Finite Group Databasebeta

← #29784

Subgroups — #29784

Order 256.

432 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic
2C2×#423C211yesmaximal, characteristic, complemented
3#2855C211yesmaximal, characteristic, complemented
4#2854C211yesmaximal, characteristic
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2067C211yesmaximal, characteristic, complemented
9#2066C211yesmaximal, characteristic, complemented
10#2065C211yesmaximal, characteristic, complemented
11#2055C211yesmaximal, characteristic, complemented
12#2055C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic
14#1894C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#468—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27#423—24no—
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#388—21no—
48#387—21no—
49#387—21no—
50#386—21no—
51#385—21no—
52#381—21no—
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4C2311yescharacteristic
67C2×SD16—22no—
68C2×SD16—41no—
69C2×SD16—41no—
70C2×M16—22no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16D414yes—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#123—44no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83#121—22no—
84#121—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42D411yescharacteristic
107C2×C42D411yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#107—41no—
112#106—41no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117#103—22no—
118#103—41no—
119#103—41no—
120#102—22no—
121#102—41no—
122#102—41no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24C2×D411yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4#4112yes—
174C22×C4C2411yescharacteristic
175SD16—44no—
176M16—44no—
177M16—44no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8C2×D412yes—
189C2×C8C2×D412yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C42—22no—
216C42—22no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23#12112yes—
235C23C22×D411yescharacteristic
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—44no—
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—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#12112yes—
285C2×C4AGL(1,ℤ/8)12yes—
286C2×C4#13712yes—
287C2×C4C22×D411yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4H(5,𝔽2)11yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C2×C42−1+411yescharacteristic
292C2×C42−1+411yescharacteristic
293C8—22no—
294C8—22no—
295C8—41no—
296C8—41no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22C2×#12111yescharacteristic
312C22#53711yescharacteristic
313C22#53711yescharacteristic
314C22C2×AGL(1,ℤ/8)11yescharacteristic
315C22C2×#13711yescharacteristic
316C22#57511yescharacteristic
317C22#57511yescharacteristic
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4#53411yescharacteristic
331C4#53611yescharacteristic
332C4#53911yescharacteristic
333C4#53911yescharacteristic
334C2—42no—
335C2—161no—
336C2#261811yescharacteristic
337C2#305511yescharacteristic
338C2#305611yescharacteristic
339C2#325111yescharacteristic
340C2#325211yescharacteristic
341C2#325411yescharacteristic
342C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#423C2semidirect
3. #2855C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
7. #2068C2semidirect
8. #2067C2semidirect
9. #2066C2semidirect
10. #2065C2semidirect
11. #2055C2semidirect
12. #2055C2semidirect
14. #1894C2semidirect
15. #1809C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4232
#28552
#28542
#20872
#20862
#20682
#20672
#20662
#20652
#20552
#18952
#18942
#18092