Finite Group Databasebeta

← #31152

Subgroups — #31152

Order 256.

396 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic
3#2451C211yesmaximal, characteristic, complemented
4#2447C211yesmaximal, characteristic
5#2086C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic
9#2042C211yesmaximal, characteristic
10#2024C211yesmaximal, characteristic, complemented
11#1969C211yesmaximal, characteristic, complemented
12#1939C211yesmaximal, characteristic, complemented
13#1934C211yesmaximal, characteristic
14#1877C211yesmaximal, characteristic
15#1863C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#474—24no—
23#464—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#398—21no—
39#397—21no—
40#397—21no—
41#395—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#390C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#387—21no—
52#385—21no—
53#381—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×Q16—41no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78#124—22no—
79#124—41no—
80C4×Q8—22no—
81C4×Q8—41no—
82C4×D4—22no—
83C4×D4—41no—
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×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—22no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—41no—
110#103—22no—
111#103—22no—
112#103—41no—
113#103—41no—
114#103—41no—
115#102—22no—
116#102—41no—
117#102D414yes—
118C4×C8—41no—
119#95—21no—
120#95—21no—
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—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—81no—
147C2×Q8#4112yes—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C2×D412yes—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—81no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171Q16—44no—
172SD16—44no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8C2×D412yes—
185C4⋊C4—22no—
186C4⋊C4—22no—
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—21no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4C2×D412yes—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—81no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23C22×D411yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—84no—
231D4—22no—
232D4—22no—
233D4—41no—
234D4—41no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
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#13512yes—
274C2×C4#13712yes—
275C2×C4C22×D411yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C2×C42−1+411yescharacteristic
280C2×C42−1+411yescharacteristic
281C8—22no—
282C8—41no—
283C8—41no—
284C8—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22#53711yescharacteristic
293C22D4211yescharacteristic
294C22#54811yescharacteristic
295C22C2×#13511yescharacteristic
296C22C2×#13711yescharacteristic
297C22#57611yescharacteristic
298C22#57811yescharacteristic
299C4—22no—
300C4—22no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4#53611yescharacteristic
313C4#53911yescharacteristic
314C4#54611yescharacteristic
315C4#56211yescharacteristic
316C2—82no—
317C2#268111yescharacteristic
318C2#324811yescharacteristic
319C2#326211yescharacteristic
320C2#327611yescharacteristic
321C2#328011yescharacteristic
322C2#336411yescharacteristic
323C2#337511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. #2451C2semidirect
5. #2086C2semidirect
6. #2068C2semidirect
7. #2056C2semidirect
10. #2024C2semidirect
11. #1969C2semidirect
12. #1939C2semidirect
15. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4642
#24512
#24472
#20862
#20682
#20562
#20552
#20422
#20242
#19692
#19392
#19342
#18772
#18632