2semidirect
11. #1936C2semidirect
12. #1924C2semidirect
13. #1864C2semidirect
14. #1853C2semidirect
15. #1847C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C2×#4762
#24402
#24202
#20872
#20852
#20772
#20712
#20572
#20332
#19362
#19242
#18642
#18532
#18472