Finite Group Databasebeta

← #22662

Subgroups — #22662

Order 256.

485 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×#435C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3C2×#427C211yesmaximal, characteristic, complemented
4#2354C211yesmaximal, characteristic
5#2352C211yesmaximal, characteristic
6#1949C211yesmaximal, characteristic, complemented
7#1948C211yesmaximal, characteristic, complemented
8#1854C211yesmaximal, characteristic, complemented
9#1854C211yesmaximal, characteristic, complemented
10#1854C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic, complemented
12#1847C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#435C418yescomplemented
24#428—24no—
25#427—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#390—21no—
41#388—21no—
42#388—21no—
43#388—21no—
44#387—21no—
45#386—21no—
46#384C2211yescharacteristic
47#382C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4—21no—
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C22⋊C4—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×D4D411yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×M16—22no—
67C2×M16—22no—
68C2×M16C2×C414yes—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8C2×C414yescomplemented
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4×D4C2×C414yes—
77C4×D4Q814yes—
78#117—22no—
79#117—22no—
80#117—22no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
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×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113#107—22no—
114#107—22no—
115#107—41no—
116#107D414yescomplemented
117#106—22no—
118#106—22no—
119#106—22no—
120#106—22no—
121#106—22no—
122#106—41no—
123C4⋊C8C2×C414yescomplemented
124C22⋊C8C2×C414yes—
125C22⋊C8C2×C414yescomplemented
126#97C2×C414yescomplemented
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135C24C2×D411yescharacteristic
136C24C2×D411yescharacteristic
137C2×D4—41no—
138C2×D4—41no—
139C2×D4C4⋊C414yes—
140C22×C4—21no—
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—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—81no—
159C22×C4C22×C412yes—
160C22×C4C22×C412yes—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×Q812yes—
166C22×C4C2×Q812yes—
167C22×C4C2411yescharacteristic
168M16—28no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—28no—
176C2×C8C22×C412yes—
177C2×C8C22×C412yes—
178C2×C8C22×C412yes—
179C2×C8C22×C412yes—
180C2×C8C2×D412yes—
181C2×C8#4112yes—
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—21no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4C4⋊C414yes—
200C4⋊C4C2×D412yes—
201C4⋊C4C2×D412yes—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4C4⋊C414yes—
210C22⋊C4C4⋊C414yes—
211C42—42no—
212C42—42no—
213C42—42no—
214C42—42no—
215C42—81no—
216C42C22×C412yes—
217C42C2×Q812yes—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23C2×C4⋊C412yes—
231C23C2×C4⋊C412yes—
232C23C22×D411yescharacteristic
233D4—44no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
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—41no—
273C2×C4—41no—
274C2×C4—84no—
275C2×C4C2×C4⋊C412yes—
276C2×C4C2×C4⋊C412yes—
277C2×C4C2×C4⋊C412yes—
278C2×C4C2×C4⋊C412yes—
279C2×C4C23×C411yescharacteristic
280C2×C4C22×D411yescharacteristic
281C2×C4C22×Q811yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C2×C42−1+411yescharacteristic
285C8—22no—
286C8—22no—
287C8—22no—
288C8C4×D414yes—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22C22×C4⋊C411yescharacteristic
304C22D4211yescharacteristic
305C22#54711yescharacteristic
306C22#57511yescharacteristic
307C22#57511yescharacteristic
308C22#57711yescharacteristic
309C22#57711yescharacteristic
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4C2×C4×D411yescharacteristic
323C4#52011yescharacteristic
324C4D4×Q811yescharacteristic
325C4#55411yescharacteristic
326C2—42no—
327C2—42no—
328C2D4×C4⋊C411yescharacteristic
329C2#291111yescharacteristic
330C2#291211yescharacteristic
331C2#326711yescharacteristic
332C2#327111yescharacteristic
333C2#334811yescharacteristic
334C2#335011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#428C2semidirect
3. C2×#427C2semidirect
6. #1949C2semidirect
7. #1948C2semidirect
8. #1854C2semidirect
9. #1854C2semidirect
10. #1854C2semidirect
11. #1853C2semidirect
12. #1847C2semidirect
13. #1809C2semidirect
14. #1807C2semidirect
15. #1776C2semidirect
23. #435C4semidirect
71. C22×C8C2×C4semidirect
116. #107D4semidirect
123. C4⋊C8C2×C4semidirect
125. C22⋊C8C2×C4semidirect
126. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C2×#4282
C2×#4272
#23542
#23522
#19492
#19482
#18542
#18532
#18472
#18092
#18072
#17762